Renormalisation as Universal-Property Flow: Categorical RG, Strong Coupling, Non-Invertible Symmetries, and Multiscale Dynamics

Renormalisation as Universal-Property Flow: Categorical RG, Strong Coupling, Non-Invertible Symmetries, and Multiscale Dynamics Andrei T. Patrascu FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] The renormalisation group (RG) is traditionally formulated as a flow of couplings driven by coarse-graining, often constrained by scalar monotonicity theorems such as the c -, a -, and F -theorems. While powerful in perturbative or weakly correlated regimes, this formulation loses predictive control in strongly coupled, cooperative, non-Markovian, or multi-scale systems, and it remains blind to categorical structures such as non-invertible symmetries. Moreover, coarse-graining in molecular, biomolecular, or cognitive systems involves multi-scale interactions that cannot be captured by a numerical flow in coupling space. In this work, we propose a categorical reformulation of the renormalisation group in terms of Universal-Property Flow (UP-RG). Instead of tracking couplings, UP-RG follows the evolution of universal structures—limits, colimits, Kan extensions, adjunctions, enriched hom-objects, and reflective subcategories—associated with a theory at each scale. We introduce the renormalisation category, whose objects encode theories together with regulator choices and coarse-graining schemes, and whose morphisms formalize RG transformations. Within this framework, we define adjoint curvature, an obstruction to perfect adjunction between coarse-graining and embedding functors, which quantifies structural irreversibility, RG anomalies, and loss of universality across scales. This shift from numerical flows to structural flows yields a number of new capabilities. Universalproperty flow extends classical monotonicity theorems to structural monotonicity, providing families of RG invariants that remain well-defined even when traditional scalar monotones fail. It offers control over RG dynamics that are non-monotonic, oscillatory, non-Markovian, or cooperative—precisely the regimes encountered in strongly coupled quantum field theories, strongly correlated electron systems, and multi-scale complex systems. Non-invertible symmetries arise naturally in this framework as adjointable monoidal endofunctors on categories of topological defects, and their RG flow is governed by Kan extensions and adjoint curvature rather than by group-theoretic automorphisms. Finally, the universal-property perspective applies beyond QFT: it provides a structural interpretation of folding funnels in biomolecular systems as universal objects, explains emergent directionality and teleological attractors in evolutionary dynamics, and offers a categorical lens on multi-scale representations in modern AI models. UP-RG thus unifies strongly coupled physics, categorical symmetry, and multi-scale biological and computational phenomena under a single mathematical paradigm, extending the scope of renormalisation far beyond its classical formulation. I. INTRODUCTION The renormalisation group (RG) is one of the central organising principles of modern theoretical physics. In its standard Wilsonian formulation, the RG is described as a flow on a space of theories, where microscopic degrees of freedom are progressively coarse-grained and their effects re-encoded in scale-dependent couplings and operators.[ 1 , 10 ] Let µ denote an energy scale and {gi ( µ ) } a (typically finite or truncated) set of running couplings parametrising an effective action Sµ [ φ ]. The RG flow is then written as dgi(µ) dln µ=βi{g(µ)},(1) where the beta functions βi encode how the couplings change under infinitesimal changes of scale. Fixed points of (1) ( βi = 0 for all i ) correspond to scale-invariant or conformal field theories, and trajectories in coupling space interpolate between such fixed points. A major conceptual advance was the recognition that, in certain situations, the RG flow is constrained by monotonicity theorems. Zamolodchikov’s c -theorem in two dimensions shows that there exists a positive function c ( {g}, µ )which decreases strictly along RG flows and is stationary at fixed points, where it coincides with the central charge of the corresponding conformal field theory.[ 11 ] In four dimensions, Komargodski and Schwimmer established the a -theorem, demonstrating that a particular trace-anomaly coefficient a plays an analogous role as a monotonically decreasing function along RG flows.[ 12 ] In three dimensions, F -theorem results based on entanglement entropy and mutual information provide monotone quantities interpolating between ultraviolet (UV) and infrared (IR) fixed points.[ 5 ] Collectively, these theorems implement the idea that the number of effective degrees of freedom decreases along RG flows. 2 Despite these successes, the traditional picture of RG as a numerical flow of couplings constrained by scalar monotones has clear limitations: •Strong coupling and truncation dependence. In strongly coupled quantum field theories and in strongly correlated many-body systems, the beta functions (1) are usually obtained via drastic truncations (e.g. polynomial truncations of the effective action in functional RG.[ 6 ]) The resulting flows are highly scheme-dependent, sensitive to regulator choices, and may exhibit oscillatory or even cyclic behaviour in coupling space which obscures the underlying physics. Scalar monotone functions may not exist, or may fail to be computable or stable under truncation. •Cooperative and multi-scale dynamics. Many systems of physical interest—such as strongly correlated electrons, spin liquids, glassy systems, or complex networks—exhibit cooperative behaviour across multiple scales. Degrees of freedom do not decouple neatly as one flows from UV to IR, and effective descriptions may involve non-local operators, emergent constraints, or collective excitations. A simple flow on a finite-dimensional space of couplings is often an inadequate description of such phenomena. •Non-Markovian and history-dependent flows. The standard RG is usually treated as Markovian in “RG time” ln µ : the flow at a given scale is assumed to depend only on the current values of the couplings. In practice, truncations, memory effects, and coarse-graining procedures can produce effective dynamics where the evolution at a given scale depends on the entire history of the flow or on multi-scale correlations that are not faithfully encoded in a local differential equation of the form (1). •Categorical and non-invertible symmetries. Modern developments in quantum field theory have revealed that symmetries are often better organised not by groups, but by higher or categorical structures. Generalised global symmetries act on extended operators and are encoded in higher-form symmetry groups and their anomalies.[ 101 ] More recently, non-invertible symmetries have emerged as a genuinely new kind of symmetry, implemented by topological defects whose fusion rules are described by fusion categories rather than ordinary groups.[ 104 ] These structures live naturally in categories (or higher categories) of defects and line/surface operators; they are not visible at the level of a finite set of couplings {gi}. •Biological and molecular systems. In molecular and biomolecular systems, such as protein folding and large biomolecular assemblies, the relevant dynamics is highly cooperative, strongly constrained, and profoundly multi-scale. The “folding funnel” picture emphasises that the effective conformational space is shaped by global constraints and energetic funnels, not by a small set of local couplings.[ 9 ] Traditional RG techniques, while suggestive, lack a systematic mathematical framework for such non-linear, teleological, multi-scale phenomena. These limitations are not merely technical; they point to a structural mismatch between the standard numerical RG picture and the kinds of systems we wish to understand. The usual formulation treats a theory at scale µ as a point in a parameter space of couplings, and an RG transformation as a vector field generating flows on that space. Scalar RG monotones are then real-valued functions on this parameter space that decrease along trajectories. In this work we adopt a different viewpoint. We observe that, in many situations, the truly robust and physically meaningful content of an RG step is not the precise numerical trajectory {gi ( µ ) } , but the way in which the theory at scale µ is characterised by certain universal properties—limits, colimits, adjunctions, Kan extensions, enriched hom-objects, and other categorical constructions that encode how the theory relates to other objects (such as regulator schemes, defect categories, or coarse-grained observables). We propose to regard the RG as a flow in the space of such universal structures, and to track the evolution of these structures under coarse-graining. We refer to this as Universal-Property RG (UP-RG). In this universal-structural perspective: • A theory at scale µ is not just a list of couplings; it is equipped with universal constructions which capture how it integrates out degrees of freedom, how it embeds into larger or more microscopic descriptions, and how its observables and defect operators are organised categorically. • An RG transformation between scales µ1 and µ2 is described by functors between appropriate categories (of theories, observables, or defects), and the key question is how these functors act on universal properties. 3 • Instead of a single scalar monotone, one obtains families of structural invariants and monotones associated to universal constructions. These invariants can remain well-defined and predictive even in regimes where couplings behave wildly, oscillate, or fail to admit a gradient-flow interpretation. • The failure of certain categorical identities—in particular, the failure of adjunctions to satisfy their triangle identities exactly—can be quantified as an adjoint curvature which measures irreversibility, information loss, anomalies, and teleological bias in the RG flow. The rest of the paper develops this viewpoint systematically. We introduce renormalisation categories and universal-property flows, define adjoint curvature as a measure of broken adjunction, and show how this framework generalises classical RG monotonicity theorems. We then explain how non-invertible and higher symmetries emerge naturally as adjointable monoidal endofunctors on defect categories, and how their RG flows are controlled at the level of universal structures. Finally, we discuss applications to strongly coupled and strongly correlated quantum field theories, to molecular and biomolecular systems, and to multi-scale learning and representation in modern AI. A. Motivations: Beyond Numerical RG The motivation for shifting from numerical to structural RG can be summarised as follows: in many of the most interesting physical, biological, and computational systems, what remains robust and predictive under coarse-graining is not a finite list of couplings, but a network of universal relationships. These are most naturally expressed in categorical language. By formulating RG directly in terms of universal properties, we obtain: 1. A framework that subsumes classical c -, a -, and F -theorems as special cases of structural monotonicity but is not limited to scalar functions on a coupling manifold. 2. A way to define and compute new RG invariants in strongly coupled and strongly correlated systems, where numerical beta functions and classical monotonicity theorems are unreliable or inapplicable. 3. A natural language for non-invertible and higher symmetries, which are inherently categorical and cannot be captured by group-valued order parameters or couplings alone. 4. A principled approach to multi-scale cooperative phenomena (e.g. protein folding funnels), where dynamical trajectories appear teleological because they are attracted to universal structures (solutions of universal mapping problems) rather than to arbitrary points in a naive state space. In the subsequent sections we will turn these motivations into precise definitions and theorems. We will show how RG transformations can be organised as functors in a renormalisation category, how universalproperty flow is defined in this setting, and how adjoint curvature naturally measures the obstruction to reversible or scheme-independent RG. We will then explore concrete examples and applications where this structural viewpoint offers new insight beyond traditional numerical RG. B. Overview of the Proposed Framework The discussion in Subsection I A suggests that the traditional formulation of RG as a flow of couplings, Eq. (1) , is often too narrow. It does not foreground the structural features that remain robust under coarse-graining, especially in strongly coupled, strongly correlated, or categorically rich systems. In this subsection we outline the central idea of this work: to reformulate RG as a flow of universal properties in suitably chosen categories associated with a theory. We call this approach Universal-Property RG (UP-RG). The standard Wilsonian picture[ 10 , 14 , 15 ] may be summarised as follows. One starts from a microscopic theory defined at a UV scale Λand constructs a family of effective theories Sµ parametrised by an IR scale µ , obtained by integrating out modes with momenta p in the shell µ < |p|< Λand rescaling fields and coordinates to preserve the cutoff. At the level of couplings, this leads to the beta functions (1) ; at the level of effective actions, to exact functional RG equations such as the Wetterich or Polchinski equations.[14, 15] 4 Within this framework, scalar monotonicity theorems—the c -, a -, and F -theorems in two, four, and three dimensions, respectively—can be interpreted as the existence of special real-valued functionals M ( µ ) on theory space satisfying dM(µ) dln µ≤0,(2) under suitable assumptions (unitarity, locality, reflection positivity, and, in some cases, conformal invariance at fixed points).[ 11 – 13 ] Equation (2) is the canonical expression of RG irreversibility in the numerical paradigm. The proposal of this paper is to retain the intuition of irreversibility and flow, but to replace the scalar monotone (2) by a richer, categorical structure. Instead of asking for a single number M ( µ )that decreases along RG trajectories, we ask: 1. What universal constructions best capture the structural content of a theory at scale µ? 2. How do these universal structures transform under coarse-graining and scheme changes? 3. Can we define structural monotonicity conditions that constrain these transformations, generalising (2)? Theories as objects in scale-dependent categories. At each scale µ we associate to the theory not only a point in coupling space, but a category (or higher category) Cµ which organises its structural content. Depending on the context, Cµmight be: • a category of observables and states (e.g. operator algebras with morphisms given by completely positive maps); • a category of effective theories obtained from a fixed microscopic theory by different regularisation and truncation prescriptions; • a monoidal (often fusion) category of topological defects, line and surface operators, capturing generalized and non-invertible symmetries;[17, 101, 104] • an enriched category where the hom-objects encode metric, probabilistic, or Hilbert-space structure. In each case, the category Cµ comes equipped with certain universal properties that characterise how objects in Cµrelate to each other via maps. Examples include: •limits and colimits of diagrams encoding coarse-graining and gluing of subsystems; •free or adjoint objects describing the “best” coarse-grained approximation of a finer structure; •reflective subcategories capturing effective theories at a given resolution; •Kan extensions describing the optimal extension of observables or symmetries across scales. Let Uµ denote, schematically, the collection of such universal constructions that we regard as physically meaningful for the theory at scale µ . One can think of Uµ as the universal essence of the theory at that scale: the part of its structure that is determined entirely by how it maps into and out of other objects in Cµ, rather than by a particular presentation in terms of couplings. RG as a family of functors between categories. An RG step from scale µ to µ0< µ induces, at the structural level, a functor Fµ→µ0:Cµ−→ Cµ0.(3) In a Wilsonian setting, Fµ→µ0 encodes integrating out modes and rescaling; in a functional RG setting, it encodes the change of the effective average action; in a defect-category setting, it encodes how topological defects in the UV induce defects in the IR across an RG domain wall.[17, 101] Given Fµ→µ0, the universal structures transform according to Uµ7−→ Uµ0:= Φµ→µ0Fµ→µ0(Uµ),(4) where Φ µ→µ0 denotes the relevant universal construction at the new scale. For example, if Uµ is defined as a limit of a diagram Dµ in Cµ , then Fµ→µ0 ( Uµ )is related to the limit of the pushed-forward diagram Fµ→µ0◦Dµ;Φµ→µ0may be the limiting cone for that diagram in Cµ0. Equation (4) is the defining relation of universal-property RG: instead of tracking a vector of couplings, we track how universal constructions transform under functors generated by RG steps. This is a flow in the space of universal properties, not in a finite-dimensional manifold. 5 Renormalisation categories and adjoint curvature. To organise these ideas, we introduce the notion of a renormalisation category R . Its objects are, schematically, pairs ( µ, Cµ )or more elaborate tuples encoding a theory together with a regularisation scheme, truncation prescription, and associated structural category. Morphisms in R correspond to RG transformations between scales and schemes, realised as functors F : Cµ→ Cµ0 together with additional coherence data. In many applications, it is natural to consider R as a 2-category, where 2-morphisms are natural transformations between RG functors (meta-RG maps, scheme changes, or interpolating families of RG flows). A central role in our framework is played by adjunctions. In an idealised setting, coarse-graining and embedding should form an adjoint pair: a functor F : Cµ→ Cµ0 describing coarse-graining and a functor G:Cµ0→ Cµdescribing a canonical (but generally lossy) embedding, with FaG⇐⇒ HomCµ0F(X), Y ∼ =HomCµX, G(Y)(5) naturally in X∈ Cµ and Y∈ Cµ0 . This adjunction expresses the idea that G is the “best possible” way to reconstruct UV data from IR data, relative to the structure encoded in Cµand Cµ0. Adjunctions come with canonical unit and counit natural transformations, η: idCµ⇒GF,  :FG ⇒idCµ0,(6) which satisfy the triangle identities. When these identities fail to hold exactly in a renormalisation category (for example, due to truncation, regulator dependence, or genuine loss of universality), we can quantify the failure by considering the composite endomorphisms ΘF:= (F)◦(Fη) : F⇒F, ΘG:= (G)◦(ηG) : G⇒G. (7) In a perfect adjunction, Θ F and Θ G are the identity transformations; in general they are nontrivial endomorphisms. We interpret the deviation from identity RF:= ΘF−idF,RG:= ΘG−idG,(8) schematically, as a measure of adjoint curvature. This curvature encodes the obstruction to viewing F and G as a perfect adjoint pair and thus measures irreversibility, loss of information, anomalies, and teleological bias in the RG flow at the structural level. In later sections we make this notion precise in appropriate enriched settings and show how RF and RG generate nontrivial dynamical effects beyond linear RG. From scalar to structural monotonicity. Within UP-RG, classical monotonicity theorems are seen as special cases of a broader principle. Instead of a single number M ( µ )satisfying Eq. (2) , we obtain families of defect functionals measuring the failure of universal properties to be perfectly preserved by RG functors. For example: • in categories enriched over ordered vector spaces or Hilbert spaces, one can define norms or order parameters associated with RFthat decrease along UP-flows; • in categories of defects and symmetries, one can define categorical invariants (centers, fusion data, dualizability) that are preserved or monotonically simplified along RG;[17, 101] • in categories of effective theories, one can track the size and complexity of reflective subcategories corresponding to accessible low-energy descriptions. We refer to such statements as structural monotonicity theorems. They generalise Eq. (2) and remain meaningful in situations where couplings oscillate or flow on manifolds too complicated for a single scalar to capture, such as strongly coupled QFTs, strongly correlated electron systems, and cooperative multiscale systems.[14, 15] Summary of contributions and applications. The rest of the paper develops the above ideas formally and explores their implications. The main contributions can be summarised as follows: • We formulate the notion of a renormalisation category and define universal-property RG (UP-RG) as a flow of universal structures Uµunder functors Fµ→µ0between categories Cµ. • We introduce adjoint curvature as a measure of the failure of coarse-graining/embedding functors to form a perfect adjunction and show how it encodes irreversibility, anomalies, scheme dependence, and teleological behaviour in RG. 6 • We generalise classical monotonicity theorems (such as the c -, a -, and F -theorems)[ 11 – 13 ] to structural monotonicity statements involving families of categorical invariants. • We explain how non-invertible and higher symmetries arise naturally as adjointable monoidal endofunctors in defect categories and how their RG flows are controlled by universal constructions (Kan extensions, centers, duals).[17, 101, 104] •We sketch applications to: – strongly coupled QFTs and phase diagrams, where UP-RG provides scheme-independent structural invariants; – strongly correlated electron systems, where functional RG data can be reorganised categorically to reveal universal structures; – molecular and biomolecular folding, where folding funnels are interpreted as teleological attractors corresponding to universal objects in configuration categories; – evolutionary dynamics, where universal-property flow captures directional, multi-scale adaptation under constraints; – multi-scale AI and representation learning, where internal representations are understood as universal constructions in categories of data and features. Detailed mathematical definitions, examples, and derivations are presented in the subsequent sections, where we make precise the notions of renormalisation categories, universal-property flows, adjoint curvature, and structural monotonicity. C. Relationship to Previous Work The present work builds on, but also departs significantly from, several strands of the renormalisation group and quantum field theory literature. A direct conceptual predecessor is the Physica Scripta paper “On the Renormalisation group, protein folding, and naturalness” the present author[ 19 ]. There, a first version of a renormalisation category was proposed: groups of RG equations, together with choices of regulators and optimisation schemes, were organised as objects and morphisms in a category, and the physical effect of changing regulators was interpreted as moving along categorical morphisms rather than along a single numerical trajectory in coupling space. This categorical viewpoint was then applied to two problems: naturalness and the apparent teleological structure of protein folding energy landscapes.[19] However, that work did not yet formulate RG directly in terms of universal properties, nor did it develop a general theory of adjoint curvature or structural monotonicity. The present paper can be viewed as a systematic extension and abstraction of those ideas: we give a precise categorical semantics to RG as a universal-property flow (UP-RG), formulate renormalisation categories and adjoint curvature explicitly, and show how these notions apply far beyond the specific contexts considered in Ref. [19]. Classical monotonicity theorems provide another crucial point of contact. In two dimensions, Zamolodchikov’s c -theorem identifies a function that decreases along RG flows; in four dimensions, Cardy proposed that a specific trace-anomaly coefficient should play an analogous role,[ 20 ] and subsequent work gave nonperturbative evidence and proofs from different angles. Holographic approaches have refined this picture by constructing gravitational duals of RG flows: in particular, Myers and Sinha showed that higher-curvature gravity theories obey a holographic c -theorem in arbitrary dimensions and related the flowing quantity to the A-type trace anomaly and universal entanglement entropy coefficients.[ 21 ] In three dimensions, Klebanov, Pufu and Safdi provided strong evidence for the F -theorem by computing the S3 free energy and demonstrating its monotonic behaviour along RG flows, even without supersymmetry.[ 22 ] All of these works share the paradigm that RG irreversibility is captured by a scalar function on theory space. Our framework instead promotes the “monotone” to a family of structural invariants derived from universal properties and adjoint curvature: the traditional scalar functions then appear as special shadows of richer categorical data. Our approach is also related to, but distinct from, continuum functional RG and holographic RG formalisms. In Polchinski’s formulation, RG is encoded as a functional differential equation for an effective action SΛ [ φ ]with a sliding cutoff Λ,[ 23 ] and in the exact renormalisation group (ERG) approach of 7 Morris the flow of the Legendre effective action is studied via derivative expansions and systematic truncations.[ 24 ] Holographic RG, as developed by de Boer, Verlinde and Verlinde, interprets RG evolution of a d -dimensional field theory as radial evolution in a ( d+ 1)-dimensional gravitational background,[ 25 ] while Heemskerk and Polchinski clarified the relation between Wilsonian RG and holographic flows in AdS/CFT.[ 26 ] In all these cases, RG is represented as a flow of functionals or couplings, possibly with a geometrical dual description. By contrast, UP-RG abstracts away from any specific functional representation and works instead with categories Cµ and their universal structures Uµ , treating functional or holographic equations as particular presentations of the underlying universal-property flow. Coarse-graining in complex systems is another area where our perspective naturally fits. Goldenfeld’s classic lectures emphasise the renormalisation group as a unifying framework for understanding equilibrium and non-equilibrium phenomena, pattern formation and scaling behaviour in a wide class of systems, from fluids to critical phenomena.[ 175 ] Goldenfeld and Kadanoff later argued that complexity can be understood as “structure with variations”, and highlighted the role of RG-type reasoning in extracting simple laws from systems with many interacting components.[ 28 ] More recently, philosophical analyses have proposed that RG methods provide a general template for explaining universality and structural similarity in complex systems across physics and beyond.[ 29 ] In all of these works, coarse-graining is central, but it is typically described in terms of flows on parameter spaces or effective equations. Our contribution is to give a categorical account in which the structures singled out in these discussions (e.g. universality classes, reduced descriptions, emergent funnels) are explicitly encoded as universal objects and universal morphisms Uµin categories Cµ, and their evolution is tracked functorially. A further point of contact is the extensive body of work on categorical and higher-categorical formulations of quantum field theory and topological defects. Moore’s analysis of rational conformal field theories already highlighted the role of modular tensor categories and related structures in organising the algebraic content of conformal field theory,[ 30 ] and subsequent developments have shown that many extended TQFTs and defect theories naturally give rise to bicategories and higher categories of phases and defects. Kapustin and Saulina, for example, described boundary conditions and line operators in abelian Chern–Simons theory in terms of a 2-category associated to the discriminant group,[ 31 ] while Carqueville, Meusburger and Schaumann showed that 3-dimensional defect TQFTs give rise to Gray categories with duals that encode fusion and duality of line and surface defects.[ 32 ] The Oberwolfach report by Bouwknegt, Freed and Schweigert surveyed the increasing use of categorical methods in string theory and QFT, emphasising the central role of fusion categories, modular tensor categories and higher-categorical structures in the modern understanding of quantum fields and strings.[ 33 ] These works provide the algebraic and categorical backdrop for our treatment of non-invertible symmetries and defect categories. The novelty here is that we do not just use categories to describe a fixed theory; we use them as the ambient space in which the RG itself becomes a functor, and where non-invertible symmetry data is tracked through universal-property flow. Within this landscape, the specific advances of the present paper can be summarised as follows: •From categorical hints to a full UP-RG formalism. Ref. [ 19 ] suggested that RG transformations and regulator changes could be treated categorically, but stopped short of defining a general framework. Here we introduce explicit renormalisation categories with objects ( µ, Cµ, Uµ )and RG functors Fµ1→µ2 , and we define universal-property RG (UP-RG) as the functorial evolution of universal structures Uµacross scales. •Adjoint curvature as a unifying notion of RG irreversibility. Classical monotonicity theorems and holographic c -functions capture irreversibility at the level of scalars.[ 20 – 22 ] Functional and holographic RG encode flows of actions or metrics.[ 23 – 26 ] We instead define an adjoint curvature RF that quantifies the failure of coarse-graining and embedding functors to form a perfect adjunction. This curvature unifies several notions of irreversibility: information loss, scheme dependence, anomalies, and teleological bias are all seen as manifestations of nontrivial RF acting on universal structures. •Structural monotonicity beyond scalar c-functions. Building on the above, we generalise the idea of an RG monotone from a single scalar function to families of structural monotones constructed from universal properties and adjoint curvature. These structural invariants constrain RG flows even in regimes where couplings oscillate, where no simple gradient-flow structure exists, or where the system is cooperative and non-Markovian—precisely the regimes that arise in strongly coupled QFT, strongly correlated electrons, complex biomolecular systems and multi-scale learning. 8 •Unified treatment of non-invertible symmetries, biomolecular systems and ML. While categorical methods have been developed for topological defects and higher-form symmetries,[ 30 – 33 ] and RG ideas have been applied to complexity and machine learning,[ 28 , 29 , 175 ] these areas have largely evolved in parallel. Our framework treats non-invertible symmetry categories, protein folding funnels, evolutionary dynamics and multi-scale representation learning as different instances of the same abstract mechanism: RG as universal-property flow in an appropriate category, with adjoint curvature controlling which aspects of structure survive coarse-graining and which are lost. In this sense, the present work is not just another reformulation of RG, but a proposal for a common categorical language in which apparently disparate uses of RG—from anomaly constraints in holography to folding pathways in proteins—can be compared, combined, and extended. II. MATHEMATICAL PRELIMINARIES In this section we fix notation and recall the basic categorical language that we will use throughout the paper. Our goal is not to be exhaustive, but to make precise what we mean by categories, functors, natural transformations, and in particular by the slogan that “objects are determined up to isomorphism by their mapping behaviour.” Standard references for these notions include Mac Lane,[ 186 ] Riehl,[ 35 ] Leinster,[178] and Borceux.[37] A. Categories, Functors, and Natural Transformations We begin with the basic definitions. Categories and morphisms. Acategory Cconsists of: •a class of objects, denoted Ob(C); •for every pair of objects X, Y ∈Ob(C), a set of morphisms (or arrows) from Xto Y, denoted HomC(X, Y )or simply Hom(X, Y )when Cis clear; •for every triple X, Y, Z ∈Ob(C), a composition map ◦: Hom(Y, Z)×Hom(X, Y )−→ Hom(X, Z),(g, f)7→ g◦f; •for every object X∈Ob(C), a distinguished identity morphism idX∈Hom(X, X), such that the following axioms hold: 1. Associativity: for all composable f, g, h, h◦(g◦f) = (h◦g)◦f. 2. Unit laws: for all f∈Hom(X, Y ), idY◦f=f=f◦idX. A morphism f∈Hom ( X, Y )is an isomorphism if there exists a morphism g∈Hom ( Y, X )such that g◦f = idX and f◦g = idY . In that case we write X∼ =Y and say that X and Y are isomorphic. Throughout, we regard isomorphic objects as “the same up to structure,” and many constructions will be invariant under replacement of an object by an isomorphic one. 9 Functors. Given two categories Cand D, a (covariant) functor F:C −→ D assigns: •to each object X∈Ob(C)an object F(X)∈Ob(D); •to each morphism f∈HomC(X, Y )a morphism F(f)∈HomDF(X), F(Y), such that: 1. Fpreserves identities: F(idX) = idF(X)for all X∈Ob(C), 2. Fpreserves composition: F(g◦f) = F(g)◦F(f)whenever g◦fis defined in C. Acontravariant functor F : C → D is formally a covariant functor F : Cop → D from the opposite category Cop (which has the same objects as C but with all morphisms reversed). Explicitly, a contravariant functor reverses the direction of arrows: f:X→Y7→ F(f) : F(Y)→F(X), while still preserving identities and turning compositions into compositions in the opposite order. Natural transformations. Let F, G : C → D be two (covariant) functors. A natural transformation η:F⇒Gassigns to each object X∈Ob(C)a morphism ηX:F(X)−→ G(X) in D , called the component of η at X , such that for every morphism f : X→Y in C the following naturality condition holds: G(f)◦ηX=ηY◦F(f).(9) In diagrammatic form, Eq. (9) states that the square F(X)F(Y) G(X)G(Y) F(f) ηXηY G(f) commutes. Natural transformations will play a central role when we discuss adjunctions and adjoint curvature: the unit and counit of an adjunction are canonical examples of natural transformations, and deviations from perfect naturality will encode the “curvature” we associate to non-ideal RG flows. Hom-functors and mapping behaviour. For an object A of a category C , we obtain canonical functors by fixing one end of the hom-set: •the covariant Hom-functor HomC(A, −) : C −→ Set sends an object Xto the set HomC(A, X)and a morphism f:X→Yto the function HomC(A, f) : HomC(A, X)−→ HomC(A, Y ), h 7→ f◦h; •the contravariant Hom-functor HomC(−, A) : Cop −→ Set sends an object Xto the set HomC(X, A)and a morphism f:X→Yto the function HomC(f, A) : HomC(Y, A)−→ HomC(X, A), h 7→ h◦f. Here Set denotes the category of sets and functions. These Hom-functors encode what we will call the mapping behaviour of the object A : they record, for every other object X , the set of morphisms from A to Xor from Xto A, and how these sets transform under preand post-composition. 16 are always well-defined natural endomorphisms of F and G in the 2-category Cat . Componentwise, for each object X∈Ob(C)and Y∈Ob(D)we have ΘF,X := F(X)◦F(ηX) : F(X)−→ F(X),(42) ΘG,Y := G(Y)◦ηG(Y):G(Y)−→ G(Y).(43) When FaG is a perfect adjunction, these composites are exactly the triangle identities (39) – (40) , and hence ΘF= idF,ΘG= idG,(44) where idFand idGdenote the identity natural transformations on Fand Grespectively. We will use the term broken adjunction to describe a situation in which one still has natural transformations ηand relating Fand G, but the triangle identities do not hold exactly, so that ΘF6= idFand/or ΘG6= idG.(45) Intuitively, F and G then behave like adjoints “up to a defect” given by Θ F and Θ G . Our goal is to interpret this defect as a kind of curvature associated to the would-be adjunction. Adjoint curvature as deviation from identity. In a general 2-category, the collection of endomorphisms of a given 1-morphism forms a monoid under vertical composition. In the case of Cat , the endomorphisms of a functor F are natural transformations F⇒F , which compose vertically in the usual way. If the 2-category is enriched over an additive category (for example, categories enriched in abelian groups or vector spaces, as in enriched category theory [ 46 ]), it often makes sense to consider linear combinations and differences of 2-morphisms. In such an enriched setting, we define the adjoint curvature of F(relative to the given ηand ) by RF:= ΘF−idF,(46) and similarly RG:= ΘG−idG.(47) Here the subtraction is to be understood in the enriched sense: if the hom-object End ( F )carries an abelian group or vector space structure, then ΘF−idFis simply the difference of these elements. Outside of a strictly additive context, it is often preferable to treat (46) – (47) as schematic and instead work with scalar or ordered invariants derived from Θ F and Θ G . For example, if C and D are enriched over normed spaces or Hilbert spaces (as in many applications to functional analysis and quantum theory [ 51 , 52 ]), then each component Θ F,X may have a norm k Θ F,X −idF(X)k , and we can define a curvature functional by taking a supremum or average over objects: kRFk:= sup X∈Ob(C) ΘF,X −idF(X) .(48) In this way, RF becomes a measure of how badly the would-be adjunction fails, quantified in a way appropriate to the enrichment. The terminology “curvature” is motivated by the analogy with parallel transport in differential geometry: there, an infinitesimal parallel transport around a small loop returns a vector to the same tangent space but not necessarily to its original value; the deviation from identity is measured by the Riemann curvature tensor. Similarly, the composite FF η −−→ FGF F −−→ F starts and ends at the same functor F but need not be the identity transformation; the deviation is precisely the adjoint curvature. Interpretation: irreversibility, information loss, anomalies, teleology. The adjoint curvature RF has several related interpretations that will be central in the rest of the paper: •Irreversibility. In an ideal adjunction describing a coarse-graining F and a reconstruction G , the triangle identity (39) means that F ( ηX )followed by F(X) leaves F ( X )unchanged. When RF6 = 0, the composite F ( ηX )– F(X) distorts F ( X )in a systematic way. This encodes a notion of irreversibility: even after applying the “best possible” reconstruction G and coarse-graining back with F, one does not return exactly to where one started. 17 •Information loss. In many physical applications, F will represent a coarse-graining or renormalisation step, and G a canonical embedding of coarse-grained data into a finer description. Perfect adjunction would mean that all information relevant to the mapping behaviour encoded by G is preserved under F . A nonzero curvature RF measures the extent to which information is lost even at the level of universal properties: it captures the part of the structure that cannot be recovered by any morphism factoring through G. •Anomalies and obstructions. In higher-categorical language, adjunctions are closely related to the theory of monads and comonads,[ 47 , 48 ] and curvature-like phenomena often appear as coherence defects or obstructions in 2-categories and bicategories.[ 49 , 50 ] In our setting, a non-vanishing RF can be interpreted as an anomaly obstructing the extension of a formal adjunction to a strict one, or as the failure of certain diagrams in a renormalisation 2-category to commute. This perspective will be particularly useful when we analyse non-invertible symmetries and their RG flows in defect categories. •Teleological bias. When we later interpret renormalisation as a dynamical process—a flow in “RG time”—the presence of a nonzero curvature RF will be seen to generate preferred directions in the space of universal structures. Roughly speaking, RF tells us in which structural directions the composite FGF pushes F away from the identity. This induces a notion of teleological or goal-directed behaviour: universal objects that minimise or annihilate curvature act as attractors or fixed points of the universal-property flow. We will make this intuition precise in later sections by constructing explicit curvature-driven dynamics on categories of theories and configurations. In summary, adjoint curvature provides a way to quantify the failure of coarse-graining and embedding functors to form a perfect adjunction. It is a structural, categorical notion that can be enriched to yield norms or orderings, and it will serve as the backbone for our generalised, structural monotonicity theorems in renormalisation. E. Enriched and Monoidal Categories (for RG and Symmetries) So far we have treated categories as mere collections of objects and morphisms with composition. For many applications to renormalisation and symmetries, this is not enough: one needs to keep track of additional structure on the Hom-sets (such as norms, probabilities, inner products) and of tensor products (fusion) of objects. This leads naturally to the notions of enriched and monoidal categories. In this subsection we recall these ideas and indicate how they will be used to describe RG flows and (non-)invertible symmetries. V-enriched categories. Let ( V,⊗, I )be a (strict) monoidal category. A V -enriched category (or V-category) Cconsists of the following data:[53, 54] •a class Ob(C)of objects; • for each pair of objects X, Y ∈Ob ( C ), an object HomC ( X, Y )of V , called the hom-object from X to Y; •for each object X, a morphism in V jX:I−→ HomC(X, X),(49) playing the role of an identity; •for each triple X, Y, Z ∈Ob(C), a composition morphism in V µX,Y,Z : HomC(Y, Z)⊗HomC(X, Y )−→ HomC(X, Z),(50) such that the obvious associativity and unit axioms hold: the diagrams expressing associativity of composition and the left/right unit laws, written in V using the associator and unitors of ⊗ , must commute for all quadruples of objects. When V = Set with the cartesian product, this reduces to the usual notion of a (locally small) category. The enriched hom-objects HomC ( X, Y )carry additional structure inherited from V (e.g. an order, metric, linear or probabilistic structure). This allows us to measure, compare or combine morphisms in a way that is sensitive to their “size” or “likelihood”, which will be crucial when we wish to quantify adjoint curvature or structural monotonicity in RG. 18 Example: Lawvere metric spaces. A particularly important example for our purposes is provided by Lawvere metric spaces.[ 54 ] Let ( R≥0∪ {∞},≥, + , 0) denote the poset of extended non-negative reals, ordered by ≥and equipped with monoidal product given by addition: d1⊗d2:= d1+d2, I := 0. This is a (strict) monoidal category in which there is at most one morphism d→d0 (namely when d≥d0 ). A category enriched in this Vis specified by: •a class of objects X; •for each pair x, y ∈X, an element d(x, y)∈R≥0∪ {∞}, playing the role of a “distance”; •the unit maps 0→d(x, x)express the inequality 0≥d(x, x), i.e. d(x, x)≤0; •the composition maps d(y, z) + d(x, y)−→ d(x, z) express the inequality d(x, z)≤d(x, y) + d(y, z),(51) which is the triangle inequality. Thus a V -enriched category for this choice of V is precisely a (possibly non-symmetric, possibly nonseparated) generalised metric space (X, d)with d(x, x)=0, d(x, z)≤d(x, y) + d(y, z)for all x, y, z. (52) In our context, such Lawvere-enriched categories can be used to encode “distances” between theories or configurations, or to quantify how much structure is lost along coarse-graining functors. Example: Hilbert and probabilistic enrichment. Other enrichments of interest include: •Hilbert enrichment: categories enriched in the monoidal category Hilb of (complex) Hilbert spaces and bounded linear maps, with tensor product as monoidal structure. Here each hom-object HomC ( X, Y )is itself a Hilbert space (e.g. a space of operators or states), and composition (50) is a bilinear (often bounded) map. Such enrichment is natural when discussing quantum channels or operator algebras from a categorical perspective.[57] •Probabilistic enrichment: categories enriched in a monoidal category of stochastic matrices, Markov kernels, or [0 , 1]-valued weights, where ⊗ encodes composition of probabilistic transitions. Here hom-objects encode conditional probabilities, and composition encodes the Chapman– Kolmogorov equations. This is relevant when RG steps are inherently stochastic or involve random coarse-graining procedures. In each case, enrichment provides a principled way to attach quantitative structure (distances, amplitudes, probabilities) to morphisms and to use that structure in defining norms or monotones, such as those used to measure adjoint curvature in Subsection II D. Monoidal categories. Amonoidal category (C,⊗, I, α, λ, ρ)consists of a category C, a bifunctor ⊗:C × C −→ C, called the tensor product, a distinguished object I(the unit), and natural isomorphisms αX,Y,Z : (X⊗Y)⊗Z∼ = −−→ X⊗(Y⊗Z)(associator),(53) λX:I⊗X∼ = −−→ X, ρX:X⊗I∼ = −−→ X(left and right unitors),(54) satisfying the coherence conditions: the famous pentagon and triangle diagrams commute for all choices of objects.[ 53 ] Mac Lane’s coherence theorem (in the ordinary, non-enriched case) shows that up to canonical isomorphism any composite of associators and unitors is uniquely determined, so that one may often suppress α, λ, ρ and work in a “strictified” monoidal category for intuition, without loss of generality. Canonical examples of monoidal categories include: 19 •(Vectk,⊗k, k), vector spaces over a field kwith the usual tensor product; •(Hilb,⊗,C), Hilbert spaces with the Hilbert-space tensor product; •representation categories Rep(G)of a group G, where ⊗is the tensor product of representations. In our setting, monoidal structure will encode the fusion or stacking of defects and symmetries: e.g. fusing two topological line operators in a QFT corresponds to taking their tensor product in an appropriate monoidal category. Fusion categories and categorical symmetries. A particularly important class of monoidal categories for quantum field theory and condensed matter physics is given by fusion categories.[ 55 ] Roughly speaking, a fusion category over an algebraically closed field k is a k -linear, semisimple, rigid monoidal category with finitely many isomorphism classes of simple objects, finite-dimensional Hom-spaces, and simple unit. More explicitly, a fusion category Fsatisfies: 1. Fis a k-linear, abelian category, with finite-dimensional Hom-spaces; 2. Fis monoidal with tensor product ⊗:F × F → F and unit object 1, and the unit is simple; 3. every object is a finite direct sum of simple objects; 4. F is rigid: each object X has a left and a right dual X∨ with appropriate evaluation and coevaluation morphisms. Fusion categories arise as categories of anyons or topological line operators in (2+1)-dimensional topological phases, and more generally as symmetry categories of QFTs. In such interpretations, objects of F correspond to defects, the tensor product X⊗Y encodes their fusion (stacking), and duality encodes orientation reversal or conjugation of defects. Invertible objects (those X such that there exists Y with X⊗Y∼ =1 ) form a group, corresponding to ordinary group-like symmetries; non-invertible objects encode genuinely non-invertible symmetries. We will return to this point in detail when we discuss non-invertible symmetries and their RG flows. Higher categories and higher symmetries. Monoidal and fusion categories already capture a rich class of extended operators and symmetries. For higher-dimensional QFTs and higher-form symmetries, however, it is often natural to work with higher categories: bicategories, tricategories, or ( ∞, n )-categories, in which 1-morphisms, 2-morphisms, and so on, represent defects of different codimensions and their junctions. Lurie’s theory of ( ∞, 1)-categories[ 181 ] and Leinster’s work on higher operads and higher categories[53] provide conceptual foundations for such structures. In this paper we will not fix a specific model of higher categories; rather, we use the language and intuition of enriched and monoidal categories as the 1-categorical shadow of a higher-categorical picture. When we refer to “higher” or “categorical” symmetries, we have in mind the idea that symmetries and defects form monoidal (or fusion) categories, possibly enhanced to higher categories, and that renormalisation acts on these symmetry categories by monoidal or enriched functors. Role in RG and symmetries. To summarise, enriched and monoidal categories provide two complementary tools that will be used throughout the rest of the paper: • Enrichment allows us to attach metric, Hilbert-space or probabilistic structure to hom-objects, enabling us to quantify notions such as distance between theories, magnitude of adjoint curvature, or probability of transitions under coarse-graining. • Monoidal (and fusion) structure allows us to describe the composition and fusion of defects, operators and symmetries, and to treat ordinary and non-invertible symmetries on the same categorical footing. Renormalisation functors will be assumed, in appropriate contexts, to be enriched and/or monoidal, so that they act compatibly on these structures. This is essential for formulating structural monotonicity theorems and for tracking non-invertible symmetries along RG flows in a precise, categorical way. III. RENORMALISATION CATEGORIES AND UNIVERSAL-PROPERTY RG In this section we move from general categorical notions to a concrete framework for renormalisation. The key idea is to package RG data (theory, regulator, truncation, and flow) into objects of a category, and to treat scheme changes and RG intertwiners as morphisms. This “renormalisation category” will be the ambient setting for universal-property RG in the subsequent subsections. 20 A. The Renormalisation Category We begin by making precise what we mean by a renormalisation object and the category whose morphisms encode structure-preserving scheme changes. Renormalisation data. In an RG analysis one typically chooses: • amicroscopic theory T , e.g. given by a space of fields Φand a bare action or Hamiltonian SΛ [ φ ]at some UV scale Λ; • aregulator or coarse-graining scheme R , such as a sharp momentum cutoff, a smooth cutoff function (as in functional RG), dimensional regularisation, lattice discretisation, or a blocking transformation in real space;[59, 174] • atruncation or approximation prescription τ , specifying which operators or couplings are retained (e.g. a derivative expansion, a polynomial truncation of the effective action, or a finite basis of operators).[60–62] We group these choices into a single object: Definition III.1 (Renormalisation datum).Arenormalisation datum (or RG scheme) is a triple S= (T, R, τ), where Tis the microscopic theory, Ra regulator/coarse-graining scheme, and τa truncation or approximation scheme. Associated to Sis a family of effective theories Tµ parametrised by a scale µ and a flow RGS µ1→µ2:Tµ1−→ Tµ2, µ1> µ2, implementing the RG evolution according to (T, R, τ). In practice, Tµ often lives in some (infinite-dimensional) space of actions or Hamiltonians and RGS µ1→µ2 is generated by beta functions dgi(µ) dln µ=βS i{g(µ)},(55) or by an exact functional RG equation (e.g. of Wetterich or Polchinski type).[59, 61, 62] Objects and morphisms of the renormalisation category. We now package renormalisation data into a category. Definition III.2 (Renormalisation category).The renormalisation category Ris defined as follows: •Objects. Objects of Rare renormalisation data S= (T, R, τ)as in Definition III.1. •Morphisms. A morphism F:S1= (T1, R1, τ1)−→ S2= (T2, R2, τ2) consists of: (i) a map FT between the corresponding theory spaces that sends effective theories of S 1 to effective theories of S2at each scale: FT(µ) : T(1) µ−→ T(2) µ, (ii) a compatibility condition with RG flows: for all µ1> µ2we require that T(1) µ1T(1) µ2 T(2) µ1T(2) µ2 RGS1 µ1→µ2 FT(µ1)FT(µ2) RGS2 µ1→µ2 (56) commutes, either strictly or up to a controlled notion of approximation (for instance, equality of correlation functions computed in a given truncation). 21 •Composition. Given morphisms F:S1→S2, G :S2→S3, their composite G◦F:S1→S3is defined by composing the underlying maps on theories: (G◦F)T(µ) := GT(µ)◦FT(µ), which again satisfies the commutation property (56) by functoriality of the RG flows. •Identities. For each object S= ( T, R, τ ), the identity morphism idS :S → Sis given by idT at each scale µ, which trivially satisfies (56). Thus R is the category whose objects are RG schemes and whose morphisms are RG intertwiners: maps that convert one scheme into another while commuting with their respective RG flows. In the idealised case where truncations are absent and the RG flows are exact, commutativity in Eq. (56) is literal; in practice, with truncations τ1, τ2 , we only require commutativity up to errors that vanish in a specified limit (e.g. as more operators are included in the truncation or as the regulator is removed).[60, 61] Examples and interpretation. Example III.3 (Change of regulator at fixed truncation) . Let S 1 = ( T, R1, τ )and S 2 = ( T, R2, τ ) correspond to the same microscopic theory and truncation, but with different regulators R1 and R2 (for example, two different cutoff functions in a functional RG equation).[ 59 , 61 ] Assuming the truncation is sufficiently rich, one can construct a map FT ( µ )that identifies the effective actions T(1) µ and T(2) µ at each scale by matching a set of physical observables (e.g. correlation functions at finite momenta). If this identification respects the flows in the sense of Eq. (56) , then F :S 1→ S 2 is a morphism in R . From the point of view of R , R1 and R2 represent different coordinate systems on the same underlying renormalisation structure. Example III.4 (Change of truncation at fixed regulator) . Let S 1 = ( T, R, τ1 )and S 2 = ( T, R, τ2 )use the same microscopic theory and regulator, but different truncation schemes (e.g. a lower-order versus a higher-order derivative expansion). Under suitable conditions, there is a natural inclusion map from the smaller truncation to the larger one (viewing the latter as extending the former), and possibly a projection in the opposite direction (e.g. integrating out higher-order operators). These maps can be arranged to form morphisms in R , and their composition reflects the compatibility of truncations with RG flows. In practice, inconsistencies between truncations correspond to a failure of Eq. (56) , which will later be measured by adjoint curvature in the categorical setting. 2-categorical refinement: RG functors and meta-RG. Definition III.2 treats RG schemes as objects and scheme-change maps as morphisms. However, each renormalisation datum S= ( T, R, τ )also gives rise to categories of its own: for instance, •a category CSof effective theories and their field redefinitions at varying scales; • a category of observables or correlation functions, with morphisms given by RG-induced maps between them; •a category of defects and symmetries, as discussed in Subsection II E. An RG step at the level of Sthen induces functors between such categories, and morphisms in R induce natural transformations between these functors. This suggests the following refinement. Definition III.5 (Renormalisation 2-category (informal)) . The renormalisation 2-category R is a (weak) 2-category whose: •objects are categories CSassociated to renormalisation data S; •1-morphisms are RG functors FS µ1→µ2:CS,µ1−→ CS,µ2, as well as scheme-change functors between different CS; 22 • 2-morphisms are natural transformations between such functors, encoding “meta-RG” operations (such as continuous deformations of regulators or truncations) and providing a higher-categorical setting for adjoint curvature. We keep this description informal for now; the precise structure of R will depend on the level of enrichment (metric, probabilistic, monoidal) one wishes to consider. What is important is that the renormalisation category R furnishes the 1-categorical backbone, while R captures the full 2-categorical picture in which RG functors and their adjunctions live. In the next subsection we will connect this structure explicitly to the notion of universal-property RG, where the primary objects of interest are not individual couplings or actions, but universal constructions Uµin the categories CS,µ, and RG acts as a flow in the space of such universal properties. B. RG Functors and RG Flows In Subsection III A we organised renormalisation data into a category R whose objects are schemes X = ( T, R, τ )and whose morphisms are structure-preserving scheme changes. We now move one level down and describe, for a fixed renormalisation datum X , how the RG flow is implemented as a family of functors between categories associated with different energy scales. The standard beta functions of perturbative renormalisation group theory will then appear as coordinate expressions of this functorial flow.[63, 64, 176] Scale-indexed categories. Fix a renormalisation datum X = ( T, R, τ )as in Definition III.2. For each scale µ > 0we associate a category CX ( µ )capturing the structure of the theory “as seen at scale µ ”. There are several natural choices, depending on the level of description: •Category of effective theories. Objects are effective actions or Hamiltonians Sµ (up to field redefinitions) compatible with the scheme ( R, τ ); morphisms are field redefinitions or reparametrisations that preserve physical observables at scale µ . This realises the heuristic notion of “theory space” as a category.[66, 176] •Category of observables at scale µ. Objects are (sub)algebras of observables AX ( µ )together with states or correlation functionals; morphisms are ∗ -homomorphisms or completely positive maps respecting the chosen structure. This choice is natural in algebraic QFT and functional RG.[67] •Category of defects and operators. When analysing symmetries, especially non-invertible and higher symmetries, it is convenient to let CX ( µ )be a monoidal (often fusion) category of topological defect operators and their junctions, with tensor product representing fusion.[68] We will write Cµinstead of CX(µ)when the renormalisation datum Xis understood. RG functors between scales. The RG flow from scale µ1to µ2< µ1induces a functor Fµ1→µ2:Cµ1−→ Cµ2,(57) which we call the RG functor from µ1 to µ2 . Intuitively, Fµ1→µ2 implements the coarse-graining dictated by the chosen scheme (R, τ): • in the effective-theory picture, Fµ1→µ2 sends an action Sµ1 to the action Sµ2 obtained by integrating out modes with µ2<|p|< µ1and rescaling fields and coordinates; •in the observable picture, Fµ1→µ2sends an observable algebra (and state) at scale µ1to its image at scale µ2under the RG transformation RGµ1→µ2 Xfrom Eq. (58); • in the defect picture, Fµ1→µ2 sends a defect (and its junctions) in the UV theory at µ1 to the corresponding effective defect in the IR theory at µ2, as realised by the RG domain wall. Definition III.6 (RG functors) . An RG functor for a renormalisation datum X is a family of functors {Fµ1→µ2}µ1>µ2>0as in (57), satisfying: 1. Semigroup (composition) property: for all µ1> µ2> µ3>0, Fµ2→µ3◦Fµ1→µ2=Fµ1→µ3, Fµ→µ= idCµ.(58) 23 2. Compatibility with observables: if Cµ is chosen so that there is a faithful functor Uµ : Cµ→Alg to a category of observable algebras, then Uµ◦Fµ1→µ2 represents the RG map RGµ1→µ2 X of Eq. (58) at the level of algebras and states. The semigroup property (58) is the functorial translation of the usual RG semigroup property (58) : coarse-graining from µ1 to µ3 is equivalent to first coarse-graining from µ1 to µ2 and then from µ2 to µ3 . The additional data encoded in Cµ —field redefinitions, symmetries, defects, enrichment—is carried along functorially. RG flows as one-parameter families of functors. For many purposes it is convenient to reparametrise the scale by t:= ln µ µ0 , t ∈R,(59) where µ0 is a reference scale. The family of RG functors can then be rewritten as a one-parameter family Φt:Ct0−→ Ct0+t, t ∈R,(60) with Φ0= id and Φt+s= Φs◦Φt,(61) whenever the domains and codomains match. Formally, { Φ t}t∈R is a (semi)group object in the endomorphism 2-category of the family {Ct}. If the renormalisation scheme is sufficiently regular, one can attempt to define an infinitesimal generator of this one-parameter family: a derivation-like operator that differentiates functors with respect to t . At a heuristic level, this generator is what becomes the beta function when we pass to coordinates on the space of theories. Beta functions as coordinate expressions of RG functors. To connect with the usual formalism, suppose we are working in the category of effective theories, and that each object Sµ∈Ob ( Cµ )admits a finitedimensional parametrisation by couplings g ( µ ) = ( g1 ( µ ) , . . . , gN ( µ )) relative to some operator basis fixed by the truncation τ. This means there is a functor Pµ:Cµ−→ RN,(62) (or to some manifold M of couplings) assigning to each effective theory its vector of couplings, and to each morphism the induced reparametrisation. Applying Pµ2to Fµ1→µ2(Sµ1)yields the couplings at scale µ2: g(µ2) = Pµ2Fµ1→µ2(Sµ1), g(µ1) = Pµ1(Sµ1).(63) In the continuum limit, take µ2 = µ e−δt with small δt > 0. Writing Sµ for the effective theory at scale µ , the RG functor acts as Sµe−δt =Fµ→µe−δt (Sµ), and the couplings change according to gi(µe−δt) = Pµe−δt,iFµ→µe−δt (Sµ).(64) Assuming differentiability in ln µ, we expand to first order: gi(µe−δt) = gi(µ)−δt βig(µ)+O(δt2),(65) where βig(µ):= −∂ ∂tPµe−t,iFµ→µe−t(Sµ)   t=0 .(66) Dividing by δt and taking the limit δt →0yields the familiar beta function equation dgi(µ) dln µ=βig(µ),(67) which we recognise as the coordinate expression of the functorial RG flow on the space of couplings.[ 63 , 66, 176] In other words, the usual RG flow in coupling space is obtained by composing the RG functor Fµ1→µ2 with a choice of coordinate (parameter) functor Pµ . Different choices of Pµ correspond to different parametrisations of theory space (e.g. different renormalisation schemes), while the underlying functorial RG structure—the family of categories Cµand functors Fµ1→µ2satisfying (58)—is more invariant. 24 Multiple categorical realisations of the same RG flow. The same physical RG flow can often be realised in several different categorical ways. For example: • In a perturbative QFT with a fixed field content, one may take Cµ to be the category of bare actions modulo field redefinitions and counterterm reparametrisations; Fµ1→µ2 acts by integrating out momentum shells and rescaling fields. • In algebraic QFT, one may take Cµ to be a category of nets of local algebras on spacetime regions at resolution µ , and Fµ1→µ2 to be a coarse-graining functor relating nets at different resolutions.[ 69 ] • In topological or conformal field theories, one may let Cµ be a (fusion) category of line or surface operators, and Fµ1→µ2a monoidal functor induced by RG domain walls.[70] In each case, the same physical content—how observables and symmetries change with scale—is encoded in different categorical languages. Universal-property RG will focus precisely on those aspects of the flow that are invariant under such choices: limits, adjunctions, Kan extensions, and other universal constructions in the categories Cµ that are preserved or transformed in a controlled way by the functors Fµ1→µ2. Summary. We have formalised RG in categorical terms as follows: • For each scale µ we have a category Cµ encoding the structure of the theory at that scale (effective actions, observables, defects, etc.). • For each pair of scales µ1> µ2 we have an RG functor Fµ1→µ2 : Cµ1→ Cµ2 forming a semigroup in RG time. • The familiar beta functions arise when we choose coordinates (couplings) on the objects of Cµ via a parameter functor Pµ, and differentiate the induced flow in coupling space. In subsequent subsections, we will describe how to lift this picture from objects to universal properties: rather than tracking individual couplings or objects, we will track how limits, adjoints, and other universal structures Uµ evolve under the functors Fµ1→µ2 . This universal-property flow will be the central object of study for our categorical renormalisation group. C. Universal-Property Flow (UP-RG): Definition In Subsection III B we described renormalisation for a fixed scheme X = ( T, R, τ )in terms of a family of categories {Cµ}µ>0and RG functors Fµ1→µ2:Cµ1−→ Cµ2, µ1> µ2,(68) forming a semigroup as in Eq. (58) . In the usual formulation, one then chooses coordinates (couplings) on the objects of Cµ and reads off the induced beta functions as in Eq. (67) . Our goal now is to shift the focus from individual objects and couplings to universal properties. The basic idea was already encapsulated informally in Eq. (4); in this subsection we make that idea precise. Universal structures at a fixed scale. Fix a scale µ and the corresponding category Cµ . As discussed in Subsection II B, many constructions in Cµ are defined by universal properties: products, limits, colimits, free objects, reflective subcategories, Kan extensions, centers, and so on. We abstract this as follows. Definition III.7 (Universal structure at scale µ).Auniversal structure at scale µconsists of: •an object Uµ∈Ob(Cµ); • auxiliary data σµ (such as cones, cocones, units/counits, natural transformations, or half-braidings) witnessing that Uµsatisfies a specified universal property in Cµ. We write ( Uµ, σµ ) ∈Univ ( Cµ ), where Univ ( Cµ )denotes the collection (or category) of chosen universal constructions in Cµ . Two universal structures ( Uµ, σµ )and ( U0 µ, σ0 µ )are regarded as equivalent if there exists an isomorphism ϕ : Uµ→U0 µ in Cµ that identifies the universal data σµ and σ0 µ via transport of structure. Concretely, (Uµ, σµ)might be: 25 • the limit of a diagram Dµ : Jµ→ Cµ encoding how degrees of freedom in different momentum shells are glued together; • the object of a reflective subcategory Eµ⊂ Cµ , with reflector LµaIµ , representing the “best” effective theory at resolution µ; • the value at a given object of a Kan extension of an observable functor along a coarse-graining functor;[71, 72] •the Drinfeld center Z(Sµ)of a symmetry category Sµ, encoding commuting defect operators.[73] In each case, the essential data is universal: it is specified by mapping properties rather than by particular coordinates or presentations. Universal-property RG data. We now package RG functors and universal structures into a single notion. Definition III.8 (Universal-property RG datum) . Auniversal-property RG (UP-RG) datum consists of: 1. a family of categories {Cµ}µ>0associated to a renormalisation datum X= (T, R, τ); 2. RG functors Fµ1→µ2 : Cµ1→ Cµ2 for all µ1> µ2> 0satisfying the semigroup property (58) , as in Definition III.6; 3. for each µ > 0, a choice of universal structure (Uµ, σµ)∈Univ(Cµ); 4. for each pair µ1> µ2, an assignment Φµ1→µ2:Univ(Cµ1)−→ Univ(Cµ2) which associates to ( Uµ1, σµ1 )a universal structure ( Uµ2, σµ2 )in Cµ2 , in such a way that ( Uµ2, σµ2 ) is a universal construction built from Fµ1→µ2(Uµ1)and the pushed-forward data Fµ1→µ2(σµ1). The family {Uµ}µ>0equipped with the maps Φµ1→µ2constitutes a universal-property flow. In particular, the universal-property flow between scales µand µ0is summarised by the assignment Uµ7−→ Uµ0:= Φµ→µ0Fµ→µ0(Uµ),(69) which is the precise version of the schematic relation anticipated in Eq. (4) . The key point is that Φ µ→µ0 is itself a universal construction: it “repairs” the pushed-forward object Fµ→µ0 ( Uµ )so that it again satisfies the defining universal property at the new scale. Examples of universal-property flow. We now spell out several concrete instances of Eq. (69). Example III.9 (Limits and colimits of diagrams) . Let Dµ : J→ Cµ be a diagram (e.g. encoding the decomposition of degrees of freedom into momentum shells or spatial blocks), and let ( Uµ, λµ )be its limit: Uµ= lim Dµ, with cone λµ,j : Uµ→Dµ ( j )satisfying the universal property (23) . Applying the RG functor Fµ→µ0 yields a diagram Fµ→µ0◦Dµ:J→ Cµ0with cones Fµ→µ0(λµ,j)from Fµ→µ0(Uµ). If Fµ→µ0preserves limits of shape J, then there is a canonical isomorphism Fµ→µ0(Uµ)∼ =limFµ→µ0◦Dµ,(70) so that we may simply take Uµ0:= Fµ→µ0(Uµ). More generally, if Fµ→µ0 does not preserve such limits, we define Uµ0 to be the actual limit of Fµ→µ0◦Dµ : Uµ0:= limFµ→µ0◦Dµ, and Φ µ→µ0 is the universal cone from Fµ→µ0 ( Uµ )to this limit. The discrepancy between Fµ→µ0 ( Uµ )and Uµ0then measures how strongly the RG functor distorts the limiting universal structure. 32 Zamolodchikov’s c -theorem in two dimensions. In two-dimensional relativistic QFT, Zamolodchikov’s c -theorem establishes that any unitary, Poincaré-invariant, local theory with a discrete spectrum admits ac-function c({gi}, µ)such that:[81, 82] 1. c ( {gi}, µ )is a function on coupling space, constructed from two-point functions of the stress-energy tensor; 2. at RG fixed points it reduces to the Virasoro central charge, c∗=cCFT; 3. along RG flows it is strictly decreasing, dc(µ) dln µ=−Gij({g})βi({g})βj({g})≤0,(94) where Gij is a positive-definite “Zamolodchikov metric” on coupling space and βi = dgi/d ln µ are the beta functions. The construction proceeds by considering the Euclidean correlation functions of the holomorphic and anti-holomorphic components T ( z ), ¯ T ( ¯z )of the stress tensor in complex coordinates, and defining three scalar functions F ( r ), G ( r ), H ( r )built from rotationally invariant combinations of the two-point function hTµν(x)Tρσ(0)i. A particular linear combination, c(r)=2F(r)−G(r)−3 8H(r), is shown to satisfy dc(r) dln r=−3 2Gij(g(r)) βi(g(r))βj(g(r)) ≤0, where r is a length scale and Gij is a positive-definite matrix defined by integrated two-point functions of the perturbing operators. Identifying r∼ 1 /µ gives Eq. (94) . The positivity of Gij follows from reflection positivity and unitarity of the underlying QFT, while the gradient form of the flow relies on conservation of the stress tensor and the structure of conformal perturbation theory.[81, 82] Thus the c -theorem provides a concrete realisation of the scalar monotonicity condition (93) with M(µ) = c(µ): RG flows can only connect CFTs of decreasing central charge, cUV > cIR. The four-dimensional a -theorem. In four-dimensional QFT, the natural analogue of the central charge is provided by the coefficients of the trace anomaly on a curved background. For a conformal field theory in d= 4, the vacuum expectation value of the trace of the stress-energy tensor takes the form hTµ µi=1 (4π)2c W2 µνρσ −a E4+bR,(95) where Wµνρσ is the Weyl tensor, E4 the Euler density, and R the Ricci scalar.[ 82 ] The a -theorem asserts, in its modern form, that there exists a function a({gi}, µ)on the space of couplings such that: •at RG fixed points, a({g}, µ)reduces to the Euler-anomaly coefficient in Eq. (95); •along RG flows, da(µ) dln µ=−χij({g})βi({g})βj({g})≤0,(96) for some positive-definite matrix χij, so that aUV > aIR. The original “weak” version of the theorem proposed that aUV > aIR for RG flows between unitary, Lorentz-invariant fixed points. A non-perturbative proof of the inequality, based on dilaton scattering and positivity of the corresponding spectral functions, was later given by Komargodski and Schwimmer; geometrical and local-RG refinements are reviewed in Ref. [ 82 ]. Just as in the two-dimensional case, Eq. (96) is of the form (93) , with a positive quadratic form in the beta functions ensuring monotonicity. 33 F -theorem in three dimensions. In three-dimensional QFT, a useful monotone is provided by the free energy on the round three-sphere. For a (super)conformal field theory on S3 , the Euclidean partition function ZS3is finite and one defines F:= −ln |ZS3|,(97) which can be interpreted as a measure of the number of degrees of freedom. The F -theorem conjectures, and in many settings proves, that Fdecreases along RG flows: FUV > FIR,∆F:= FUV −FIR >0.(98) In three-dimensional N = 2 supersymmetric theories, supersymmetric localisation makes ZS3 exactly computable, and F can be expressed in terms of the R-charges of fields. This led to the proposal of an F -maximisation principle and strong evidence for the F -theorem in a broad class of models.[ 83 ] More generally, holographic and entropic arguments support the monotonicity of F in non-supersymmetric theories, under the assumptions of unitarity, locality and appropriate boundary conditions. From the perspective of Eq. (93) , M ( µ )is here the finite part of the sphere free energy, with monotonicity encoded in the inequality (98) between fixed points. Entropic monotones and the data processing inequality. The above monotones are closely tied to information-theoretic quantities defined from the vacuum state and its deformations. A central role is played by relative entropy. Given two states ρ and σ on a quantum system, the quantum relative entropy is S(ρkσ) := Tr ρ(ln ρ−ln σ), whenever the support of ρis contained in that of σ.[123] It satisfies: •S(ρkσ)≥0, with equality if and only if ρ=σ; •data processing inequality (DPI): for any completely positive, trace-preserving (CPTP) map E, SE(ρ)kE(σ)≤S(ρkσ).(99) Eq. (99) is a general expression of the fact that quantum channels cannot increase the distinguishability of states; it can be derived from the joint convexity and monotonicity properties of trace functions.[ 86 , 123 ] In relativistic QFT, entanglement entropy and relative entropy for regions of spacetime provide powerful tools to prove RG monotonicity theorems. For instance: • In 1+1 dimensions, Casini and Huerta constructed an entropic c -function from the entanglement entropy of intervals and used strong subadditivity plus Lorentz invariance to prove an alternative version of the c-theorem purely in terms of entropy.[84] • In higher dimensions, entanglement and mutual information across spheres or other symmetric regions can be related to sphere free energies or anomaly coefficients, leading to entropic proofs of Fand a-type monotonicity under suitable assumptions.[85] All these entropic constructions rely heavily on: •Positivity and reflection positivity: to ensure that relative entropy and correlation matrices are non-negative and that the Zamolodchikov-type metrics are positive-definite. •Convexity and strong subadditivity: to control the behaviour of entanglement entropies under geometric manipulations (splitting and joining regions) and to derive inequalities for entropic c-functions. •Decoupling and locality: to guarantee that massive degrees of freedom decouple in the IR and that UV divergences can be subtracted in a universal way, leaving finite, scheme-independent monotones. 34 Summary and motivation for the UP-RG extension. The classical c -, a -, and F -theorems, together with entropic monotones based on relative entropy, provide a powerful but essentially scalar picture of RG irreversibility: •There exists a function M(µ)on theory space such that dM/d ln µ≤0along RG flows. • This function is typically constructed from correlation functions or entropic quantities associated with the stress-energy tensor. •Monotonicity crucially relies on unitarity/reflection positivity, locality, convexity, and the absence of exotic behaviours such as limit cycles or non-decoupling of massive modes. While extremely successful in relativistic QFT, this framework is less adapted to strongly coupled, categorically rich or cooperative systems, where: •RG flows may be non-Markovian or involve oscillatory behaviour in coupling space; •non-invertible symmetries and higher-categorical structures play a central role; • natural observables are structural (defects, centers, universal limits) rather than purely numerical. These limitations motivate the universal-property RG (UP-RG) developed in the previous sections, where irreversibility is captured not only by scalar monotones but by structural monotonicity of universal constructions and adjoint curvature. B. Limitations in Strong Coupling and Cooperative Systems The scalar monotonicity theorems reviewed in Subsection IV A rely on a set of structural assumptions that are often violated, or at least extremely difficult to verify, in strongly coupled and cooperative systems. These assumptions include: positivity and reflection positivity (unitarity), convexity of effective actions, locality and decoupling of massive modes, and a gradient-flow structure on coupling space. In this subsection we highlight several classes of systems in which these assumptions break down: non-convex effective actions, oscillatory or cyclic RG flows, strongly correlated electrons and non-Fermi liquids, and systems with long-range entanglement or cooperative behaviour. Non-convex effective actions and functional RG. In the Wilsonian picture, integrating out fluctuations above a scale k yields an effective average action Γ k [ φ ]whose exact infrared limit Γ k→0 is a convex functional of the fields. In practice, however, approximate RG equations—such as those obtained from the functional renormalisation group (FRG) with truncations in derivative or operator expansions— often produce non-convex effective potentials, especially in strongly coupled regimes or near first-order transitions.[88, 89] For example, in a scalar theory with bare double-well potential, the exact effective potential Uk→0 ( φ )is convex (the Maxwell construction), but low-order truncations of the FRG flow may retain a double-well shape with a non-convex region. This signals that the approximated RG flow does not respect the convexity constraints required by a standard variational or entropic interpretation. In such regimes: • the effective potential may possess multiple local minima separated by barriers that change nonmonotonically with scale; • susceptibility matrices and correlation functions can become non-positive in truncated approximations; • candidate scalar monotones constructed from curvature of the free energy or integrated correlators may lose their positivity properties. Thus the conditions underlying the existence of a smooth, strictly decreasing function M ( µ )of the type (2) become fragile or ill-defined. This motivates a shift in focus from scalar quantities derived from a specific truncation to structural properties of the RG flow that are less sensitive to non-convex artefacts. 35 Oscillating beta functions and RG limit cycles. Another obstruction to scalar monotonicity arises when RG flows exhibit limit cycles or more complicated recurrent behaviour. In a limit cycle, the couplings return to their original values (up to a symmetry) after a finite change in RG “time”. Formally, a set of couplings gi(µ)exhibits a limit cycle of period T > 0if gi(µeT) = gi(µ)for all µand all i. (100) This behaviour has been explicitly realised in quantum theories with an infinite tower of bound states (Efimov physics) and in certain renormalisation schemes for singular potentials.[90, 91] In the presence of a limit cycle, no non-constant scalar function M ( µ )can satisfy a strict monotonicity condition of the form dM(µ) dln µ<0 along the entire flow, since this would force M to change by a finite amount over each period T , contradicting the periodicity gi ( µeT ) = gi ( µ )together with the assumption that M depends smoothly on the couplings. At best, M can be constant on the cycle or monotone only on segments of the flow outside the recurrent region. More complicated oscillatory or chaotic RG trajectories present similar obstacles: they lack the gradient-flow structure that underpins the c-, a-, and F-theorems. These examples show that conventional scalar monotones are intrinsically tied to flows that are asymptotically gradient-like and do not exhibit recurrent behaviour. In contrast, the universal-property RG framework can in principle track structural features (such as the behaviour of defect categories or centers) along cycles, even when scalar functions fail to capture irreversibility. Strongly correlated electrons and non-Fermi liquids. Interacting fermion systems in d≥ 2provide another important class of examples where standard conditions for scalar monotonicity become problematic. In Landau Fermi liquids, low-energy excitations near the Fermi surface are described in terms of weakly interacting quasiparticles, and RG flows near the Fermi surface can often be organised in terms of marginal and irrelevant interactions.[ 92 ] However, in many strongly correlated materials (e.g. highTc cuprates, heavy fermions) the Fermi-liquid paradigm appears to break down, leading to non-Fermi liquid behaviour: anomalous scaling of response functions, absence of well-defined quasiparticles, and singular self-energies.[93] In such systems: • RG flows in the space of couplings defined on the Fermi surface (e.g. in the BCS, forward-scattering, and umklapp channels) can head towards strong-coupling regimes where perturbative control is lost; • competing instabilities (superconductivity, density waves, spin liquids) lead to flows that are highly sensitive to initial conditions and to the precise choice of truncation and scheme; • the low-energy effective theory may include emergent gauge fields or topological degrees of freedom not easily captured by a simple local stress tensor or entanglement entropy functional. In these circumstances, the construction of a single scalar monotone M ( µ )with clear physical interpretation and robust positivity properties is highly non-trivial. Instead, one expects that different universal structures (e.g. symmetry categories, emergent gauge sectors, or Fermi-surface patches) may exhibit different forms of structural monotonicity, as encoded in the universal-property RG framework. Long-range entanglement and cooperative phases. Topologically ordered phases, quantum spin liquids, fracton phases, and other systems with long-range entanglement provide striking counterexamples to the assumptions underlying many entropic monotonicity arguments. In gapped topological phases in two spatial dimensions, the ground state exhibits a characteristic pattern of long-range entanglement, reflected in a universal contribution −γto the entanglement entropy of a disk, S(A) = α|∂A| − γ+· · · ,(101) where γ is the topological entanglement entropy encoding the total quantum dimension of the underlying anyon theory.[ 94 , 95 ] This term is not easily interpreted as a local density of degrees of freedom, and its behaviour under continuous deformations can be subtle: it is robust under any RG flow that preserves the topological order, but can jump discontinuously across phase transitions. More generally, in cooperative systems with long-range entanglement or constraints: • entanglement entropies may obey modified “area laws” with subleading universal terms that do not behave smoothly under RG; 36 • integrating out degrees of freedom can generate emergent gauge constraints and topological sectors that are invisible in local correlation functions but crucial for the global phase structure; • positivity and strong subadditivity hold at the level of entropic quantities, but the mapping between these quantities and simple scalar monotones like c,a, or Fcan break down. In such settings, it is more natural to describe universality in terms of categorical data: fusion categories of anyons, centers of symmetry categories, and higher-form symmetries. Scalar monotones based solely on stress tensors or entanglement entropies may fail to capture the full structural content of the phases and their RG flows. Implications for universal-property RG. The examples above illustrate that, in strongly coupled and cooperative systems: • scalar monotones of the form (2) may fail to exist, or may exist only in restricted domains of theory space; • even when they exist, they may be difficult to compute or interpret in the presence of non-convexities, limit cycles, strong correlations, or long-range entanglement; • the most robust pieces of structure are often categorical: universal properties of defect categories, symmetry centers, or reflective subcategories. This motivates the universal-property RG approach developed in Section III, in which irreversibility is encoded in structural monotonicity of universal constructions and in adjoint curvature, rather than in a single scalar function on coupling space. By lifting RG from the level of coordinates to the level of universal properties, one gains tools that remain meaningful in regimes where classical monotonicity theorems are silent. C. UP-RG Perspective on c-/a-/F-Theorems The scalar monotonicity theorems reviewed in Subsection IV A assert the existence of functions M ( µ )on theory space that decrease along RG flows, cf. Eq. (93) . In the universal-property RG (UP-RG) framework of Section III, we interpret these classical monotones as shadows of deeper structural monotonicity relations of the type (76) , applied to specific universal structures built from the stress tensor and its correlation functions, local RG data, and partition functions on special manifolds. In other words, the c -, a - and F -functions can be seen as particular numerical measures of universal objects Uµ in the categories Cµ , and their monotonicity is a consequence of structural defect inequalities. Zamolodchikov’s c -theorem as structural monotonicity. In two dimensions, Zamolodchikov’s c -theorem states that there exists a c-function c(µ)satisfying dc(µ) dln µ=−Gij({g})βi({g})βj({g})≤0,(102) with Gij a positive-definite metric on coupling space constructed from two-point functions of local operators; see Eq. (94). From the UP-RG viewpoint, we can identify a universal structure at scale µby U(c) µ:= Oi, Tµν, Gij (µ),(103) where: • {Oi}is a basis of local operators appearing in the perturbation of the UV CFT; •Tµν is the conserved stress tensor, considered as the generator of local translations (a universal property in the category of local QFTs); •Gij ( µ )is the Zamolodchikov metric on coupling space, defined by suitably normalised two-point functions of Oiat scale µ. The metric Gij ( µ )is itself determined by a universal mapping property: it is the unique positive-definite bilinear form on the space of couplings such that the RG flow is a gradient flow of c ( µ )in conformal perturbation theory.[96] 37 In terms of the structural defect framework of Subsection III D, define Defect(c) µ(U(c) µ) := Gij({g(µ)})βi({g(µ)})βj({g(µ)})≥0.(104) Then Eq. (102) can be rewritten as dc(µ) dln µ=−Defect(c) µ(U(c) µ)≤0,(105) which is precisely of structural monotonicity form: the derivative of c ( µ )is the negative of a non-negative defect functional measuring how far the coupling vector {gi ( µ ) } is from a fixed point (where all βi = 0). The structure here is the triplet ( Oi, Tµν, Gij )and the universal role of Tµν as the generator of translations and conformal transformations, not merely the numerical value of c(µ). In this sense, the c -function is a particular scalar probe of a more refined universal object: the positive-definite Zamolodchikov metric and its interaction with the RG vector field βi. The a -theorem and local RG as structural data. In four dimensions, the a -theorem can be written in the form da(µ) dln µ=−χij({g})βi({g})βj({g})≤0,(106) with χij a positive-definite matrix constructed from local RG data and Weyl anomaly coefficients, cf. Eq. (96) . Here the relevant universal structure is richer: it involves not only correlation functions of the stress tensor, but also the way the theory couples to curved backgrounds and responds to Weyl transformations. We may define a universal structure U(a) µ:= Tµ µ,{AI}, χij(µ),(107) where: •Tµ µ is the trace of the stress tensor, whose expectation value encodes the Weyl anomaly on curved backgrounds; • {AI} denote background metric and sources for marginal operators, appearing in the local renormalisation group analysis; •χij ( µ )is the positive-definite metric on coupling space obtained from correlators of the trace anomaly and sources, and which governs the gradient-flow structure of a(µ).[97] Define Defect(a) µ(U(a) µ) := χij({g(µ)})βi({g(µ)})βj({g(µ)})≥0.(108) Then Eq. (106) becomes da(µ) dln µ=−Defect(a) µ(U(a) µ)≤0,(109) again of the structural monotonicity form (76) . The quantity Defect(a) µ measures the failure of the theory at scale µ to satisfy the Weyl-invariance universal property characteristic of a conformal fixed point, as quantified by local RG and the anomaly coefficients. The a -function is thus a scalar summary of a deeper structure: the local RG connection and its positive-definite metric on coupling space.[98] The F -theorem and partition functions as universal functors. In three dimensions, the F -theorem states that the finite part of the free energy on S3 , F = −ln |ZS3| , decreases along RG flows, Eq. (98) . From a structural point of view, a (extended) QFT can be viewed as a symmetric monoidal functor Z: Bord3−→ Vect, from the 3-dimensional bordism category to the category of vector spaces. Evaluating Z on the closed manifold S3 yields a distinguished vector Z ( S3 ), whose norm or phase encodes universal information about the theory. 38 We can therefore define a universal structure U(F) µ:= Zµ, S3,(110) where Zµ is the partition-function functor at scale µ (possibly defined via a regularised path integral). The scalar quantity F ( µ ) = −ln |Zµ ( S3 ) | is a numerical evaluation of this structure on a specific object S3 of the bordism category. Renormalisation changes Zµ into Zµ0 via the RG functor Fµ→µ0 on the underlying QFT category, and the F-theorem states that this transformation decreases F(µ). An entropic derivation of the F -theorem views F ( µ )as proportional to a relative entropy between the vacuum state of the CFT and some reference state, and uses strong subadditivity and the data processing inequality.[ 99 ] In the UP-RG language, this is precisely the setting of Subsection III D: the universal structure U(F) µ includes a distinguished “universal” state (the CFT vacuum) and a coarse-graining channel induced by RG, and the defect functional is a relative entropy that decreases under coarse-graining. The number F(µ)is a particular scalar invariant extracted from this structural data. Embedding classical monotones into UP-RG. The examples above illustrate a general pattern: • each classical monotone ( c , a , F ) can be associated with a universal structure Uµ (stress-tensor data, local RG geometry, partition-function functor) in the sense of Definition III.7; •the monotone satisfies an equation of the form dM(µ) dln µ=−Defectµ(Uµ), where Defectµ(Uµ)≥0is a structural defect functional in the sense of Definition III.13; • the scalar function M ( µ )is thus a numerical measure of a deeper structural quantity: a metric on coupling space, an anomaly coefficient, a special value of a functor Z , or a relative entropy between distinguished states. From the UP-RG perspective, the classical c -, a - and F -theorems are special cases of structural monotonicity: they arise when one chooses particularly simple universal structures Uµ and defect functionals Defectµ , and then compresses the resulting information into a single scalar function M ( µ ). The more general framework developed in Section III allows one to: • consider families of structural monotones associated with many different universal structures (limits, centers, Kan extensions, adjunctions) simultaneously; • define monotones even in regimes where conventional scalar functions cease to exist or are difficult to compute (strong coupling, limit cycles, non-invertible symmetries, long-range entanglement); • interpret RG irreversibility in terms of adjoint curvature and structural defects, rather than solely in terms of a gradient flow on a finite-dimensional coupling manifold. In this sense, UP-RG does not replace classical monotonicity theorems but embeds them in a richer categorical and structural framework. The classical scalar monotones are recovered when one evaluates structural defects on specific universal objects (stress tensor, anomaly functional, sphere partition function), whereas UP-RG keeps track of the full web of universal properties that underlie these constructions. D. Structural Monotonicity: Theorems Beyond the c-Theorem Subsection III D introduced structural monotonicity: instead of a single scalar monotone M ( µ )as in Eq. (2), one considers defect functionals Defectµ:Univ(Cµ)−→ R≥0(111) on universal structures Uµat each scale µ, and requires that Defectµ0(Uµ0)≤Defectµ(Uµ)for all µ0>0, µ > µ0,(112) in the spirit of Eq. (76) . This subsection formulates a general structural monotonicity theorem, valid for any class of universal constructions that is suitably preserved or contracted by the RG functors Fµ→µ0 : Cµ→ Cµ0 of Subsection III B. The c -, a - and F -theorems appear as special cases when the universal structures Uµ are built from the stress tensor, local RG geometry, or partition functions on special manifolds (Subsection IV C). 39 Setting: preserved universal constructions. Let ( Cµ, Fµ→µ0, Uµ )be a UP-RG datum as in Definition III.8, so that Uµ0∼ =Φµ→µ0Fµ→µ0(Uµ)(113) for a suitable universal construction Φ µ→µ0 : Univ ( Cµ ) →Univ ( Cµ0 ), as in Eq. (69) . Fix a class K of universal constructions (limits of certain shapes, Kan extensions along certain functors, adjunctions between certain pairs, centers of symmetry categories, etc.) and assume that: •for each µ, the universal structure Uµbelongs to K; • the RG functors Fµ→µ0 preserve K up to canonical comparison morphisms, in a sense made precise below. Concretely, suppose that for each Uµ∈ K there is a universal property characterised by a family of isomorphisms of hom-objects, such as HomCµ(X, Uµ)∼ =Ψµ(X),(114) natural in X , where Ψ µ is some functor built from the rest of the data (a diagram, an inclusion, a functor defining a Kan extension, etc.). After applying Fµ→µ0, one obtains comparison morphisms κµ→µ0(X) : HomCµ0X, Uµ0−→ HomCµ0X, Fµ→µ0(Uµ),(115) which are isomorphisms if Fµ→µ0 strictly preserves the relevant universal construction. In general, these comparison maps need not be invertible, and their “defect from invertibility” is what we wish to control en route from µto µ0. Enriched control: metrics, orders, and norms. Assume that each category Cµ is enriched over a monoidal category V (Subsection II E) that carries an order or metric structure: for each hom-object HomCµ(X, Y )∈ V we have either: •an order relation compatible with composition (e.g. for Va poset or Lawvere metric space); •or a norm/metric k·ksuch that composition is non-expansive. Assume also that the RG functors Fµ→µ0 are V -functors and non-expansive with respect to this structure, in the sense that: •in the ordered case, fgimplies Fµ→µ0(f)Fµ→µ0(g); • in the metric case, kFµ→µ0 ( f ) −Fµ→µ0 ( g ) k≤kf−gk for all morphisms f, g between fixed objects. These are natural assumptions in many physical situations: for instance, when V is a category of normed or Hilbert spaces and morphisms are bounded linear maps, or when V is a category of stochastic kernels with the usual order or total-variation metric. A general structural monotonicity theorem. We are now ready to state a general structural monotonicity theorem, which subsumes the special cases discussed in Subsection IV C. Theorem IV.1 (Structural monotonicity for preserved universal structures) . Let ( Cµ, Fµ→µ0, Uµ )be a UP-RG datum as in Definition III.8, with each Cµ enriched in a monoidal category V with order or metric as above. Let Kbe a class of universal structures in the categories Cµsuch that: 1. For all µ,Uµ∈ K and Uµ0is given by the universal-property flow Uµ0∼ =Φµ→µ0Fµ→µ0(Uµ), µ > µ0>0,(116) as in Eq. (69). 2. For each Uµ∈ K , the universal property is expressed by a family of isomorphisms of hom-objects as in Eq. (114) , and Fµ→µ0 comes equipped with canonical comparison maps κµ→µ0 ( X )as in Eq. (115) . 3. The comparison maps are “contractive” in the enrichment: there exists a functional δµ on homobjects such that for all X, δµ0 κµ→µ0(X)≤δµ idHomCµ(X,Uµ),(117) where δµis orderor norm-compatible with composition and with the V-enrichment. 40 Then there exist non-negative defect functionals Defect(K) µ:K(Cµ)−→ R≥0(118) such that, for all µ>µ0>0, Defect(K) µ0(Uµ0)≤Defect(K) µ(Uµ).(119) In particular, each preserved universal structure Uµ∈ K yields a structural monotone Defect(K) µ ( Uµ )that is non-increasing along the RG flow. Outline of proof. We sketch the main steps; detailed versions follow standard enriched categorical arguments and the philosophy of functorial semantics.[100] 1. From universal properties to canonical comparison maps. For Uµ∈ K , the universal property is encoded in an isomorphism of hom-functors of the form (114) . Applying the RG functor Fµ→µ0 and using functoriality, one obtains induced maps between hom-objects in Cµ0 , and the universal structure Uµ0 is defined by a fresh instance of the same universal property in Cµ0 . The naturality of the universal property yields canonical comparison maps κµ→µ0 ( X )as in Eq. (115) , fitting into commutative squares that express the compatibility of Uµand Uµ0with Fµ→µ0. 2. Defect as enriched size of comparison maps. Using the enrichment in V , we define the defect of Uµat scale µby Defect(K) µ(Uµ) := sup X∈Ob(Cµ) δµ κµ→µ(X),(120) where κµ→µ ( X )is the identity comparison in the ideal case, and δµ measures deviation from being an isomorphism (e.g. a norm of κ−id or an order-theoretic distance). In the perfect preservation case, κµ→µ(X)is the identity, so Defect(K) µ(Uµ)=0. 3. Contractivity under RG. For µ > µ0 , the UP-RG condition (116) and naturality of the comparison maps imply that the composite comparison from Uµ to Uµ0 factors as a product of individual comparison maps, each controlled by the contractivity property (117) . Monoidal compatibility of δµwith composition (or, in the metric case, a triangle inequality) then yields δµ0 κµ0→µ0(X)≤δµ κµ→µ(X),(121) for all X, from which Eq. (119) follows upon taking the supremum. 4. Non-negativity and vanishing at universality. By construction, Defect(K) µ ( Uµ ) ≥ 0, and it vanishes if and only if all comparison maps κµ→µ ( X )achieve their universal ideal (identity or isomorphism), which is equivalent to Fµ→µ0 preserving the universal property of Uµ exactly across scales. Thus, structural monotonicity detects genuine universality: the flow reduces the failure of universality but cannot increase it. Examples and special cases. The theorem encompasses several important examples: • If K consists of limits of diagrams encoding integrated-out modes (Example III.9), then Defect(K) µ ( Uµ ) measures how far the RG functor Fµ→µ0 is from preserving these limits; monotonicity expresses that successive coarse-graining cannot reintroduce lost limiting structure. • If K consists of reflective hulls of effective theories (Example III.10), then Defect(K) µ ( Uµ )measures the failure of Fµ→µ0 to commute with the reflector; monotonicity says that the “distance” between a theory and its best effective approximation does not increase under RG. • If K consists of centers of symmetry categories (Example III.12), then Defect(K) µ ( Uµ )measures deviations from braided commutativity in the defect sector; monotonicity implies that non-invertible symmetries and anyonic structures evolve in a controlled, irreversibility-compatible manner along the flow. In all cases, the key ingredients of the proof are: (i) functoriality of RG, (ii) naturality of universal properties, (iii) canonical comparison maps arising from preservation of universal constructions, and (iv) enrichment by an ordered or metric structure that renders these comparison maps contractive. 41 Beyond scalar monotones. Theorem IV.1 situates classical RG monotonicity theorems as special cases of a broader structural picture. When K is chosen so that Uµ encodes the stress tensor and its two-point structure, and the defect functional is the quadratic form Gijβiβj , one recovers the c - and a -theorems in the form discussed in Subsection IV C. When K encodes partition functions on curved manifolds and entropic quantities, one recovers F -type theorems and relative entropy monotones. More generally, K may encode any class of universal constructions relevant to the physics at hand, yielding families of structural monotones that persist in regimes where scalar monotones cease to be available. The universal-property RG framework thus elevates RG irreversibility from a statement about particular numbers to a statement about the evolution of universal structures. The scalar functions c ( µ ), a ( µ )and F(µ)appear as coarse numerical shadows of this richer structural hierarchy. V. NON-INVERTIBLE SYMMETRIES AND CATEGORICAL SYMMETRY UNDER UP-RG A. Defects and Symmetries We now specialise the universal-property RG framework of Section III to the study of generalized global symmetries—including non-invertible ones—realised by topological defects in quantum field theory. Throughout this subsection, T denotes a fixed d -dimensional QFT (or a suitable universality class), and we review how its topological defects organise into monoidal (and, in general, higher) categories that encode both invertible and non-invertible symmetries. Topological defects in d -dimensional QFT. Let T be a d -dimensional QFT on a spacetime manifold Md . A codimension-1topological defect (or ( d− 1)-dimensional defect) for T is an interface supported on a(d−1)-dimensional submanifold W⊂Msuch that: •the QFT on either side of Wis (locally) the same theory T; • correlation functions are invariant under smooth deformations of W that do not cross operator insertions or other defects. The “topological” property means that the defect can be freely deformed without affecting correlators, as long as it is not dragged across charged operators or other defects. This makes defects natural carriers of symmetry: they implement non-trivial transformations on local operators and states when moved around them.[101, 104] More generally, one can consider a stratification of M by compatible defect worldvolumes of various codimensions: codimension-1defects (interfaces), codimension-2junctions of interfaces, codimension-3 junctions of junctions, and so on. The local topological behaviour of these stratifications is encoded in a hierarchy of (higher) categories of defects.[ 102 , 103 ] For the moment we concentrate on codimension-1 defects and their junction operators, which already form a monoidal category. The defect category CT = End ( T ).Consider all topological codimension-1defects in T that separate Tfrom itself; these are often called endodefects, since they are interfaces from Tto T. We write CT:= End(T)(122) for the associated defect category, defined as follows: •Objects D∈Ob ( CT )are topological codimension-1defects of T from T to T . Physically, these are domain walls across which the QFT is still T , but along which local fields may pick up non-trivial monodromies or transformations. •Morphisms f : D→D0 are junction operators, i.e. local operators inserted at codimension-2loci where defects D and D0 meet. Composition of morphisms corresponds to concatenation of junctions along the common defect worldvolume. •Tensor product is given by stacking defects in parallel, i.e. fusing them along codimension-2 intersections. We denote this monoidal product by ⊗:CT× CT−→ CT,(D1, D2)7−→ D1⊗D2.(123) Diagrammatically, D1⊗D2 is obtained by bringing the two defects infinitesimally close and viewing them as a single composite defect. 48 Summary: structural invariants beyond couplings. The invariants sketched above—center size, adjoint curvature norms, fusion defects, rigidity defects—are all structural: • They depend only on the categorical symmetry data and the universal constructions in CT , not on a particular choice of local couplings. • They are either constant (topological invariants) or non-increasing along RG flows in regimes where the relevant universal structures are preserved or contractive, in the sense of Theorem IV.1. • They remain meaningful in strongly coupled or highly entangled systems where scalar monotones built from stress tensors or entropies are difficult to define or compute. In this way, non-invertible symmetries provide a natural arena in which to see the full power of universal-property RG: irreversibility and universality are expressed not as the monotonic behaviour of a single number, but as the robust behaviour of categorical structures under RG functors. VI. STRONGLY COUPLED AND STRONGLY CORRELATED SYSTEMS A. Why Standard RG Diagnostics Fail Here The classical RG toolbox reviewed in Section IV is at its most powerful in weakly coupled, relativistic, local QFTs with a small number of couplings and a clear separation of scales. In such settings, betafunctions are often perturbatively accessible, flows in coupling space are gradient-like, and scalar monotones such as c ( µ ), a ( µ )or F ( µ )provide sharp measures of irreversibility. By contrast, many of the most interesting physical systems are strongly coupled and strongly correlated: non-Fermi liquids, Kondo and heavy-fermion systems, Mott transitions, frustrated spin systems, Berezinskii–Kosterlitz–Thouless (BKT) transitions, and RG flows with limit cycles. In these systems, standard RG diagnostics often fail or become ambiguous. We now explain why. Non-Fermi liquids and quantum critical metals. In Landau Fermi liquids, low-energy excitations near the Fermi surface are described in terms of quasiparticles with a well-defined dispersion, and RG flows can be organised around marginal and irrelevant interactions. Many strongly correlated metals, however, exhibit non-Fermi liquid behaviour: self-energies with anomalous frequency dependence, absence of sharp quasiparticle peaks, and unconventional power-law scaling in transport and response functions.[ 115 , 116 ] From an RG standpoint: • there may be no small expansion parameter controlling interactions on the Fermi surface (e.g. in the vicinity of a quantum critical point); • fermions coupled to gapless bosonic modes (order-parameter fluctuations, emergent gauge fields) lead to highly non-local effective actions in frequency-momentum space; • beta-functions for patch theories of the Fermi surface are strongly scheme-dependent and can display non-monotonic behaviour, signalling competition between different channels (pairing, density waves, etc.). In such circumstances, it is difficult to construct a single scalar monotone M ( µ )with a clear physical meaning; instead, one often resorts to partial diagnostics, such as anomalous dimensions of specific operators or scaling of transport coefficients, which do not admit a simple gradient-flow interpretation. Kondo effect and overscreening. The Kondo effect provides a paradigmatic example of how strong coupling emerges from an innocuous-looking weak-coupling problem. A magnetic impurity coupled to a Fermi sea via an antiferromagnetic exchange interaction undergoes a flow from a weakly coupled UV fixed point to a strong-coupling IR fixed point describing a Kondo singlet.[ 117 ] In the single-channel case, the IR fixed point is a Fermi liquid, but in multichannel Kondo models, the impurity can be overscreened, producing non-Fermi liquid fixed points with fractional residual entropy and anomalous exponents. Key features from the RG point of view: • the flow is strongly driven towards strong coupling; perturbative beta-functions cease to be reliable near the IR fixed point; • different regularisation schemes and truncations can yield qualitatively different intermediate flows, even if they agree on the existence of the IR fixed point; 49 • the local physics (impurity screening, residual entropy) is encoded in boundary conformal field theories or in emergent fusion rules for impurity degrees of freedom, rather than in a simple bulk scalar monotone. This emphasises that in impurity and boundary problems, universal quantities are often encoded in categorical data (boundary conditions, fusion of boundary fields) that are poorly captured by naive scalar RG functionals. Mott transitions and dynamical mean-field theory. Mott metal–insulator transitions in Hubbard-type models are quintessential examples of strong-correlation phenomena in which the effective interactions are of order of the bandwidth or larger.[ 118 ] Standard weak-coupling RG expansions around a non-interacting Fermi surface are typically inadequate. Instead, non-perturbative approaches such as dynamical mean-field theory (DMFT) reveal a rich structure: • coexistence regions with multiple locally stable solutions (metallic and insulating), indicating a first-order transition ending at a critical end-point; • non-trivial frequency dependence of local self-energies and redistribution of spectral weight between Hubbard bands and quasiparticle peaks; •emergent local moments and Kondo-like screening phenomena in lattice settings. In this context, the effective action is highly non-local in time, and its functional space has many relevant directions. Attempting to capture this with a small number of couplings and a single scalar monotone M ( µ )is inadequate: the interesting universal structure lies in the dynamical and local correlations, not in a simple flow on a finite-dimensional coupling manifold. Strongly correlated spin systems and frustration. Frustrated magnets and quantum spin liquids exhibit long-range entanglement and emergent gauge structures.[ 119 ] The low-energy effective theories often involve: •compact gauge fields coupled to gapless matter; •deconfined spinon excitations and emergent fermions or bosons; •topological degeneracies sensitive to global boundary conditions. Lattice RG procedures (block spins, decimations) can drastically alter the local constraints and emergent gauge structures, making the RG flow highly non-local in the original microscopic variables. Scalar monotones derived from, say, local correlation functions or entanglement entropies can be highly nonuniversal, depending sensitively on how the coarse-graining respects (or fails to respect) the emergent gauge constraints. Moreover, the phase diagrams of such systems are often characterised by competing orders and neardegenerate manifolds of states. As a result, RG flows can wander through extended crossover regions where no single effective description dominates, and any attempt to assign a simple M ( µ )becomes ambiguous. BKT transitions and essential scaling. Berezinskii–Kosterlitz–Thouless (BKT) transitions in twodimensional systems (such as the XY model or superfluid films) are controlled by the unbinding of topological defects (vortices) and exhibit essential, rather than power-law, scaling.[ 120 , 121 ] The RG flows of the vortex fugacity and stiffness can be written as dK−1 d` ∼y2,dy d` ∼(2 −πK)y, with `= ln(µ0/µ)the RG “time”. These flows exhibit: •lines of marginal fixed points corresponding to the low-temperature quasi-ordered phase; •essential singularities in correlation lengths, ξ∼expconst./p|T−Tc|; •extreme sensitivity to initial conditions near the BKT transition. While one can sometimes define an effective monotone along specific slices of parameter space, there is no single scalar function M ( µ )with a simple, universal gradient-flow interpretation valid across the entire BKT transition. The natural universal structures are instead encoded in the topology of the vortex configurations and in the renormalised stiffness, which are more naturally described in terms of dual defects and symmetry categories. 50 Limit cycles and multi-scale flows. As already emphasised in Subsection IV B, RG limit cycles and more complicated recurrent flows present an intrinsic obstruction to scalar monotonicity. In strongly coupled few-body systems with large scattering lengths or in certain lattice models, RG trajectories can spiral around cycles in coupling space, exhibiting discrete scale invariance. Any strictly monotone scalar function M(µ)would be incompatible with such recurrence. Even without strict cycles, multi-scale systems with competing interactions can exhibit flows where couplings oscillate or pass near several approximate fixed points before settling in an IR regime. In these cases: •M ( µ ), if it exists at all, may exhibit long plateaux, sharp crossovers, or non-universal behaviour that obscures its interpretation; • different choices of RG scheme (regulator, truncation, coarse-graining) can produce qualitatively different trajectories, making M(µ)highly scheme-dependent; • universal quantities are often associated with structural features (emergent symmetries, defect categories, centers, dualities) rather than with simple scalar measures. Consequences for UP-RG. The examples above highlight common themes: •Non-monotonic and non-gradient flows: couplings may oscillate, approach multiple fixed points, or follow trajectories incompatible with a global gradient structure. •Strong scheme dependence: different coarse-graining choices can yield markedly different flows, even when they agree on certain qualitative features (existence of an IR phase or universality class). •Non-perturbative structures: essential singularities, emergent gauge fields, and topological sectors are not well captured by finite truncations of local couplings. •Absence of simple scalar monotones: constructing a globally defined, physically meaningful M(µ)is often impossible or artificial. These limitations motivate the universal-property RG approach: instead of tracking flows of scalar couplings and searching for scalar monotones, one tracks flows of universal structures (defect categories, centers, duals, Kan extensions, adjunctions) and their associated structural monotones and adjoint curvatures. This shift from numbers to structures is particularly well-suited for the strongly coupled and strongly correlated systems discussed above. B. UP-RG as a Diagnostic Tool The failures of standard RG diagnostics in strongly coupled and strongly correlated systems, discussed in Subsection VI A, suggest that numerical flows of couplings and scalar monotones are not the right observables in these regimes. The universal-property RG (UP-RG) framework proposes a different strategy: Instead of tracking bare couplings, track the evolution of universal structures and their defect functionals. In this subsection we make this idea concrete and explain how UP-RG can be used as a diagnostic tool for non-perturbative phenomena in strongly correlated systems. Step 1: identify stable universal properties. The first step is to identify, for each class of systems, universal structures that remain meaningful even when perturbation theory breaks down. Examples include: •Operator algebras: algebras of observables Aµ (e.g. C∗ -algebras of local operators or von Neumann algebras associated to regions) at scale µ , together with their inclusion maps Aµ,→ Aµ0 . These algebras encode locality, commutation relations, and superselection sectors.[122] •Defect categories: monoidal categories CTµ of defects at scale µ (Section V), capturing ordinary and non-invertible symmetries, emergent gauge fields, and topological order. •Symmetry fusion rules: fusion coefficients Nk ij ( µ )and associated Grothendieck rings, which may remain well-defined and quantised even in strongly coupled regimes (Subsection V D). 51 •Kan extensions and reflective subcategories: universal constructions such as left Kan extensions of observables along coarse-graining functors or reflective subcategories of “effective” theories (Subsection III C). These structures are robust: they are defined by universal mapping properties and are often insensitive to the particular parametrisation of couplings. Their RG evolution can therefore serve as a diagnostic of deep structural changes (e.g. emergence of new symmetries, topological sectors, or dynamical constraints). Step 2: define structural defect functionals. For each universal structure Uµ we define a defect functional Defectµ(Uµ)as in Subsection III D. In strongly correlated systems, natural choices include: •Operator-algebraic defects: distances (in an appropriate operator-norm or metric) between Aµ and its RG image Fµ→µ0 ( Aµ ), or between actual states and universal reference states (e.g. KMS states) measured by relative entropies. Contractivity of CPTP maps ensures structural monotonicity.[123] •Defect-category defects: invariants such as the change in the number of simple objects in the center Z(CTµ), or norms of adjoint curvatures for tensoring functors LD(Subsection III E). •Fusion-rule defects: quadratic measures of the deviation of Nk ij ( µ )from reference fusion data (UV or IR), as in Eq. (144). •Rigidity defects: counts of objects that lose duals under RG (Eq. (145) ), signalling breakdown or emergence of categorical symmetry structures. By construction, these defect functionals are non-negative and vanish precisely when the relevant universal property is satisfied exactly at scale µ. Step 3: use adjoint curvature to detect strong-coupling anomalies. Adjoint curvature, introduced in Subsections II D and III E, provides a way to detect when strong-coupling effects introduce irreversibility or anomalies at the level of universal structures. In strongly correlated systems: •RG functors Fµ→µ0may fail to be part of exact adjunctions between UV and IR descriptions (e.g. between fine-grained and coarse-grained operator algebras); • domain walls between different scales or phases may induce functors on symmetry categories CTµ with non-trivial loop curvature (Eq. (92)), signalling path dependence of RG in scheme space. Non-zero adjoint curvature RFµ→µ0 can then be interpreted as a structural anomaly of the RG flow: a precise measure of the failure to reconstruct universal structures when flowing down and then back up in scale. In practical terms, such curvature can be bounded by comparing correlation functions, operator norms, or categorical invariants along different RG paths. Step 4: enrichment as a probe of correlation strength. Enriched categories enter naturally when we wish to capture not only the existence of morphisms (maps, defects, junctions) but also their size, probability, or correlation strength. For strongly correlated systems, it is natural to enrich: • the categories Cµ of theories and observables in Hilbert spaces or operator algebras (Hilbert-enriched or C∗-enriched categories); • the defect categories CTµ in fusion categories with additional metric or probability structure on hom-spaces (e.g. norms of junction operators). In such an enriched setting, RG functors Fµ→µ0 become non-expansive maps in the enrichment (e.g. contractions in operator norm or monotone maps in a partial order). This directly feeds into the hypotheses of Theorem IV.1: defect functionals can be defined by assigning to each universal structure Uµ a size δµ in the enrichment (norm of a comparison morphism, relative entropy, etc.), and RG contractivity guarantees structural monotonicity of the form Defectµ0(Uµ0)≤Defectµ(Uµ). Physically, this means that the “strength” of certain correlations (as measured by norms, entropies, or categorical sizes) can be made to obey monotone behaviour even when individual couplings and correlation functions exhibit wild, non-monotonic dependence on scale. 52 UP-RG diagnostics in practice. Putting these steps together, UP-RG suggests a practical diagnostic strategy for strongly correlated systems: 1. Identify the relevant universal structures (operator algebras, defect categories, symmetry fusion rules, Kan extensions). 2. Construct an RG functor Fµ→µ0 acting on these structures (e.g. via coarse-graining maps, partial trace channels, or domain walls). 3. Define enriched defect functionals and adjoint curvatures to measure deviations from ideal universal behaviour. 4. Study their behaviour along numerical or analytical RG flows: •invariance of certain defects diagnoses preserved symmetries or topological sectors; • monotone decrease of others signals structural irreversibility, even when scalar couplings oscillate; •non-zero loop curvature or Kan obstructions reveal anomalies or path dependence in RG. In this way, UP-RG provides a systematic framework to extract meaningful, structural information from strongly coupled and strongly correlated systems where conventional scalar RG diagnostics are insufficient. C. New RG Invariants in Strong Coupling In strongly coupled and strongly correlated systems, the most robust indicators of universality are often not scalar functions of couplings, but structural invariants: quantities built from universal properties of symmetry categories, operator algebras, or tensor-network fixed points that remain constant (or change in controlled ways) under RG flows and across different regulators and truncation schemes. In the UP-RG framework, such invariants arise naturally as scale-independent features of universal structures Uµ and of the renormalisation category (Subsection III A). In this subsection we describe three broad families of such invariants: •invariants extracted from tensor-network coarse-graining, •center-based invariants detecting topological order, •curvature-based invariants measuring the obstruction to merging different truncation schemes. Definition VI.1 (Strong RG Invariant) . Let {Cµ, Fµ→µ0} be a UP–RG datum in the sense of Definition III.8, and let Uµ∈Univ ( Cµ )be a universal structure at scale µ . We say that a family of universal structures {Iµ}is a strong RG invariant if the following conditions hold: 1. RG preservation: For every pair µ > µ0the RG functor induces an isomorphism Fµ→µ0(Iµ)∼ =Iµ0, i.e. the invariant is preserved exactly under coarse–graining. 2. Scheme independence: For any renormalisation schemes S1, S2 and any loop γ in the renormalisation category Rbased at scale µ , the induced scheme-loop functor Kγ (as in Eq. (146) ) satisfies Kγ(Iµ)∼ =Iµ, so that Iµis unaffected by regulator or truncation choices. 3. Structural stability: For any universal-property flow Uµ7→ Uµ0 induced by Eq. (73) , the invariant remains unchanged: Iµ0∼ =Iµ. Thus the invariant survives even when other universal structures evolve non-trivially. Strong RG invariants therefore represent universal structures that are preserved under all RG functors, under all scheme changes, and under all canonical universal reconstructions. They generalise classical invariants (e.g. topological order, modular data, Drinfeld centers, or tensor-network fixed-point classes) and remain well-defined even in strongly coupled or non-perturbative regimes where scalar monotonicity fails. 53 Tensor-network invariants as categorical fixed-point data. Tensor-network renormalisation methods (MERA, TNR, PEPS coarse-graining, etc.) are concrete realisations of RG functors on categories of states or channels: objects are tensors (or tensor networks) and morphisms are local contractions or isometric embeddings.[ 124 , 125 ] A scale-invariant fixed point of such a scheme is a tensor T∗ (or a finite set of tensors) that is invariant under coarse-graining up to gauge transformations: Fµ→µ0(T∗)∼ =T∗, where Fµ→µ0 is the tensor-network RG functor and ∼ = denotes equivalence under local basis changes and internal symmetries. Formally, one can organise tensor-network data into a monoidal category T: •objects are tensors (e.g. MPS, PEPS, MERA layers) with given boundary legs; •morphisms are local tensor contractions compatible with the physical and virtual indices; •the monoidal product is given by stacking or concatenation of networks. Gauge transformations act as automorphisms in T ; fixed points are then universal objects determined only up to such automorphisms. The isomorphism class [ T∗ ]of a scale-invariant fixed-point tensor is therefore a categorical RG invariant: it is preserved under further coarse-graining and under changes of local gauge. In symmetry-protected and topologically ordered phases, [ T∗ ]captures non-trivial structural information: • For symmetry-protected topological (SPT) phases, the projective representations carried by the virtual legs of an MPS or PEPS tensor define cohomology classes (e.g. in H2 ( G, U (1)) for 1D SPTs) that remain invariant under symmetric RG flows.[126, 127] • For topologically ordered phases, fixed-point PEPS or MERA tensors encode the fusion and braiding structure of emergent anyons; these structures are stable under local perturbations and tensor-network coarse-graining, and thus provide RG invariants of the phase. Within UP-RG, these data appear as universal structures Uµ in a tensor-network category, whose universal-property flow stabilises in a given phase. The resulting invariants are categorical (projective classes, fusion rules) rather than purely numerical. Centers and topological order as strong-coupling invariants. As discussed in Subsection V D, the Drinfeld center Z ( CT )of the defect category CT plays a central role in characterising topological order: non-trivial centers encode long-range entangled degrees of freedom and emergent anyonic excitations. Given an RG flow between gapped phases described by theories TUV →TIR , the induced functor on centers, Z(FR) : Z(CTUV )−→ Z(CTIR ), is expected to be an equivalence if the topological order is preserved, and to change only when the system passes through a phase transition. Thus, in a given topological phase, the following are RG invariants: •the braided equivalence class of the center Z(CT), •modular data (e.g. Sand Tmatrices) extracted from Z(CT), •total quantum dimension and fusion rules of the emergent anyons. These invariants are intrinsically strong-coupling objects: they survive far beyond the domain of perturbation theory, remain stable under coarse-graining, and are insensitive to many microscopic details. In the UP-RG picture they are special cases of structural invariants associated with universal structures Uµ built from centers and fusion categories. 54 Scheme curvature and compatibility of truncation schemes. A distinctive feature of UP-RG is that it treats renormalisation schemes and truncations themselves as objects and morphisms in a renormalisation category R (Subsection III A). Different regulators or truncation schemes S(1) , S(2) for the same microscopic theory give rise to different RG functors F(1) µ→µ0:C(1) µ→ C(1) µ0, F(2) µ→µ0:C(2) µ→ C(2) µ0, together with scheme-change functors H12 µ : C(1) µ→ C(2) µ and H21 µ : C(2) µ→ C(1) µ . Traversing a loop γ in scheme/scale space as in Eq. (91) yields a composite functor Kγ:C(1) µ−→ C(1) µ, whose deviation from the identity measures the obstruction to merging the two schemes into a single, scheme-independent RG description. Following Subsection III E, we define the scheme curvature of the loop γby Kγ:= Kγ−idC(1) µ,(146) which may be evaluated on universal structures Uµ (e.g. centers, fusion rules, or tensor-network fixed points). A natural RG invariant associated with scheme curvature is then Cscheme(µ) := sup γ∈Loopsµ Kγ(Uµ) ,(147) where the supremum runs over loops in R at scale µ , and k · k is an appropriate norm or size function in the enrichment (e.g. operator norm or categorical dimension). The quantity Cscheme(µ)is: • universal in the sense that it depends only on the action of scheme loops on universal structures Uµ ; • scheme-robust: if different regulators and truncations are genuinely equivalent in the IR, then Cscheme(µ)tends to zero as µ→0; • diagnostic of universality: a non-zero limit of Cscheme ( µ )in the IR signals residual scheme dependence and may indicate either an incomplete choice of universal structure or the presence of subtle anomalies. In tensor-network language, Cscheme ( µ )measures how far two different coarse-graining circuits (e.g. different MERA or TNR implementations) fail to produce equivalent fixed-point tensors and scaling data; in holographic terms, it is related to how different discretizations approximate the same emergent geometry.[124, 125, 128] Summary. In strongly coupled regimes, UP-RG promotes RG invariants from scalars to categorical structures and their derived quantities: •fixed-point classes [T∗]of tensor networks, •centers and modular data Z(CT)of symmetry categories, •scheme-curvature invariants such as Cscheme(µ). These invariants are constant (or change in controlled ways) along RG flows within a phase, insensitive to many microscopic details and regulator choices, and capable of detecting emergent topological order and structural properties of fixed points. They illustrate how universal-property RG extends the notion of RG invariants beyond the classical c -, a -, and F -type scalars, providing a richer language to diagnose universality at strong coupling. D. Implications for Phase Diagrams and Critical Phenomena In the traditional picture, phase diagrams are drawn in a coupling space M , and universality classes are identified with RG fixed points or orbits in this space: each point g∈ M defines a microscopic Hamiltonian or action, and RG flows partition Minto basins of attraction of fixed points, which define 55 universality classes of critical phenomena.[ 129 ] This viewpoint is powerful in weakly coupled regimes, but in strongly coupled and strongly correlated systems the geometry of M becomes complicated and scheme-dependent, and fixed points can be difficult to identify directly. The UP-RG framework offers a complementary perspective: instead of focusing on points in coupling space, one focuses on universal structures and their equivalence classes. Phases and universality classes then become, in a precise sense, homotopy classes of universal structures rather than isolated points in M. Phase diagrams from structural data. Let M be a parameter space of microscopic models (couplings, disorder, lattice geometry, etc.). To each point g∈ M we associate a renormalisation datum S(g)=(Tg, Rg, τg)∈Ob(R), where Ris the renormalisation category of Definition III.2. Fix a structural RG invariant I:R −→ D, as in Definition VI.1, where D is a category of universal structures (e.g. fusion categories, modular tensor categories, tensor-network phases, operator algebras). The composite map M −→ R I −−→ D assigns to each ga universal structure I(S(g)), well-defined up to isomorphism in D. Two points g, g0∈ M lie in the same structural phase if there exists a continuous path in M , together with compatible RG flows and scheme changes in R , such that all induced morphisms map to isomorphisms in Dunder I. Equivalently: I(S(g)) ∼ =I(S(g0)) in D. Thus, phases are identified with connected components (or, more generally, homotopy classes) in the image of I, not with regions of constant numerical couplings. Phase transitions then correspond to loci in M where the isomorphism class of I (S( g )) changes: categorical centers jump, tensor-network fixed points change class, duals are lost or gained, or loop curvatures become non-trivial. In particular: • In conventional symmetry-breaking transitions, the pattern of symmetry breaking encoded in CT and its center changes; • In topological and symmetry-protected topological (SPT) transitions, the modular data or SPT invariants extracted from I(S(g)) change discontinuously.[130–132] Universality classes as homotopy classes of universal structures. In the usual RG language, a universality class is the set of all microscopic models whose flows end at the same fixed point, modulo irrelevant directions. In UP-RG, the focus shifts from the fixed point in coupling space to the universal structure at the fixed point. Let U be the (2-)category whose objects are universal structures (e.g. defect categories with centers and duals, tensor-network fixed points), whose 1-morphisms are structure-preserving functors, and whose 2-morphisms are natural isomorphisms. The structural RG invariant I can be viewed as landing in U up to equivalence. Two models are in the same universality class if the corresponding universal structures are equivalent in the homotopy category Ho(U): [I(S(g))] = [I(S(g0))] in Ho(U). In this sense: universality classes are homotopy classes of universal structures, and phase diagrams can be thought of as partitions of Minduced by the map into Ho(U). This perspective has several advantages: •It naturally accommodates situations where different microscopic models flow to fixed points that are not literally the same point in coupling space but have equivalent universal structures (e.g. dual descriptions, emergent symmetries, deconfined quantum critical points).[133] • It provides a clear criterion for when two apparently different critical points belong to the same universality class: their universal structures must be equivalent in Ho ( U ), even if their Lagrangians or Hamiltonians look different. 56 Jumps in categorical data and criticality. Critical phenomena are thus characterised by qualitative changes in universal structures. For instance: • At a conventional second-order phase transition, the defect category CT may gain or lose objects corresponding to domain walls between distinct ordered phases, or its fusion rules may change continuously but its center may change discretely. • At a topological phase transition, Z ( CT )may change from a trivial center (no topological order) to a non-trivial modular tensor category (non-abelian anyons), with associated jumps in the total quantum dimension or modular S-matrix. • At a deconfined quantum critical point, the universal structure may exhibit an emergent higher symmetry or anomaly that is absent on either side of the transition, and which can be detected via RG curvature or defect-category invariants.[133] In all these cases, the critical point is a locus where one or more categorical invariants change, often in a way that cannot be fully captured by local order parameters or simple correlation-length divergences. New classification schemes and hidden universality. Because structural invariants are insensitive to many microscopic details and to regulator choices, they provide the basis for new classification schemes: •Classification of phases: Phases can be classified by the equivalence classes of their universal structures, such as defect categories with centers and duals, modular data, and tensor-network fixed points. This recovers and extends existing classifications of topological and SPT phases (by K-theory, cohomology, or modular categories)[ 130 – 132 ] while placing them in a unified categorical RG framework. •Hidden universality in strong coupling: Strongly coupled models that appear unrelated at the level of couplings may share the same universal structure in U , revealing hidden universality. For instance, different lattice realisations of deconfined criticality or non-Fermi liquids can flow to fixed points with the same emergent symmetry category or tensor-network invariant, even if their Lagrangian descriptions differ.[129, 133] •Refined notions of multicriticality: Multicritical points can be characterised by intersections of strata in Ho ( U ), where multiple universal structures collide or where symmetry categories enhance. This can distinguish qualitatively different multicritical behaviours that would look similar in a purely numerical coupling-space description. In summary, UP-RG lifts phase diagrams and universality classes from the level of couplings to the level of universal structures and their homotopy classes. Phase transitions correspond to jumps in categorical data; fixed points and universality classes are structural objects in U ; and strongly coupled critical phenomena can be analysed and classified using robust categorical invariants, even when conventional scalar RG diagnostics fail. VII. MOLECULAR AND BIOMOLECULAR SYSTEMS: FOLDING, COOPERATION, AND TELEOLOGY A. Multi-Scale Cooperative Structure in Biomolecules Biomolecules such as proteins, RNA, and large molecular complexes provide a canonical example of systems with rich multi-scale structure and strong cooperativity. Their dynamics and function emerge from the interplay between local chemical interactions (at the level of individual residues or nucleotides) and global collective organisation (at the level of domains, folds, and assemblies). In this subsection we review these features, with an eye towards their reinterpretation in the universal-property RG (UP-RG) framework. Hierarchy of structural scales in proteins. A protein is specified by a primary structure: a sequence s= (a1, a2, . . . , aN), where each ai is an amino acid drawn from a finite alphabet. This sequence determines, via local covalent connectivity and side-chain chemistry, a hierarchy of higher-level structures:[134] 57 •Secondary structure: local motifs such as α -helices and β -strands, stabilised by backbone hydrogen bonds and local packing constraints. •Tertiary structure: the global three-dimensional fold of a single polypeptide chain, arising from long-range hydrophobic interactions, electrostatics, and steric packing. •Quaternary structure: assemblies of multiple chains (subunits) into complexes, often with emergent binding interfaces and allosteric communication pathways. This hierarchy can be schematically expressed as a sequence of (highly nonlinear) maps s7−→ Sec(s)7−→ Ter(s)7−→ Qua(s)7−→ Func(s),(148) where Sec ( s )denotes an arrangement of local motifs, Ter ( s )a folded tertiary structure, Qua ( s )a complex of chains, and Func ( s )the functional ensemble (binding, catalysis, signalling). Importantly, each arrow in (148) is not a simple local coarse-graining but a global, cooperative reorganisation of many degrees of freedom. Energy landscapes and folding funnels. The modern statistical-mechanical view of protein folding describes the process in terms of a high-dimensional free-energy landscape F ( x )over conformational coordinates x (backbone dihedrals, side-chain rotamers, solvent coordinates, etc.).[ 135 , 136 ] For many natural proteins, this landscape is thought to have a funnel-like structure: • At high energies, many unfolded and partially folded conformations coexist, separated by modest barriers. • As the free energy decreases, the number of accessible conformations narrows, guiding the system towards the native basin of attraction. • Residual frustration (competing interactions) creates local minima and kinetic traps, but an overall bias towards the native state remains. Mathematically, one can think of a coarse-grained order parameter Q (e.g. fraction of native contacts) and an effective free energy F ( Q )obtained by integrating out microscopic details. Folding then corresponds to a biased diffusion in Q on a rugged but funnel-shaped free-energy profile, with transition rates between basins determined by barrier heights and entropic factors.[136, 137] Cooperativity across scales. A hallmark of biomolecular systems is cooperativity: transitions in conformation or binding that involve the coordinated rearrangement of many residues across multiple length scales. Examples include:[138] • highly cooperative two-state folding, where a small change in temperature or denaturant leads to an abrupt transition between folded and unfolded ensembles, with many residues participating; • allosteric transitions, in which ligand binding at one site induces conformational changes at distant sites, mediated by networks of coupled interactions; • assembly/disassembly of oligomeric complexes, where multiple subunits and interfaces rearrange collectively. In physical terms, these processes are controlled by collective coordinates that are not easily expressible as a small number of local couplings or order parameters. Instead, they involve: •long-range correlations in fluctuation patterns, •non-additivity of interaction free energies (cooperative binding), •strong coupling between local and global modes (e.g. hinge motions affecting local active sites). Failure of simple one-parameter RG descriptions. Standard RG intuition suggests that near a continuous phase transition, the behaviour of a system can often be captured in terms of a small number of relevant couplings and a single scalar monotone controlling flow towards an IR fixed point. In many biomolecular systems, however, such a description is inadequate: •No single control parameter: folding and functional transitions depend simultaneously on many environmental and internal parameters (temperature, denaturant, pH, crowding, post-translational modifications). There is no unique “RG time” or single coupling whose flow summarises the physics. 64 For teleological dynamics, one can consider the pair F:= Tt, G := Et, where Tt is the forward dynamics and Et is a coarse-grained or reconstruction functor that attempts to “pull back” states toward the universal object U∗ (e.g. via relaxation, error correction, or evolutionary selection). An ideal teleological adjunction would satisfy triangle identities aligning Tt and Et with the universal maps uX. The teleological curvature is then the endomorphism Rtel Tt:= ΘTt−idTt,(158) evaluated on relevant universal structures (e.g. native basins, functional modules). A norm or spectral radius of Rtel Tt can be interpreted as a measure of deviation from teleological alignment: zero curvature corresponds to perfect adjoint alignment, non-zero curvature to path-dependence and irreversibility. Non-Markovian memory and evolutionary constraints. The curvature (158) encodes two important aspects of biological dynamics: •Non-Markovian memory. If the effective dynamics on a coarse-grained category (e.g. metastable states or functional modules) were exactly Markovian and compatible with the universal property of U∗, then the loop XTt −→ Tt(X)Et −−→ X would be (approximately) the identity on universal structures, and Rtel Tt would vanish there. Deviations from this behaviour—e.g. memory effects where previous history influences the outcome of Et◦Tt—manifest as non-zero curvature. •Evolutionary constraints. Over evolutionary timescales, the constraint functor C and the universal object U∗ themselves evolve. Not all universal structures are reachable by mutation and selection; the genotype-to-phenotype map is highly constrained.[ 149 , 150 ] In categorical terms, this means the adjunction between genotype space and phenotype space is only approximate, and its curvature encodes evolutionary path-dependence and historical contingency. The teleological curvature then measures how far the realised evolutionary trajectories are from idealised universalproperty optima. In both cases, adjoint curvature provides a quantitative object that captures how far real dynamics is from a hypothetical, perfectly teleological dynamics in which universal structures are instantaneously and reversibly realised. Summary: teleology without future causes. The universal-property perspective thus offers a way to reconcile teleological language with mechanistic, time-forward dynamics:[147, 148] • “Goals” correspond to universal objects selected by constraint functors and universal properties (limits, colimits, adjunctions). • Teleological dynamics is dynamics whose structural defects relative to these universals (the teleological defects TeleDefectt) decrease over time, as guaranteed by structural monotonicity. • Deviations from perfect teleology—non-Markovian memory, evolutionary limitations, noisy environments—are encoded in adjoint curvature, which measures the failure of forward and backward maps to form perfect adjunctions on universal structures. In this way, the UP-RG framework allows one to speak rigorously about teleology-like behaviour in biomolecular and biological systems: it is not an extra physical principle, but a manifestation of universal-property flow in enriched categorical dynamics. VIII. FURTHER APPLICATIONS: EVOLUTION, COGNITIVE SYSTEMS, AND AI/ML A. Evolution as Universal-Property Flow Biological evolution is often pictured as a stochastic walk in a space of genotypes, guided by mutation, recombination, drift, and selection acting through phenotypes in specific environments.[ 151 , 152 ] From 65 the UP-RG viewpoint, evolution can be reformulated as a flow in a space of universal structures— developmental channels, regulatory networks, and phenotypic attractors—that solve joint genotype– environment constraint problems. In this subsection we outline a categorical formalisation of this idea. Categories of genotypes, phenotypes, and environments. We introduce three basic categories: •Gen : the category of genotypes. Objects are genotypes (e.g. DNA sequences, regulatory-network architectures); morphisms are mutations, recombinations, and other heritable transformations. •Phen : the category of phenotypes. Objects are phenotypic states (morphologies, physiological states, behaviours); morphisms are developmental or functional transformations between phenotypes. •Env : the category of environments. Objects are environment states (ecological conditions, resource distributions); morphisms represent environmental changes or perturbations. Development maps genotypes-in-environments to phenotypes. Categorically, this is a bifunctor D:Gen ×Env −→ Phen,(g, e)7−→ D(g, e),(159) which is functorial in each argument. For fixed g∈Gen, the restriction Dg:= D(g, −) : Env −→ Phen is the developmental channel of g, mapping environments to phenotypic outcomes. For fixed e∈Env, De:= D(−, e) : Gen −→ Phen is the genotype–phenotype map at environment e.[155] To incorporate viability and fitness, we introduce a constraint functor C:Phen −→ Vfit, where Vfit is an ordered or metric-enriched category (e.g. R≥0 with order by increasing fitness cost, or a space of viability scores). The composite C◦D:Gen ×Env −→ Vfit encodes how genotype–environment pairs satisfy fitness and viability constraints. Universal phenotypes as attractors in phenotype space. For a fixed pair ( g, e ), consider the full subcategory Reach ( g, e ) ⊂Phen of phenotypes reachable from the developmental dynamics of genotype g in environment e (e.g. via stochastic dynamics, reaction–diffusion processes, regulatory network dynamics). We assume that Reach(g, e)is enriched in Vfit via C. Astable phenotypic attractor for ( g, e )is then modelled as a universal object P∗ ( g, e ) ∈Reach ( g, e ) solving the joint development–fitness constraint problem. One convenient way to express this is as a terminal object for a diagram of constraints: P∗(g, e)∼ =limC|Reach(g,e),(160) meaning that P∗ ( g, e )minimises (or suitably extremises) the fitness cost among reachable phenotypes and is universal with respect to maps from other phenotypes into it. In dynamical terms, P∗ ( g, e ) corresponds to an attractor of the developmental dynamics whose basin of attraction dominates the long-time behaviour.[153, 154] More generally, rather than choosing a single phenotype, we may consider a developmental channel as a universal structure: Ug:= Dg:Env −→ Phen,(161) together with a natural transformation selecting attractors P∗ ( g, − ) : Env →Phen . This Ug encodes how g projects the space of environments into a subset of phenotypes; it is thus a universal structure in the sense of Definition III.7. 66 Evolutionary dynamics as flow in the space of universal structures. Standard population-genetic models describe the dynamics of genotype frequencies pt ( g )under mutation, recombination, and selection. In discrete time, pt+1(g) = Fpt, D, C, Et(g), where Et summarises the environment at time t and F is a replicator–mutator update functional.[ 151 , 152 ] From the UP-RG standpoint, we instead focus on how the associated universal structures Ugevolve. Define the evolution–development functor I:Gen −→ Uevo, g 7−→ Ug,(162) where Uevo is a category whose objects are developmental channels and regulatory networks (functors Env →Phen , possibly with additional structure), and whose morphisms are natural transformations preserving developmental and regulatory structure. Mutations g→g0 induce morphisms Ug→Ug0 in Uevo. Over evolutionary time, population-genetic dynamics and environmental change together induce a stochastic flow on Uevo: Ug7−→ U0 g:= Eg0∼pt+1 I(g0),(163) where the expectation symbolically represents the population-weighted mixture of developmental channels present at time t + 1. In a coarse-grained limit, Eq. (163) becomes a deterministic flow in the space of universal structures, analogous to the UP-RG flows Uµ→Uµ0in Section III. Thus, rather than viewing evolution solely as movement in a genotype space, UP-RG emphasises the induced movement in the space of developmental channels, regulatory architectures, and phenotypic attractors—the universal structures that mediate genotype–environment interactions.[155] Structural defect functionals and evolutionary canalisation. Within Uevo , we can define structural defect functionals that measure how well a developmental channel Ug satisfies viability and robustness constraints. For example: • aviability defect Defectvia ( Ug )measuring the fraction of environments in which Ug fails to produce any viable attractor P∗(g, e); • acanalisation defect Defectcan ( Ug )measuring the variability of P∗ ( g, e )under perturbations of e (small canalisation defect corresponds to Waddington’s notion of canalised development[153]); • acomplexity defect Defectcomp ( Ug )penalising unnecessarily complicated or fragile regulatory networks. Assuming selection favours genotypes with lower defects, Theorem IV.1 suggests that, under suitable non-expansivity assumptions on mutation and selection operators, these defect functionals can behave as structural evolutionary monotones: EDefect(Ugt+1 )≤EDefect(Ugt), even when genotypes themselves wander non-monotonically and the fitness landscape is rugged. In this way, evolution can be viewed as a universal-property flow that tends to reduce defects in developmental channels and regulatory architectures, pushing populations toward universal structures that solve increasingly stringent joint genotype–environment constraints. The resulting picture integrates classical population genetics, the genotype–phenotype map, and evo–devo notions such as canalisation and developmental bias into a single categorical RG framework.[153–155] B. Multi-Scale Representations in AI Modern deep neural networks—including convolutional networks, transformers and protein-structure predictors such as AlphaFold—learn a hierarchy of internal representations distributed across layers.[ 156 , 157 ] These representations can be viewed as multi-scale features: early layers capture local patterns, intermediate layers capture motifs or parts, and deeper layers capture global structure and task-specific abstractions. In the universal-property RG framework, these learned representations appear as objects and functors in suitable categories, and training can be interpreted as a universal-property flow toward structures that solve information-theoretic and symmetry constraints. 67 Layers as functors between representation categories. Let Data denote a category of inputs and labels. A convenient model is to take: •objects: finite datasets S={(xi, yi)}sampled from a distribution on X × Y; • morphisms: data augmentations or preprocessing maps α : S→S0 (e.g. geometric transformations, noise injections) that preserve label semantics. For a given neural architecture with layers ` = 1 , . . . , L , denote by R` the representation space at layer ` (typically a Euclidean or Hilbert space), with R0≃ X the input space and RL the output space (e.g. logits). A trained network with parameters θdefines, for each layer `, a map f`,θ :R`−1−→ R`, and by evaluation on a dataset San induced map on finite sets f`,θ(S) := {f`,θ(xi)|(xi, yi)∈S}. This yields a functor F`,θ :Data −→ Rep`, S 7−→ f`,θ(S),(164) where Rep` is a category of representations at layer ` (objects: finite subsets of R` with labels; morphisms: induced maps under data augmentation or further processing). Composition of layers corresponds to composition of functors: FL,θ ◦ · · · ◦ F1,θ :Data −→ RepL. This categorical description emphasises that each layer realises a structured map from data to representations, respecting certain invariances and equivariances under data augmentations.[ 158 ] Training then adjusts the functors F`,θ to satisfy additional constraints, such as predictive sufficiency and invariance to nuisance factors. Universal properties: minimal sufficient and invariant representations. In supervised learning with input random variable Xand label Y, a representation Z(at some layer) is sufficient for Yif I(Z;Y) = I(X;Y), and minimal sufficient if, among all sufficient Z , it minimises I ( Z ; X ).[ 159 ] In classical statistics, sufficient statistics carry all information about the parameter or label; minimal sufficient statistics are those from which all other sufficient statistics factor through a measurable map.[ 160 ] Categorically, one can formalise this as follows. Consider the category Stat whose objects are Markov kernels T : X→Z (stochastic maps) that are sufficient for Y , and whose morphisms are post-processing maps h : Z→Z0 such that T0 = h◦T is also sufficient. Then a minimal sufficient statistic T∗:X→Z∗is a terminal object in Stat: ∀T:X→Zsufficient,∃!hT:Z−→ Z∗with T∗=hT◦T. (165) This is precisely a universal property: Z∗ is determined up to unique isomorphism by the requirement that all sufficient representations factor uniquely through it. In deep networks, intermediate-layer representations often approximate such minimal sufficient statistics for the labels with respect to the training distribution, especially when regularised (e.g. via noise, weight decay, bottlenecks).[ 159 , 161 ] From the UP-RG viewpoint, the universal structure associated with a layer is the isomorphism class of its minimal sufficient representation functor, viewed as an object of a category of stochastic maps. Similarly, invariances and equivariances can be expressed by universal properties. Let G be a group acting on X (e.g. translations, rotations). A G -equivariant representation is a functor F : Data →Rep equipped with a natural action of Gsuch that F(g·S)∼ =g· F(S), g ∈G. Group-equivariant networks[ 158 ] can be seen as constructing functors that are initial or terminal in appropriate categories of G-equivariant maps, thus realising universal properties of invariance. 68 UP-RG for AI: training as universal-property flow. Training by gradient descent or related algorithms induces a time-dependent family of parameters θt and hence a family of layer functors F`,θt . For each layer `and time twe can associate a universal structure U`,t capturing, for example: •the minimal sufficient representation realised by that layer, •the invariant or equivariant structure with respect to a group of augmentations, • internal categorical constraints (e.g. factorisations, commutative diagrams linking different tasks or views of the data). We thus obtain a universal-property flow U`,0 Tt −−→ U`,t, t ≥0,(166) for each layer ` , analogous to the UP-RG flows Uµ→Uµ0 in Section III. The “time” parameter is training time (or optimisation steps), and the flow takes place in a space of universal structures (sufficient statistics, equivariant representations), not merely in parameter space. To quantify progress, we can define layer-wise defect functionals measuring violation of the desired universal properties. For example: • asufficiency defect Defectsuff `,t measuring the gap I ( X ; Y ) −I ( Z`,t ; Y )between the full information about Yand that carried by the layer representation Z`,t; • aminimality defect Defectmin `,t := I ( Z`,t ; X ) −I ( Z∗ ` ; X )relative to a reference minimal sufficient statistic Z∗ `; • an invariance defect Defectinv `,t measuring deviation from G -equivariance (e.g. averaged norm of F`,θt(g·x)−g· F`,θt(x)over g∈Gand xin the data): Defectinv `,t := Ex,gh F`,θt(g·x)−g· F`,θt(x)  2i.(167) Under suitable assumptions (e.g. appropriate loss functions, regularisation, and optimisation dynamics), these defects tend to decrease during training, in line with Theorem IV.1: training drives representations towards universal structures that are more sufficient, more minimal, and more invariant.[159, 161] Potential invariants and structural understanding of generalisation. Some aspects of the learned representation hierarchy appear remarkably stable: for instance, early-layer filters in vision models and mid-layer structures in language models generalise across tasks and datasets, and fine-tuning often changes only a small subspace of the representation manifold.[ 161 ] In UP-RG terms, this suggests the presence of structural invariants: • equivalence classes of layer functors F`,θ under reparametrisations and symmetries that remain unchanged during fine-tuning; •categorical relations (e.g. commutative diagrams linking multiple tasks, modalities, or views) that persist across training regimes; • universal sufficient statistics for broad families of tasks (e.g. language modelling, protein-structure prediction) that act as attractors in the space of representations. Identifying and formalising these invariants within the UP-RG framework could yield a structural understanding of why deep networks generalise: rather than memorising training data, they converge to universal-property attractors (minimal sufficient, invariant representations) that are largely determined by task symmetries and information constraints, not by the specific optimiser trajectory. This connects representation learning in AI/ML to the broader theme of this work: dynamics and optimisation as flows in the space of universal structures. 69 IX. TELEOLOGY, NONLINEARITY, AND ADJOINT CURVATURE A. From Adjoint Curvature to Nonlinear Dynamics In Sections II D and III E we introduced the notion of adjoint curvature: for an approximate adjunction FaG between functors F : C → D and G : D → C , with unit η : idC⇒GF and counit  : FG ⇒idD , we defined the endomorphism ΘF:= (F)◦(Fη) : F⇒F, RF:= ΘF−idF,(168) and interpreted RF as a measure of the failure of the triangle identities, i.e. as an obstruction to perfect adjunction. In this subsection we explain how such broken adjunctions naturally generate nonlinear corrections to effective evolution equations, and how this connects to nonlinear quantum mechanics, state-dependent geometry, and emergent arrows of time. Linear dynamics, coarse-graining, and approximate adjunction. Let C be a category of “fine-grained” states, enriched over Banach spaces or Hilbert spaces (e.g. density matrices with completely positive maps). Suppose we have a linear dynamical semigroup {Tt}t≥0on C, generated by a linear operator L: d dtxt=L(xt), xt∈Ob(C),(169) with Tt=etL in an appropriate sense. Now introduce a coarse-graining functor F : C → D (e.g. a partial trace, spatial averaging, or projection to relevant observables) and a reconstruction functor G : D → C (e.g. an embedding of effective states back into the fine description). The pair ( F, G )is typically only an approximate adjunction: there are natural transformations η : idC⇒GF ,  : FG ⇒idD but the triangle identities hold only approximately, with curvature RFas in (168). The naive coarse-grained dynamics on Dis defined by yt:= F(xt),d dtyt=FL(G(yt))=: Leff (yt),(170) where xt := G ( yt )is a chosen fine representative of the coarse state yt . If FaG were an exact adjunction with vanishing curvature, this construction would lead to a closed linear evolution equation on D , at least when L commutes appropriately with F and G . In practice, however, GF 6 = idC and the effective evolution on D is not a semigroup: composing coarse-graining and time evolution in different orders does not commute, and memory and nonlinear effects appear, just as in projection-operator approaches to statistical mechanics.[162–164] Adjoint curvature RFprovides a compact way to encode this failure. Curvature-induced corrections to effective generators. Consider two short time steps t and s . On the fine space we have Tt+s=Tt◦Ts. On the coarse space, starting from y0∈Ob(D), we can compare: •the direct coarse evolution: ydirect t+s:= F◦Tt+s◦G(y0), •the iterated coarse evolution: yiter t+s:= F◦Tt◦G◦F◦Ts◦G(y0) = F◦Tt◦(GF)◦Ts◦G(y0). If GF = idC, these coincide. In general, however, the difference is ∆t,s(y0) := yiter t+s−ydirect t+s=F◦Tt◦(GF −idC)◦Ts◦G(y0).(171) Using the unit η and the curvature RF of F one can rewrite GF −idC in terms of ( GRF )and higher-order terms; schematically, GF −idC≃GRFF+· · · . Substituting into (171) and taking the limit of small t, s (so that Tt≃id + tL , Ts≃id + sL ), one obtains an effective generator on Dof the form d dtyt=Llin(yt) + NR(yt) + memory terms,(172) 70 where Llin := F◦L◦G, NR(y)≃F◦L◦GRFG(y),(173) and the ellipsis denotes higher-order and non-local (in time) corrections. The term NR is generically nonlinear in y once one passes to a reduced description in which different fine states mapped to the same y are identified; the curvature RF probes how the failure of adjunction depends on the state, and this state-dependence manifests as nonlinear drift and memory at the coarse level. This picture is closely related to the emergence of nonlinear and dissipative terms in the Mori–Zwanzig formalism, where projection operators that are not exact projectors give rise to memory kernels and nonlinear corrections in the effective dynamics.[ 162 – 164 ] Here, the adjoint curvature plays the role of a geometric measure of such deviations. State-dependent geometry and nonlinear quantum mechanics. A familiar setting in which coarse descriptions become nonlinear is quantum mechanics on projective Hilbert space. Pure states form rays in a Hilbert space H ; quotienting by global phase yields the complex projective space P ( H ), endowed with the Fubini–Study metric. Schrödinger evolution is linear on H but corresponds to Hamiltonian flow on P ( H )with respect to this metric.[ 165 , 166 ] More general, state-dependent geometries or modifications of the connection on the projective bundle lead to nonlinear Schrödinger equations.[167, 168] In categorical terms, passing from H to P ( H )corresponds to a quotient functor F : CH→ D that forgets global phase, together with a (local) section G that picks representatives in each ray. This pair ( F, G )is only an approximate adjunction when restricted to physically relevant subcategories (e.g. when additional constraints such as normalisation, gauge choices, or coarse-grained observables are imposed), and the resulting adjoint curvature RF measures the state-dependence of the induced connection on P ( H ). The curvature then contributes additional state-dependent terms to the effective Hamiltonian vector field, which can be interpreted as nonlinear corrections to Schrödinger dynamics, as in the frameworks of Białynicki-Birula & Mycielski and Weinberg.[167, 168] More broadly, any situation in which: 1. a linear or affine dynamics is defined on a “lifted” space, 2. one passes to a quotient or coarse-grained space via a functor Fwith approximate adjoint G, will generically produce curvature-driven corrections of the form (172) , encoding state-dependent geometry of the effective state space. Curvature, irreversibility, and emergent arrows of time. Adjoint curvature also provides a natural language for emergent arrows of time. In Section III E we saw that curvature of RG functors measures irreversibility and scheme-dependence of coarse-graining; the same holds for dynamical adjunctions. If FaG is an exact adjunction with vanishing curvature, one can often construct time-reversal symmetries or microscopic reversibility at the level of universal structures. When RF6 = 0, the effective dynamics on D: •fails to form an exact semigroup (coarse Markov property is broken), •exhibits memory and path-dependence encoded in ∆t,s from (171), • admits structural monotones (entropy-like functionals) that decrease along the flow, as in Theorem IV.1. In many physical settings, such as open quantum systems coupled to baths or coarse-grained hydrodynamic variables, this manifests as dissipation and irreversibility: entropy production functionals and relative entropies become structural monotones along the coarse dynamics.[ 169 ] Adjoint curvature provides a unifying, geometric measure of the deviation from micro-reversible dynamics induced by coarse-graining and the selection of universal structures. Summary. To summarise, broken adjunctions and their adjoint curvature: • encode the failure of coarse-graining and reconstruction functors to be perfectly compatible with dynamics; • generate nonlinear and memory-dependent corrections to effective evolution equations, as in (172) ; • realise state-dependent geometries whose curvature contributes additional drift terms, linking to nonlinear quantum mechanics and geometric formulations of dynamics; 71 • quantify irreversibility and emergent arrows of time in coarse descriptions, connecting to entropy production and structural monotonicity. In subsequent subsections we will apply this perspective to concrete examples from RG flows, biomolecular dynamics, and learning dynamics in AI systems. B. Teleological Attractors as Universal Solutions In Subsection VII D we introduced the idea that teleological behaviour arises when dynamics is attracted to objects that solve universal constraint problems. There we focused on concrete examples such as protein-folding funnels. Here we formulate this notion abstractly and clarify how universal objects act as teleological attractors for universal-property flows. Universal objects as solutions of constraint problems. Let C be a complete V -enriched category, and let D:J−→ C be a diagram encoding a family of constraints (e.g. energetic, geometric or functional). A universal solution of this constraint problem is a limit or colimit of D: U∗:= lim Dor U∗:= colim D, (174) together with the universal cone or cocone structure. By universal property, any other object X∈Ob ( C ) that satisfies the same pattern of constraints admits a unique mediating morphism to (or from) U∗ ; U∗ is therefore determined up to unique isomorphism by the constraint diagram. Examples include: • equilibrium thermodynamic states as minimisers of free energy subject to constraints (Legendretransform and convex duality diagrams); • maximum-entropy distributions subject to moment constraints, where U∗ is the exponential-family distribution that all admissible statistics factor through; • stable folds or regulatory networks obtained as limits or colimits of compatibility diagrams in biomolecular and biological categories (Sections VII and VIII). Universal-property flow towards teleological attractors. Suppose we have a dynamical universal-property flow on C, given by a family of endofunctors Tt:C −→ C, t ≥0, with T0 = idC and Tt+s = Tt◦Ts , and assume that each Tt preserves the shape of the constraint diagram D (i.e. Tt◦D is naturally isomorphic to D up to a re-labelling of objects). For any initial object X0 , we obtain a trajectory Xt:= Tt(X0). Following Subsection VII D, we define a structural teleological defect functional TeleDefectt(X0) := δXt, U∗≥0,(175) where δ is a defect measure that vanishes if and only if Xt satisfies the universal property of U∗ (e.g. is isomorphic to U∗ or lies in its isomorphism class), and is defined via the enrichment (metric, order, or information-theoretic divergence). In many cases, δ can be realised as a Lyapunov functional—for instance, a relative entropy or free-energy difference that is minimal at U∗.[170, 171] Definition IX.1 (Teleological attractor) . The universal object U∗ is a teleological attractor for the flow {Tt}if, for every X0∈Ob(C), TeleDefectt0(X0)≤TeleDefectt(X0)for t0≥t, lim t→∞ TeleDefectt(X0)=0.(176) Under mild compactness and continuity assumptions, the monotonicity and convergence conditions in (176) imply that U∗ is a (global) attractor in the sense of dynamical systems theory: trajectories starting from any initial X0 converge to the isomorphism class of U∗ in the topology induced by the enrichment.[170, 171] This makes explicit the teleological interpretation: system trajectories converge, not to arbitrary fixed points, but specifically to the universal solutions of the constraint problem encoded by D. 72 Initial conditions versus universal constraints. In a conventional dynamical system, evolution is determined entirely by the initial condition and a local rule (e.g. a vector field or generator). In the universal-property perspective, the long-time behaviour is additionally restricted by global constraints expressed categorically by Dand its limit U∗. Teleology can then be phrased as: Dynamics is guided both by initial conditions and by universal constraints; the latter select a distinguished universal solution U∗that acts as the attractor of the universal-property flow. Different initial objects X0 may follow very different transient trajectories (their “histories”) but all are funnelled onto the same universal solution up to isomorphism. This is the categorical analogue of synergetic phenomena and order-parameter reduction, in which many microscopic degrees of freedom collapse onto a low-dimensional manifold of macroscopic order parameters that solve suitable extremal principles.[171, 172] Role of adjoint curvature and imperfect teleology. In Subsection VII D we also introduced a teleological curvature Rtel Tt , see Eq. (158) , measuring the failure of the dynamical functors Tt to respect the universal constraint maps uX : X→U∗ from Eq. (155) . When this curvature vanishes, the universal maps are perfectly compatible with the dynamics and the system realises the universal solution U∗ in an optimally efficient way: trajectories are everywhere aligned with the gradient of the defect functional and converge monotonically. When Rtel Tt6= 0, teleology is only approximate: •trajectories may orbit or wander near the universal structure before settling into it; •multiple metastable universal-like structures may coexist, each solving a subset of the constraints; • non-Markovian memory, evolutionary path dependence, or learning history can bias which universal solution is ultimately reached. The size and structure of Rtel Tt thus quantify how strongly the actual dynamics is aligned with the universal constraint U∗, providing a graded notion of teleology rather than a binary one. Examples and interpretations. This abstract picture encompasses several examples discussed earlier: • In biomolecular folding, the native basin Ns acts as a universal solution for the constraint functor Cs ; folding dynamics is a universal-property flow on Conf ( s )with Ns as teleological attractor (Subsection VII B). • In biological organisation, motifs, domains, complexes and cellular modules appear as universal structures across the biological RG ladder, with structural defects decreasing across scales (Subsection VII C). • In learning systems and AI, minimal sufficient and invariant representations are universal solutions in appropriate statistical categories; gradient-based training can be viewed as a universal-property flow toward these attractors (Subsection VIII B).[173] In each case, what appears as “goal-directed” or teleological behaviour is re-expressed as convergence toward objects determined uniquely (up to isomorphism) by how they fit into a diagram of constraints. The universal property selects the “goal”, while structural monotonicity and adjoint curvature govern the strength and quality of the teleological pull. X. DISCUSSION AND OUTLOOK A. Summary of the Framework In this work we have proposed a categorical reformulation of renormalisation and multi-scale dynamics in terms of universal-property flow. Classical renormalisation group (RG) theory describes how couplings flow under coarse-graining and how critical phenomena are organised into universality classes.[ 174 – 176 ] Our framework lifts this picture from the level of numerical parameters to the level of universal structures in suitable categories, and introduces adjoint curvature as a geometric measure of irreversibility, anomaly, and teleological alignment. 73 RG as universal-property flow. Instead of a single coupling space, we consider for each scale µ a category Cµof theories, observables, or defect data, and RG steps as functors Fµ→µ0:Cµ−→ Cµ0(µ > µ0). Within each Cµ we identify universal structures Uµ —limits, colimits, adjunctions, Kan extensions, centers, free objects—as in the discussion of universal properties in Subsection II B. The renormalisation group is then reinterpreted as a flow Uµ7−→ Uµ0 in the space of universal constructions, organised by a renormalisation category R (Section III). Classical beta-functions and couplings become derived quantities: they parameterise how objects in Cµ move under Fµ→µ0, but the primary dynamical objects are the universal structures themselves. Adjoint curvature as irreversibility and anomaly. Whenever coarse-graining and reconstruction functors F, G form an approximate adjunction FaG , the unit and counit define composites whose deviation from the identity is measured by adjoint curvature RF (Subsection II D). In RG settings this curvature quantifies: •irreversibility of coarse-graining (loss of information and scheme dependence); •anomalies and obstructions in matching UV and IR descriptions; •path-dependence in scheme space (curvature of RG loops). The same construction applies to dynamical adjunctions (Section IX): adjoint curvature then measures non-Markovian memory, state-dependent geometry, and emergent arrows of time in coarse-grained dynamics. Structural monotonicity versus scalar monotones. Classical RG monotonicity theorems, such as the c -, a -, and F -theorems, identify scalar functionals that decrease along RG flows in specific dimensions.[ 174 , 176 ] We generalised this notion to structural monotonicity (Section IV): given a family of universal structures Uµand an enrichment (metric, order, probability), one defines defect functionals Defect(Uµ)≥0 that measure violation of a universal property (e.g. failure of a limit to exist exactly, loss of duals, breaking of adjunctions). Under mild positivity and contractivity assumptions on RG functors, these defects are non-increasing along the flow (Theorem IV.1). Scalar monotones such as c ( µ )appear in this framework as particular defect functionals—numerical shadows of richer structural quantities (e.g. metrics on coupling spaces, stress-tensor structures). Structural monotonicity thus strictly extends classical scalar monotonicity: it provides families of monotones associated with each preserved universal structure and remains meaningful in regimes where scalar monotones are difficult to define. Non-invertible symmetries and strong coupling. A key advantage of the universal-property approach is that it naturally accommodates non-invertible symmetries and strongly coupled systems. Symmetry operators and topological defects form monoidal (often fusion) categories CT attached to a theory T (Section V). RG domain walls induce monoidal functors FR:CTUV −→ CTIR , and universal constructions in CT —centers, duals, module categories—provide robust invariants of the flow. Non-invertible symmetries are adjointable rather than invertible objects; their behaviour under RG is controlled by adjoint curvature and Kan-extension obstructions, yielding new invariants for symmetry preservation, breaking, and anomaly flow. In strongly coupled and strongly correlated systems (Section VI), classical scalar diagnostics often fail: flows are non-monotonic, beta-functions are scheme-dependent, and topological or entangled structures dominate. Our framework replaces flows of couplings by flows of universal structures (defect categories, centers, tensor-network fixed points) and structural defects (fusion-rule deviations, rigidity defects, scheme curvatures), providing new, scheme-robust invariants that capture universality at strong coupling. 80 IMA Volumes in Mathematics and its Applications, Vol. 152, Springer, New York, pp. 105–191 (2010). [51] C. Heunen and J. Vicary, Categories for Quantum Theory: An Introduction, Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford (2019). [52] S. Abramsky and B. Coecke, “A categorical semantics of quantum protocols,” in: Proceedings of the 19th IEEE Symposium on Logic in Computer Science (LICS 2004), IEEE Computer Society, Los Alamitos, CA, pp. 415–425 (2004), arXiv:quant-ph/0402130. [53] T. Leinster, Higher Operads, Higher Categories, London Mathematical Society Lecture Note Series, Vol. 298, Cambridge University Press, Cambridge (2004). [54] F. W. Lawvere, “Metric spaces, generalized logic, and closed categories,” Rend. Sem. Mat. Fis. Milano 43 , 135–166 (1973); reprinted in Repr. Theory Appl. Categ. 1, 1–37 (2002). [55] P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Tensor Categories, Mathematical Surveys and Monographs, Vol. 205, American Mathematical Society, Providence, RI (2015). [56] J. Lurie, Higher Topos Theory, Annals of Mathematics Studies, Vol. 170, Princeton University Press, Princeton, NJ (2009). [57] B. Coecke and A. Kissinger, Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning, Cambridge University Press, Cambridge (2017). [58] K. G. Wilson, “The renormalization group: Critical phenomena and the Kondo problem,” Rev. Mod. Phys. 47, 773–840 (1975). [59] C. Wetterich, “Exact evolution equation for the effective potential,” Phys. Lett. B 301, 90–94 (1993). [60] C. Bagnuls and C. Bervillier, “Exact renormalization group equations. An introductory review,” Phys. Rept. 348, 91–157 (2001), arXiv:hep-th/0002034. [61] T. R. Morris, “Elements of the continuous renormalization group,” Prog. Theor. Phys. Suppl. 131 , 395–414 (1998), arXiv:hep-th/9802039. [62] J. Polonyi, “Lectures on the functional renormalization group,” Central Eur. J. Phys. 1 , 1–71 (2003), arXiv:hep-th/0110026. [63] M. Gell-Mann and F. E. Low, “Quantum electrodynamics at small distances,” Phys. Rev. 95 , 1300–1312 (1954). [64] L. P. Kadanoff, “Scaling laws for Ising models near Tc,” Physics 2, 263–272 (1966). [65] J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge Lecture Notes in Physics, Cambridge University Press, Cambridge (1996). [66] F. J. Wegner, “The critical state, general aspects,” in: C. Domb and M. S. Green (eds.), Phase Transitions and Critical Phenomena, Vol. 6, Academic Press, London, pp. 7–124 (1976). [67] K. Rejzner, “Locality and renormalization in quantum field theory,” Phil. Trans. R. Soc. A 373 , 20140241 (2015). [68] P. Etingof, D. Nikshych and V. Ostrik, “On fusion categories,” Ann. Math. 162 , 581–642 (2005), arXiv:math/0203060 [math.QA]. [69] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2, 2nd ed., Springer, Berlin (1997). [70] J. Fröhlich, C. King, A. Recknagel and C. Schweigert, “The holographic principle and non-commutative geometry,” in: Y. J. Ng (ed.), Quantum Theory and Symmetries, World Scientific, Singapore, pp. 151–177 (2000), and related developments summarised in: J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, “Duality and defects in rational conformal field theory,” Nucl. Phys. B 763, 354–430 (2007), arXiv:hep-th/0607247. [71] R. Street and R. Walters, “Yoneda structures on 2-categories,” J. Algebra 50, 350–379 (1978). [72] J. Adámek and J. Rosický, Locally Presentable and Accessible Categories, London Mathematical Society Lecture Note Series, Vol. 189, Cambridge University Press, Cambridge (1994). [73] M. Müger, “On the structure of modular categories,” Proc. London Math. Soc. (3) 87 , 291–308 (2003), arXiv:math/0201017 [math.CT]. [74] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge (2000). [75] V. Vedral, “The role of relative entropy in quantum information theory,” Rev. Mod. Phys. 74 , 197–234 (2002). [76] I. Jack and H. Osborn, “Analogs for the c -theorem in four dimensions,” Nucl. Phys. B 343 , 647–688 (1990). [77] H. Osborn, “Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories,” Nucl. Phys. B 363, 486–526 (1991). [78] Y. Nakayama, “Scale invariance vs. conformal invariance,” Phys. Rept. 569 , 1–93 (2015), arXiv:1302.0884 [hep-th]. [79] D. F. Litim, “Optimized renormalization group flows,” Phys. Lett. B 486 , 92–99 (2000), arXiv:hepth/0005245. [80] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed., International Series of Monographs on Physics, Vol. 113, Oxford University Press, Oxford (2002). [81] M. R. Gaberdiel, “An Introduction to Conformal Field Theory,” Rept. Prog. Phys. 63 , 607–667 (2000), arXiv:hep-th/9910156. doi:10.1088/0034-4885/63/4/203 [82] G. M. Shore, The c and a -Theorems and the Local Renormalisation Group, SpringerBriefs in Physics, Springer, Cham (2017), arXiv:1601.06662 [hep-th]. [83] D. L. Jafferis, I. R. Klebanov, S. S. Pufu and B. R. Safdi, “Towards the F -Theorem: N= 2Field Theories on the Three-Sphere,” JHEP 06, 102 (2011), arXiv:1103.1181 [hep-th]. doi:10.1007/JHEP06(2011)102 81 [84] H. Casini and M. Huerta, “A Finite Entanglement Entropy and the c -Theorem,” Phys. Lett. B 600 , 142–150 (2004), arXiv:hep-th/0405111. doi:10.1016/j.physletb.2004.08.072 [85] H. Casini and M. Huerta, “Entanglement Entropy in Free Quantum Field Theory,” J. Phys. A 42 , 504007 (2009), arXiv:0905.2562 [hep-th]. doi:10.1088/1751-8113/42/50/504007 [86] T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed., Wiley Series in Telecommunications and Signal Processing, John Wiley & Sons, Hoboken, NJ (2006). [87] M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer, Berlin (1993). [88] B. Delamotte, “An introduction to the nonperturbative renormalization group,” in: A. Schwenk and J. Polonyi (eds.), Renormalization Group and Effective Field Theory Approaches to Many-Body Systems, Lecture Notes in Physics, Vol. 852, Springer, Berlin (2012), pp. 49–132. [89] P. Kopietz, L. Bartosch and F. Schütz, Introduction to the Functional Renormalization Group, Lecture Notes in Physics, Vol. 798, Springer, Berlin (2010). [90] S. D. Głazek and K. G. Wilson, “Limit cycles in quantum theories,” Phys. Rev. Lett. 89, 230401 (2002). [91] E. Braaten and H.-W. Hammer, “Universality in few-body systems with large scattering length,” Phys. Rept. 428, 259–390 (2006), arXiv:cond-mat/0410417. [92] R. Shankar, “Renormalization-group approach to interacting fermions,” Rev. Mod. Phys. 66 , 129–192 (1994). [93] C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams and A. E. Ruckenstein, “Phenomenology of the normal state of Cu-O high-temperature superconductors,” Phys. Rev. Lett. 63, 1996–1999 (1989). [94] X.-G. Wen, Quantum Field Theory of Many-Body Systems, Oxford University Press, Oxford (2004). [95] A. Kitaev and J. Preskill, “Topological entanglement entropy,” Phys. Rev. Lett. 96 , 110404 (2006), arXiv:hepth/0510092. [96] A. Cappelli, D. Friedan and J. I. Latorre, “C Theorem and Spectral Representation,” Nucl. Phys. B 352 , 616–670 (1991). [97] D. Z. Freedman, S. S. Gubser, K. Pilch and N. P. Warner, “Renormalization Group Flows from Holography– Supersymmetry and a c-Theorem,” Adv. Theor. Math. Phys. 3, 363–417 (1999), arXiv:hep-th/9904017. [98] A. Schwimmer and S. Theisen, “Entanglement Entropy, Trace Anomalies and Holography,” Nucl. Phys. B 801, 1–24 (2008), arXiv:0802.1017 [hep-th]. [99] T. Nishioka, “Entanglement Entropy: Holography and Renormalization Group,” Rev. Mod. Phys. 90 , 035007 (2018), arXiv:1801.10352 [hep-th]. [100] F. W. Lawvere, “Functorial Semantics of Algebraic Theories,” Proc. Natl. Acad. Sci. USA 50 , 869–872 (1963). [101] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, “Generalized Global Symmetries,” J. High Energy Phys. 2015, 172 (2015), arXiv:1412.5148 [hep-th]. [102] N. Carqueville, I. Runkel and G. Schaumann, “Orbifolds of n -Dimensional Defect TQFTs,” Geom. Topol. 23, 781–864 (2019), arXiv:1705.06085 [math.QA]. [103] L. Bhardwaj, L. E. Bottini, S. Schäfer-Nameki and A. Tiwari, “Non-Invertible Higher-Categorical Symmetries,” SciPost Phys. 14, 007 (2023), arXiv:2204.06564 [hep-th]. [104] S.-H. Shao, “What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries,” arXiv:2308.00747 [hep-th] (2023). [105] P. Etingof and V. Ostrik, “Finite tensor categories,” Mosc. Math. J. 4 , 627–654 (2004), arXiv:math/0301027 [math.QA]. [106] C. L. Douglas, C. Schommer-Pries and N. Snyder, “Dualizable Tensor Categories,” Mem. Amer. Math. Soc. 268, no. 1308 (2020), arXiv:1312.7188 [math.QA]. [107] L. Kong, “Anyon Condensation and Tensor Categories,” Nucl. Phys. B 886 , 436–482 (2014), arXiv:1307.8244 [cond-mat.str-el]. [108] D. Gaiotto and T. Johnson-Freyd, “Condensations in higher categories,” Selecta Math. 28 , 35 (2022), arXiv:1905.09566 [math.CT]. [109] Z. Komargodski and N. Seiberg, “From Finite to Continuous Non-Invertible Symmetries,” SciPost Phys. 11 , 072 (2021), arXiv:2002.04287 [hep-th]. [110] F. Apruzzi and S. Schäfer-Nameki, “RG Flows, Product Symmetries, and Categories,” Phys. Rev. Lett. 130 , 021602 (2023), arXiv:2208.07373 [hep-th]. [111] A. Davydov, D. Nikshych and V. Ostrik, “On the structure of the Witt group of braided fusion categories,” Selecta Math. 16, 1–25 (2010), arXiv:0909.3140 [math.QA]. [112] E. C. Rowell and Z. Wang, “Mathematics of topological quantum computing,” Bull. Amer. Math. Soc. 55 , 183–238 (2018), arXiv:1705.06206 [math.QA]. [113] M. A. Levin and X.-G. Wen, “String-net condensation: A Physical mechanism for topological phases,” Phys. Rev. B 71, 045110 (2005), arXiv:cond-mat/0404617. [114] J. Fuchs, I. Runkel and C. Schweigert, “TFT construction of RCFT correlators I: Partition functions,” Nucl. Phys. B 646, 353–497 (2002), arXiv:hep-th/0204148. [115] C. M. Varma, “Singular Fermi liquids,” Phys. Rev. Lett. 83 , 3538–3541 (1999); see also: C. M. Varma, “Quantum criticality in heavy fermion metals,” Phys. Rev. B 55, 14554–14580 (1997). [116] P. A. Lee, N. Nagaosa and X.-G. Wen, “Doping a Mott insulator: Physics of high-temperature superconductivity,” Rev. Mod. Phys. 78, 17–85 (2006). [117] A. C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press, Cambridge (1993). [118] M. Imada, A. Fujimori and Y. Tokura, “Metal-insulator transitions,” Rev. Mod. Phys. 70 , 1039–1263 (1998). [119] L. Balents, “Spin liquids in frustrated magnets,” Nature 464, 199–208 (2010). 82 [120] V. L. Berezinskii, “Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. Classical systems,” Sov. Phys. JETP 32, 493–500 (1971). [121] J. M. Kosterlitz and D. J. Thouless, “Ordering, metastability and phase transitions in two-dimensional systems,” J. Phys. C 6, 1181–1203 (1973). [122] R. Haag, Local Quantum Physics: Fields, Particles, Algebras, 2nd ed., Springer, Berlin (1996). [123] M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer, Berlin (1993). [124] G. Vidal, “Entanglement Renormalization,” Phys. Rev. Lett. 99, 220405 (2007), arXiv:cond-mat/0512165. [125] G. Evenbly and G. Vidal, “Tensor Network Renormalization,” Phys. Rev. Lett. 115 , 180405 (2015), arXiv:1412.0732 [cond-mat.str-el]. [126] Z.-C. Gu and X.-G. Wen, “Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order,” Phys. Rev. B 80, 155131 (2009), arXiv:0903.1069 [cond-mat.str-el]. [127] X. Chen, Z.-C. Gu and X.-G. Wen, “Complete classification of one-dimensional gapped quantum phases in interacting spin systems,” Phys. Rev. B 84, 235128 (2011), arXiv:1103.3323 [cond-mat.str-el]. [128] B. Swingle, “Entanglement Renormalization and Holography,” Phys. Rev. D 86 , 065007 (2012), arXiv:0905.1317 [cond-mat.str-el]. [129] S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press, Cambridge (2011). [130] A. Kitaev, “Periodic table for topological insulators and superconductors,” AIP Conf. Proc. 1134 , 22–30 (2009), arXiv:0901.2686 [cond-mat.mes-hall]. [131] X. Chen, Z.-C. Gu, Z.-X. Liu and X.-G. Wen, “Symmetry-protected topological orders in interacting bosonic systems,” Science 338, 1604–1606 (2012). [132] X.-G. Wen, “Colloquium: Zoo of quantum-topological phases of matter,” Rev. Mod. Phys. 89 , 041004 (2017), arXiv:1610.03911 [cond-mat.str-el]. [133] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M. P. A. Fisher, “Deconfined Quantum Critical Points,” Science 303, 1490–1494 (2004), arXiv:cond-mat/0311326. [134] B. Alberts, A. Johnson, J. Lewis, D. Morgan, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, 6th ed., Garland Science, New York (2015). [135] J. D. Bryngelson and P. G. Wolynes, “Spin glasses and the statistical mechanics of protein folding,” Proc. Natl. Acad. Sci. USA 84, 7524–7528 (1987). [136] J. N. Onuchic, Z. Luthey-Schulten and P. G. Wolynes, “Theory of protein folding: The energy landscape perspective,” Annu. Rev. Phys. Chem. 48, 545–600 (1997). [137] K. A. Dill and H. S. Chan, “From Levinthal to pathways to funnels,” Nat. Struct. Biol. 4, 10–19 (1997). [138] C. M. Dobson, “Protein folding and misfolding,” Nature 426, 884–890 (2003). [139] J. D. Bryngelson, J. N. Onuchic, N. D. Socci and P. G. Wolynes, “Funnels, pathways, and the energy landscape of protein folding: A synthesis,” Proteins 21, 167–195 (1995). [140] K. A. Dill and S. Bromberg, Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology, 2nd ed., Garland Science, New York (2010). [141] D. Thirumalai, E. P. O’Brien, G. Morrison and C. Hyeon, “Theoretical Perspectives on Protein Folding,” Annu. Rev. Biophys. 39, 159–183 (2010). [142] H. Frauenfelder, B. H. McMahon, R. H. Austin, K. Chu and J. T. Groves, “The role of structure, energy landscape, dynamics, and allostery in the enzymatic function of myoglobin,” Proc. Natl. Acad. Sci. USA 100, 8615–8620 (2003). [143] G. S. Ayton, W. G. Noid and G. A. Voth, “Multiscale modeling of biomolecular systems: in serial and in parallel,” Curr. Opin. Struct. Biol. 17, 192–198 (2007). [144] C. Clementi, “Coarse-grained models of protein folding: toy models or predictive tools?” Curr. Opin. Struct. Biol. 18, 10–15 (2008). [145] H. Kitano, “Systems biology: a brief overview,” Science 295, 1662–1664 (2002). [146] A.-L. Barabási and Z. N. Oltvai, “Network biology: understanding the cell’s functional organization,” Nat. Rev. Genet. 5, 101–113 (2004). [147] E. Schrödinger, What is Life? The Physical Aspect of the Living Cell, Cambridge University Press, Cambridge (1944). [148] J. Monod, Chance and Necessity: An Essay on the Natural Philosophy of Modern Biology, Alfred A. Knopf, New York (1971). [149] M. Eigen, “Selforganization of matter and the evolution of biological macromolecules,” Naturwissenschaften 58, 465–523 (1971). [150] S. A. Kauffman, The Origins of Order: Self-Organization and Selection in Evolution, Oxford University Press, Oxford (1993). [151] R. A. Fisher, The Genetical Theory of Natural Selection, Clarendon Press, Oxford (1930). [152] S. Wright, “The roles of mutation, inbreeding, crossbreeding and selection in evolution,” in Proceedings of the Sixth International Congress of Genetics (1932), pp. 356–366. [153] C. H. Waddington, The Strategy of the Genes, Allen & Unwin, London (1957). [154] M. J. West-Eberhard, Developmental Plasticity and Evolution, Oxford University Press, Oxford (2003). [155] A. Wagner, Arrival of the Fittest: Solving Evolution’s Greatest Puzzle, Penguin, New York (2014). [156] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser and I. Polosukhin, “Attention Is All You Need,” Adv. Neural Inf. Process. Syst. 30 (2017). [157] J. Jumper et al., “Highly accurate protein structure prediction with AlphaFold,” Nature 596 , 583–589 (2021). 83 [158] T. Cohen and M. Welling, “Group equivariant convolutional networks,” Proc. 33rd Int. Conf. Mach. Learn. (ICML 2016), arXiv:1602.07576 [cs.LG]. [159] N. Tishby and N. Zaslavsky, “Deep Learning and the Information Bottleneck Principle,” Proc. 2015 IEEE Information Theory Workshop (ITW), pp. 1–5, arXiv:1503.02406 [cs.LG]. [160] E. L. Lehmann and H. Scheffé, “Completeness, similar regions, and unbiased estimation. I,” Sankhy¯a 10 , 305–340 (1950). [161] A. Achille and S. Soatto, “Emergence of Invariance and Disentangling in Deep Representations,” J. Mach. Learn. Res. 19, 1–34 (2018), arXiv:1706.01350 [cs.LG]. [162] R. Zwanzig, “Ensemble Method in the Theory of Irreversibility,” J. Chem. Phys. 33, 1338–1341 (1960). [163] H. Mori, “Transport, Collective Motion, and Brownian Motion,” Prog. Theor. Phys. 33, 423–455 (1965). [164] H. Grabert, Projection Operator Techniques in Nonequilibrium Statistical Mechanics, Springer, Berlin (1982). [165] T. W. B. Kibble, “Geometrization of Quantum Mechanics,” Commun. Math. Phys. 65, 189–201 (1979). [166] A. Ashtekar and T. A. Schilling, “Geometrical Formulation of Quantum Mechanics,” in A. Harvey (ed.), On Einstein’s Path, Springer, New York (1999), arXiv:gr-qc/9706069. [167] I. Białynicki-Birula and J. Mycielski, “Nonlinear Wave Mechanics,” Ann. Phys. 100, 62–93 (1976). [168] S. Weinberg, “Testing Quantum Mechanics,” Ann. Phys. 194, 336–386 (1989). [169] H. Spohn, “Entropy Production for Quantum Dynamical Semigroups,” J. Math. Phys. 19 , 1227–1230 (1978). [170] J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia (1976). [171] H. Haken, Synergetics: An Introduction, 3rd ed., Springer, Berlin (1983). [172] J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1982). [173] K. Friston, “The free-energy principle: a unified brain theory?” Nat. Rev. Neurosci. 11, 127–138 (2010). [174] K. G. Wilson, “The renormalization group: Critical phenomena and the Kondo problem,” Rev. Mod. Phys. 47, 773–840 (1975). [175] N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Addison–Wesley, Reading, MA (1992). [176] J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, Cambridge (1996). [177] J. C. Baez and J. Dolan, “Higher-dimensional algebra and topological quantum field theory,” J. Math. Phys. 36, 6073–6105 (1995). [178] T. Leinster, Basic Category Theory, Cambridge University Press, Cambridge (2014). [179] K. Costello, Renormalization and Effective Field Theory, Mathematical Surveys and Monographs, Vol. 170, American Mathematical Society, Providence, RI (2011). [180] K. Costello and O. Gwilliam, Factorization Algebras in Quantum Field Theory, Vol. 1, Cambridge University Press, Cambridge (2021); see also arXiv:1311.4099 [math.QA]. [181] J. Lurie, “On the Classification of Topological Field Theories,” Current Developments in Mathematics 2008 , 129–280 (2009), arXiv:0905.0465 [math.CT]. [182] R. Orús, “A practical introduction to tensor networks: Matrix product states and projected entangled pair states,” Ann. Phys. 349, 117–158 (2014), arXiv:1306.2164 [cond-mat.str-el]. [183] L. Savary and L. Balents, “Quantum spin liquids: a review,” Rep. Prog. Phys. 80 , 016502 (2017), arXiv:1601.03742 [cond-mat.str-el]. [184] M. AlQuraishi, “End-to-End Differentiable Learning of Protein Structure,” Cell Systems 8 , 292–301.e3 (2019). [185] C. Zhang, S. Bengio, M. Hardt, B. Recht and O. Vinyals, “Understanding deep learning requires rethinking generalization,” International Conference on Learning Representations (ICLR) (2017), arXiv:1611.03530 [cs.LG]. [186] S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer, New York (1998). [187] J. C. Baez and M. Stay, “Physics, Topology, Logic and Computation: A Rosetta Stone,” in D. Coecke (ed.), New Structures for Physics, Lecture Notes in Physics, Vol. 813, Springer, Berlin (2010). [188] R. W. Batterman, “Multiple realizability and universality,” Br. J. Philos. Sci. 51, 115–145 (2000). [189] J. Butterfield, “Reduction, Emergence and Renormalization,” in R. Batterman (ed.), The Oxford Handbook of Philosophy of Physics, Oxford University Press, Oxford (2013). [190] G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems, Wiley, New York (1977). [191] G. F. R. Ellis, “Top-down causation and emergence: some comments on mechanisms,” Interface Focus 2 , 126–140 (2012).