Structural Infinity: A Coalgebraic Unification of Infinite Phenomena

Structural Infinity: A Coalgebraic Unification of Infinite Phenomena Hironao Nakamura Independent Researcher [email protected]aseda.jp | @hirochu_io December 2025 Abstract We introduce a categorical framework that unifies several heterogeneous notions of infinity—Dedekind infinity, analytic divergence, continuum refinement, and the higher-dimensional identity tower of Homotopy Type Theory (HoTT)—under a single coalgebraic principle. Given an endofunctor 𝐺∶𝒞→𝒞and a coalgebra 𝛾 ∶ 𝑋 → 𝐺𝑋, structural infinity is defined by the non-invertibility of all transition maps 𝛿𝑛∶ 𝐺𝑛𝑋 → 𝐺𝑛+1𝑋. We show that many classical and homotopical infinitary phenomena are precisely those coalgebras whose unfolding never stabilizes at any finite depth, reflecting a generative perspective in which mathematical structures emerge from an underlying field of potentiality. 1. Introduction Mathematical practice features a wide variety of notions of infinity: Dedekind-infinite sets in classical set theory, the unbounded progress of sequences in analysis, endlessly refinable structures in continuum theory, and the infinite tower of identity types in Homotopy Type Theory (HoTT). Although these notions share an intuitive sense of “never ending,” they arise in distinct domains and are rarely treated within a single unifying framework. This paper approaches these phenomena from a generative perspective that views mathematical objects not as fixed discrete collections, but as structures that emerge from an underlying pre-structured field of potentiality. From this standpoint, what matters is not size but how an object continues to unfold or stabilize. This viewpoint suggests that a unified account of infinity should capture non-stabilizing generative behavior rather than cardinal or ordinal properties. Traditional approaches describe infinity either as size (cardinality), as asymptotic behavior (limits), or as structural complexity (higher-dimensional homotopy). However, no categorical principle is currently available that captures all of these phenomena at once. In this paper, we propose a unified and purely categorical account of infinity based on coalgebraic iteration. The central thesis is: Infinity is not a matter of size, but of structure: an object is infinite when its coalgebraic unfolding never becomes invertible at any finite stage. Concretely, given an endofunctor 𝐺∶𝒞→𝒞and a coalgebra 𝛾 ∶ 𝑋 → 𝐺𝑋, we consider the canonical transition maps 𝛿𝑛∶= 𝐺𝑛(𝛾) ∶ 𝐺𝑛𝑋 → 𝐺𝑛+1𝑋. We say that (𝑋, 𝛾) exhibits structural infinity when 𝛿𝑛is not an isomorphism for all 𝑛 ∈ ℕ. In the homotopical setting, “isomorphism’ ’ is read as equivalence of types. This expresses infinity as the non-stabilization of coalgebraic unfolding, shifting the viewpoint from cardinality or ordinals to a genuinely structural criterion. This simple but powerful principle yields a striking unification: •Dedekind infinity: injective non-surjective maps 𝑓 ∶ 𝑋 → 𝑋 give rise to non-invertible transitions. •Analytic infinity: the successor function 𝑠∶ℕ→ℕis injective but not surjective. 1 •Continuum infinity: strict refinement functors on partitions of [0, 1] produce ever finer structures that cannot be reversed. •Homotopical infinity: for non-truncated types, the identity-tower transitions Id𝑛(𝐴) → Id𝑛+1(𝐴) are never equivalences. Despite their distinct origins, these examples share the same abstract pattern: the transitions 𝐺𝑛(𝛾) are never invertible, and thus the unfolding process never stabilizes. Contributions The contributions of this paper are as follows: 1. Definition of structural infinity. We introduce a categorical definition of infinity based solely on the non-invertibility of transition maps in coalgebraic iteration. 2. Unification of heterogeneous infinite phenomena. We show that Dedekind, analytic, continuum, and homotopical infinities all arise as instances of structural infinity. 3. Coalgebraic contextualization. We relate structural infinity to the Adámek chain and universal coalgebra, identifying non-stabilization as a fundamental organizing principle. 4. Foundational implications. Our framework supports categorical foundations, HoTT, higher structures, and future work on epistemic toposes and categorical cognition models. Outline of the paper Section 2 reviews categorical preliminaries. Section 3 introduces the definition of structural infinity and develops its basic properties. Section 4 demonstrates that several major forms of infinity are captured by this definition. Section 5 clarifies the relationship between structural infinity and universal coalgebra. Section 6 reviews related work, and Section 7 concludes. 2. Preliminaries This section reviews the categorical background required for the development of structural infinity. We assume basic familiarity with category theory, functors, and natural transformations. Throughout, 𝒞denotes an arbitrary category. For HoTT-based examples, we read “isomorphism’ ’ as “equivalence of types’ ’ and interpret 𝒞as the (∞, 1)- category of ∞-groupoids. 2.1 Categories and isomorphisms Acategory 𝒞consists of objects, morphisms, identity maps, and associative composition. A morphism 𝑓 ∶ 𝑋 → 𝑌 in 𝒞is an isomorphism if there exists 𝑔 ∶ 𝑌 → 𝑋 such that 𝑔 ∘ 𝑓 = id𝑋, 𝑓 ∘ 𝑔 = id𝑌. In HoTT, where objects may be interpreted as ∞-groupoids, the corresponding notion is an equivalence of types. We write 𝑋 ≅ 𝑌 for an isomorphism (or equivalence) between 𝑋and 𝑌. Standard categorical background can be found in Awodey’s textbook (Awodey 2010). 2 2.2 Endofunctors An endofunctor on 𝒞is a functor 𝐺 ∶ 𝒞 → 𝒞. Typical examples include: • The identity functor 𝐺(𝑋) = 𝑋. • Polynomial functors such as 𝐺(𝑋) = 𝐴 × 𝑋 + 𝐵. • Refinement functors on partitions of spaces. • In HoTT, the identity-type functor 𝐺(𝐴) = Id(𝐴) or loop space operators Ω𝐴. Iteration of 𝐺is defined by 𝐺0=Id𝒞, 𝐺𝑛+1 = 𝐺 ∘ 𝐺𝑛. 2.3 Coalgebras and coalgebra morphisms Acoalgebra for 𝐺is a pair (𝑋, 𝛾) with 𝛾 ∶ 𝑋 → 𝐺𝑋. A morphism of 𝐺-coalgebras ℎ ∶ (𝑋, 𝛾) → (𝑌 , 𝛿) is a morphism ℎ ∶ 𝑋 → 𝑌 in 𝒞such that the following diagram commutes: 𝑋𝛾 −→ 𝐺𝑋 ↓ℎ↓𝐺(ℎ) 𝑌𝛿 −→ 𝐺𝑌 . Coalgebras serve as a general language for unfolding, infinite structures, streams, automata, and non-wellfounded recursion. 2.4 The Adámek chain and stabilization Given an endofunctor 𝐺, the Adámek chain (Adámek 1974) is the transfinite sequence 0 → 𝐺0 → 𝐺20 → ⋯ used to construct the initial algebra of 𝐺. Dually, inverse chains starting at a terminal object 1construct terminal coalgebras. The chain stabilizes at stage 𝑛when the canonical map 𝐺𝑛0≅ −→ 𝐺𝑛+10 is an isomorphism. Our notion of structural infinity concerns non-stabilization of the chain arising from a coalgebra (𝑋, 𝛾): 𝑋𝛿0 −→ 𝐺𝑋 𝛿1 −→ 𝐺2𝑋𝛿2 −→ ⋯ , where 𝛿𝑛= 𝐺𝑛(𝛾). 3 2.5 Examples of functor iteration The following examples illustrate the breadth of coalgebraic iteration: •Self-maps. 𝐺 = Id and 𝛾 ∶ 𝑋 → 𝑋. Transition maps are powers of the same morphism. •Successor iteration. 𝐺 = Id and 𝛾=𝑠∶ℕ→ℕ. •Refinement functors. 𝐺sends finite partitions to strictly finer partitions. •Identity towers in HoTT. 𝐺(𝐴) = Id(𝐴) and 𝐺𝑛(𝐴) = Id𝑛(𝐴) generate higher identity types. These will be revisited in Section 4 as instances of structural infinity. Summary We have introduced the categorical ingredients—endofunctors, coalgebras, transition maps, and stabilization—that form the basis of our framework. The next section presents the formal definition of structural infinity and its immediate properties. For general background on coalgebra and its role in modeling state-based systems, see also Jacobs’ textbook (Jacobs 2016). 3. Structural Infinity This section introduces the central concept of the paper: structural infinity, a categorical notion of infinity based on the non-invertibility of transition maps in coalgebraic iteration. Given an endofunctor 𝐺∶𝒞→𝒞and a coalgebra (𝑋, 𝛾), recall the transition maps 𝛿𝑛∶= 𝐺𝑛(𝛾) ∶ 𝐺𝑛𝑋 → 𝐺𝑛+1𝑋. Each transition 𝛿𝑛describes one further “unfolding” of the structure encoded by (𝑋, 𝛾). A coalgebra is infinite precisely when none of these steps collapses to an isomorphism. 3.1 Definition Definition 3.1 (Structural Infinity) Let 𝐺∶𝒞→𝒞be an endofunctor and (𝑋, 𝛾) a𝐺-coalgebra. We say that (𝑋, 𝛾) exhibits structural infinity if ∀𝑛 ∈ ℕ, 𝛿𝑛∶ 𝐺𝑛𝑋 → 𝐺𝑛+1𝑋is not an isomorphism. In HoTT, “isomorphism’ ’ is interpreted as equivalence of types, so the condition becomes: 𝛿𝑛is not an equivalence for all 𝑛. Intuitively, no finite amount of unfolding exhausts the structure of the coalgebra. The process never stabilizes, and no finite stage yields a reversible step. 4 3.2 Intuitive meaning If for some finite 𝑘the transition map 𝛿𝑘were an isomorphism, then 𝐺𝑘𝑋 ≅ 𝐺𝑘+1𝑋, and all subsequent stages would also be isomorphic. The unfolding would “close” at stage 𝑘, indicating finiteness in a structural sense. Structural infinity requires that every transition introduces irreducible new structure. Thus infinity is reinterpreted as a non-stabilization property of coalgebraic iteration. This shifts the viewpoint: • not as cardinality, • not as ordinal divergence, • but as a structural, process-based, and coalgebraic phenomenon. 3.3 Alternative formulation via images (optional) In categories with (regular epi, mono)-factorizations, structural infinity can be characterized through stabilization of images. Let 𝛾(𝑛) ∶= 𝛿𝑛−1 ∘⋯∘𝛿0∶ 𝑋 → 𝐺𝑛𝑋 and write 𝑋 ↠ 𝐼𝑛↪ 𝐺𝑛𝑋 for the image factorization of 𝛾(𝑛). Proposition 3.2 (Image formulation) Suppose: 1. each morphism in 𝒞admits an image factorization, and 2. 𝐺preserves monomorphisms. Then the following implications hold: 1. (Structural finiteness ⇒image stabilization) If (𝑋, 𝛾) is structurally finite—i.e., if some transition 𝛿𝑘∶ 𝐺𝑘𝑋 → 𝐺𝑘+1𝑋 is an isomorphism—then the image sequence stabilizes from stage 𝑘onward: 𝐼𝑛≅ 𝐼𝑛+1 for all 𝑛 ≥ 𝑘. 2. (Image non-stabilization ⇒structural infinity) If 𝐼𝑛≇ 𝐼𝑛+1 for all 𝑛 ∈ ℕ, then (𝑋, 𝛾) is structurally infinite. In particular, strict non-stabilization of the images (𝐼𝑛)is a sufficient condition for structural infinity, and this perspective is especially useful when the objects 𝐺𝑛𝑋remain isomorphic (as in the case of Dedekind infinite sets) while the images strictly shrink. 5 3.4 Structural finiteness Definition 3.3 (Structurally finite coalgebra) A𝐺-coalgebra (𝑋, 𝛾) is structurally finite if there exists 𝑘 ∈ ℕ such that 𝛿𝑘∶ 𝐺𝑘𝑋 → 𝐺𝑘+1𝑋 is an isomorphism. Then 𝐺𝑘𝑋is a stable stage, and the unfolding becomes finite beyond 𝑘. 3.5 Basic lemmas Lemma 3.4 (Monic transitions) If each 𝛿𝑛is monic but not iso, then (𝑋, 𝛾) is structurally infinite. Idea. No information is lost (monic), but new information is continually added (non-iso). Lemma 3.5 (Functoriality) Let ℎ ∶ (𝑋, 𝛾) → (𝑌 , 𝛿) be a coalgebra morphism. If (𝑋, 𝛾) is structurally infinite and 𝐺(ℎ) preserves non-invertibility, then (𝑌 , 𝛿) is also structurally infinite. This shows that structural infinity is stable under coalgebra morphisms that preserve the direction of unfolding. 3.6 Relationship to coalgebraic theory Structural infinity sits naturally within coalgebraic semantics: • It generalizes non-well-founded coalgebras. • It coincides with non-stabilizing segments of the Adámek chain, where no finite stage yields a fixed point. • It applies to infinite data structures,streams,infinite trees, and the identity tower in HoTT. However, the crucial conceptual shift is that non-invertibility of transitions —not cardinality or ordinal height—plays the central role. 3.7 Summary Structural infinity can be summarized as: A coalgebra is infinite when every transition in its unfolding is non-invertible. This unified principle will allow us, in the next section, to reinterpret classical and homotopical infinities as instances of structural infinity. 6 4. Classical Infinities as Structural Infinity This section shows that several major mathematical notions of infinity—Dedekind infinity, analytic divergence, continuum refinement, and the homotopical identity tower—are all instances of structural infinity as defined in Section 3. Each of these arises from a coalgebra (𝑋, 𝛾) whose transition maps never become invertible. 4.1 Dedekind Infinity We begin with the classical notion of Dedekind-infinite sets. Definition 4.1 (Dedekind infinite set) A set 𝑋is Dedekind infinite if there exists a morphism 𝑓 ∶ 𝑋 → 𝑋 that is injective but not surjective. This is equivalent to the existence of a bijection 𝑋 ≅ 𝑋 ∖ {𝑥0}for some element 𝑥0, but we adopt the coalgebraically natural formulation above. Proposition 4.2 If 𝑋is Dedekind infinite, then the coalgebra (𝑋, 𝑓) for the identity functor 𝐺 = IdSet is structurally infinite. Proof. For 𝐺 = Id, the transition maps are 𝛿𝑛= 𝑓 ∶ 𝑋 → 𝑋. Since 𝑓is injective but not surjective, it is not an isomorphism. Thus each 𝛿𝑛is non-invertible, and by Definition 3.1, (𝑋, 𝑓) is structurally infinite. Remark 4.3 The object 𝐺𝑛𝑋 = 𝑋 remains unchanged for all 𝑛, but the structural infinite descent is captured entirely by the non-invertibility of 𝑓. This illustrates the usefulness of the transition-map formulation over the object-level formulation 𝐺𝑛𝑋 ≇ 𝐺𝑛+1𝑋. 4.2 Analytic Infinity (Successor iteration) Analytic “infinity” often appears through the unbounded successor process 𝑛 ↦ 𝑛 + 1. Definition 4.4 (Successor coalgebra) Let 𝐺 = IdSet and let 𝑠 ∶ ℕ → ℕ, 𝑠(𝑛) = 𝑛 + 1. Then (ℕ, 𝑠) is a 𝐺-coalgebra. 7 Proposition 4.5 The successor coalgebra (ℕ, 𝑠) is structurally infinite. Proof. The transition maps are all 𝛿𝑛= 𝑠 ∶ ℕ → ℕ. Since 𝑠is injective but not surjective (0 is not in its image), it is not an isomorphism. Thus (ℕ, 𝑠) is structurally infinite. Remark 4.6 This captures the analytic intuition of 𝑛 → ∞ as the ever-progressing, never-invertible successor operation. 4.3 Continuum Infinity (Strict partition refinement) We now turn to the “infinitary” nature of the continuum. Let 𝒫be the category whose: • objects are finite partitions of the interval [0, 1], and • morphisms are refinements:𝑃 → 𝑄 if 𝑄subdivides 𝑃. Define an endofunctor 𝐺 ∶ 𝒫 → 𝒫 that performs strict refinement, e.g., by subdividing each interval of a partition into two subintervals. Definition 4.7 (Refinement coalgebra) For a fixed partition 𝑃0, define 𝛾 ∶ 𝑃0→ 𝐺(𝑃0) as the refinement map sending each part of 𝑃0to its refined version. Proposition 4.8 The refinement coalgebra (𝑃0, 𝛾) is structurally infinite. Proof. For each 𝑛, 𝛿𝑛∶ 𝐺𝑛(𝑃0) → 𝐺𝑛+1(𝑃0) is the strict refinement map. Such a map is: • monic (each part of 𝐺𝑛(𝑃0)embeds into 𝐺𝑛+1(𝑃0)), but • not an isomorphism (new intervals are created; the partition becomes strictly finer). Thus 𝛿𝑛is never invertible, and (𝑃0, 𝛾) is structurally infinite. Remark 4.9 This captures the classical geometric intuition that the continuum can be subdivided indefinitely without reaching any “finest” stage. 8 4.4 Homotopical Infinity (Identity tower in HoTT) In Homotopy Type Theory (HoTT) (Univalent Foundations Program 2013), a type 𝐴carries an infinite hierarchy of higher identity types. Let 𝒞denote the (∞, 1)-category of ∞-groupoids. Define the endofunctor 𝐺(𝐴) = Id(𝐴) or, more precisely, the loop-space operator Ω𝐴 (the simplified identity-type functor suffices for our purpose). Let 𝛾 ∶ 𝐴 → 𝐺(𝐴) assign to each element 𝑎the trivial path refl𝑎. Definition 4.10 (Identity-tower coalgebra) The coalgebra (𝐴, 𝛾) generates the iterated identity tower 𝐴 → Id(𝐴) → Id2(𝐴) → Id3(𝐴) → ⋯ . Proposition 4.11 (Identity-tower transitions) Let 𝒞denote the (∞, 1)-category of ∞-groupoids, and let 𝐺(𝐴) = Id(𝐴) (or more generally the loop-space operator Ω𝐴). Let 𝛾 ∶ 𝐴 → 𝐺(𝐴) map each point to its reflexivity path. If for every 𝑛 ∈ ℕ the canonical transition 𝛿𝑛∶Id𝑛(𝐴) → Id𝑛+1(𝐴) is not an equivalence of types, then the identity tower (𝐴, 𝛾) is structurally infinite. Proof. If none of the transitions 𝛿𝑛is an equivalence, then by Definition 3.1, no stage of the coalgebraic unfolding stabilizes up to equivalence. Therefore (𝐴, 𝛾) is structurally infinite. Remark 4.12 (Typical HoTT situation) A common—and mathematically important—instance where the above condition holds is when a type 𝐴has nontrivial higher homotopy in arbitrarily high dimensions, for example when 𝐴is not 𝑛-truncated for any finite 𝑛in the sense of HoTT. In such cases the higher identity types Id𝑛(𝐴) do not collapse to lower ones, and the transition maps Id𝑛(𝐴) → Id𝑛+1(𝐴) fail to be equivalences for each 𝑛. Thus, non-truncated types provide natural examples of structural infinity: they exhibit an identity tower that never stabilizes up to equivalence. Summary of Section 4 All classical and homotopical forms of infinity considered here— Dedekind, analytic, continuum, and higher identity structures— are unified under the same coalgebraic condition: Their transition maps never become invertible. The next section connects this concept with universal coalgebra and Adámek’s chain construction. 9