Observation is a Topos, Not a Type: A Two-Layer Topos-HoTT Framework for Grounding and Conceptual Representation

Observation is a Topos, Not a Type: A Two-Layer Topos-HoTT Framework for Grounding and Conceptual Representation Hironao Nakamura Independent Researcher [email protected]aseda.jp | @hirochu_io December 2025 Abstract Observations and concepts do not exist from the outset as two independent entities. Within a predifferentiated field in which world and self are not yet separated, a structural fluctuation gives rise to a complementary emergence of “the world as observation” and “the self as conceptual structure.” This paper models this generative process using a two-layer mathematical architecture: observations are represented in a presheaf topos 𝐸obs, concepts in a higher topos 𝐸con whose internal logic models Homotopy Type Theory (HoTT), and the grounding geometric morphism 𝑔 ∶ 𝐸obs → 𝐸con captures the structural core of their interaction. The adjunction 𝑔∗⊣ 𝑔∗expresses how prediction (world-to-self) and abstraction (self-to-world) arise as two directions of a unified structure. We introduce three design principles for grounding—Monotonicity, Sufficiency, and Minimality— and formulate the Grounding Representation Principle, which asserts that any coherent observation embeds into the pullback of an appropriate concept type. A finite verification in a 3×3 grid world confirms that this interactional structure operates consistently even in minimal settings. By viewing observations as fields, concepts as types, and consciousness as residing in the relation between them, this framework provides a new structural foundation for symbol grounding, conceptual change, cognitive dynamics, HoTT-based AGI architectures, and future work on the mathematics of consciousness. 1 Introduction We experience the world as observation. Yet the structure of observation itself remains a fundamental problem across artificial intelligence, cognitive science, quantum measurement theory, and consciousness studies. Most contemporary frameworks take it for granted that observations can be treated as “data’ ’—as types, tensors, or vectors fed directly into a model. Observation is assumed to be an input, a typed piece of information, a numerical array delivered to a system. This assumption is rarely questioned. However, observations appear as fields with locality and relational structure, and they occupy a structural layer different from that of concepts. The passage from observation to concept is not merely a function: it is an interpretive process that arises in the interaction between world and self. Under this assumption, several difficulties systematically resist formalization: • observations possess local and relational structure, • observations and concepts inhabit different structural layers, 1 • the mapping from observations to concepts constitutes the formation of meaning, • learning involves changes to the type universe itself. These issues surface concretely in existing work. Harnad’s Symbol Grounding Problem(Harnad 1990) leaves unresolved how symbols become connected to the world, and Potapov and Bogdanov(Potapov and Bogdanov 2022) identify three obstacles—grounded types, dynamic type change, and heuristics—that arise when HoTT is used as an internal language for AGI. Although these challenges appear unrelated, we argue that they share a common underlying assumption. Observation is not a type The central claim of this paper is: The assumption that observations can be represented as types is itself the source of the difficulty. Observations should not be treated as types but as fields of appearances endowed with locality and relational structure. From this perspective, observations are naturally represented as objects of a presheaf topos 𝐸obs. Observation as a topos — toward a two-layer architecture We therefore separate observation and conceptual structure into two mathematical layers: observations as objects of the presheaf topos 𝐸obs, concepts as types in a higher topos 𝐸con whose internal logic models Homotopy Type Theory (HoTT). The two layers are connected by the geometric morphism 𝑔 ∶ 𝐸obs → 𝐸con. The inverse image 𝑔∗expresses prediction from world to concept, while the right adjoint 𝑔∗expresses abstraction from observation to concept. The adjunction 𝑔∗⊣ 𝑔∗provides a unified mathematical account of these two complementary directions. Generative co-emergence of world and self This structure highlights a deeper view. World and self are not pre-existing entities; they arise co-emergently from fluctuations within a more primitive field of potential distinctions. Recognition emerges not inside either layer but in the relation between them: in the morphism that links observational structure to conceptual structure. The two-layer topos architecture provides a mathematical form for this generative complementarity. Contributions of this paper The aims of the present work are as follows: 1. To define a two-layer structure consisting of the observation topos 𝐸obs and the concept topos 𝐸con. 2. To introduce the grounding morphism 𝑔and clarify its structural role. 3. To formulate three design principles for good grounding—Monotonicity, Sufficiency, and Minimality. 2 4. To present the Grounding Representation Principle, formalizing how observations become “embedded’ ’ into appropriate concepts. 5. To illustrate the framework using a finite grid-world example, providing an intuitive picture of the two-layer structure. This paper focuses exclusively on presenting the abstract framework. Applied questions—such as • learning dynamics, • conceptual change (theory change), • heuristics and probabilistic updating, • the resolution of Potapov and Bogdanov’s three obstacles to HoTT-based AGI(Potapov and Bogdanov 2022), • and detailed analyses of consciousness— are left to companion papers. 2 Preliminaries This section summarizes the minimal background required for the framework developed in this paper. For comprehensive treatments of topos theory, higher topos theory, and Homotopy Type Theory (HoTT), see Johnstone(Johnstone 2002), Lurie(Lurie 2009), and the HoTT Book(The Univalent Foundations Program 2013), respectively. Here we restrict attention to the material directly needed in what follows. 2.1 Presheaf Topos Let 𝐶be a finite small category that serves as a site of observations. Objects of 𝐶represent observation points, and morphisms represent relations between them (e.g., adjacency). Using the category Set of sets, we form the presheaf topos 𝐸obs =Set𝐶𝑜𝑝 . Each object 𝐹 ∈ 𝐸obs is a presheaf, assigning to every observation point 𝑐 ∈ 𝐶 a set 𝐹(𝑐) of possible local observations. For the purposes of this paper, we take the canonical (trivial) Grothendieck topology on 𝐶, sufficient for the finite grid-world example. When needed, coherent gluing of such data may be handled through sheafification. 2.2 Homotopy Type Theory and Higher Toposes (Concept Layer) Homotopy Type Theory (HoTT) is a univalent internal logic in which types are interpreted as spaces or homotopical objects. We do not construct an explicit (∞, 1)-topos model of HoTT in this paper. Instead, we assume a highertopos-like conceptual layer whose internal logic behaves HoTT-like, leaving precise model construction to future work. We adopt the following assumptions: • concepts are interpreted as types and the internal logic of 𝐸con behaves in a HoTT-like manner (types = concepts; paths = identifications; univalence holds). 3 Under this view, the observational layer and the conceptual layer are represented by mathematically distinct structures. 2.3 Geometric Morphisms The observation topos 𝐸obs and the concept topos 𝐸con are connected by a geometric morphism 𝑔 ∶ 𝐸obs → 𝐸con. A geometric morphism consists of an adjoint pair 𝑔∗∶ 𝐸con ⇄ 𝐸obs ∶ 𝑔∗, where 𝑔∗is the inverse image functor, a left exact (finite-limit-preserving) functor, and 𝑔∗is its right adjoint. Throughout this paper we interpret these functors cognitively: •𝑔∗(prediction / concretization): for a concept 𝐴,𝑔∗(𝐴) represents the space of observational patterns compatible with 𝐴; •𝑔∗(abstraction / interpretation): for an observation presheaf 𝐹,𝑔∗(𝐹) expresses how 𝐹is abstracted into conceptual structure. The adjunction 𝑔∗⊣ 𝑔∗thus captures the structural duality between prediction and interpretation. 2.4 Finite Distributions To model uncertainty in observations, we use the finite distribution monad 𝐷 ∶ Set →Set, which assigns to each finite set the set of all finite probability distributions on it. Given an observation presheaf 𝐹 ∶ 𝐶𝑜𝑝 →Set, a natural transformation 𝑝 ∶ 𝐹 ⇒ 𝐷 ∘ 𝐹 assigns to each observation point a finite probability distribution over its possible local observations. We do not appeal to general measure theory (e.g., the Giry monad); only finite distributions are used in this paper. 3 The Two-Layer Topos–HoTT Framework This section formalizes the central structure of this paper: atwo-layer epistemic model consisting of • the observation topos 𝐸obs, 4 • the concept topos 𝐸con, and • the grounding geometric morphism 𝑔 ∶ 𝐸obs → 𝐸con which connects the two. The purpose of this decomposition is to avoid forcing observations and concepts into a single level of representation (e.g., a single type universe), and instead to separate them as mathematically distinct structures, placing the essence of cognition in the mappings between these layers. 3.1 Observation Topos 𝐸obs Let 𝐶be a finite small category serving as the site of observations. Objects of 𝐶represent observation points, and morphisms represent relations among them (e.g., adjacency or inclusion). Using the category Set of sets, we define the observation topos 𝐸obs =Set𝐶𝑜𝑝 . Each 𝐹 ∈ 𝐸obs is a field of observations, assigning to each 𝑐 ∈ 𝐶 a set 𝐹 (𝑐) of possible local observations. When necessary, coherent gluing of local data can be handled via sheafification. In this paper we restrict attention to finite presheaf topoi, rather than arbitrary Grothendieck topoi, in order to keep the framework minimal while preserving its essential structure. 3.2 Concept Topos 𝐸con As in Section 2.2, we do not assume a fully constructed (∞, 1)-topos model of HoTT here. Rather, 𝐸con is taken to be a conceptual layer with HoTT-like internal logic, with formal model construction deferred to future work. We adopt the following assumptions: • the internal logic of 𝐸con behaves in a HoTT-like way (types correspond to concepts, paths correspond to identifications, univalence holds); • an object 𝐴 ∈ 𝐸con represents an abstract concept; • morphisms represent structural correspondences or proofs between concepts. This abstraction lets us describe observations and conceptual structure at mathematically distinct layers. 3.3 Grounding Geometric Morphism 𝑔 ∶ 𝐸obs → 𝐸con The two layers are connected by a geometric morphism 𝑔 ∶ 𝐸obs → 𝐸con. A geometric morphism consists of an adjoint pair 𝑔∗∶ 𝐸con ⇄ 𝐸obs ∶ 𝑔∗, 5 where 𝑔∗is the inverse image functor (a left exact, finite-limit-preserving functor) and 𝑔∗is its right adjoint. We interpret these functors through a cognitive metaphor: •𝑔∗(prediction / concretization, top-down): for a concept 𝐴,𝑔∗(𝐴) represents the space of observational patterns compatible with 𝐴. •𝑔∗(abstraction / interpretation, bottom-up): for an observation presheaf 𝐹,𝑔∗(𝐹) expresses how 𝐹is lifted into conceptual structure. The adjunction 𝑔∗⊣ 𝑔∗thus provides a structural account of the duality between prediction and interpretation. 3.4 Grounding Axioms (Design Principles for Good Grounding) Not every geometric morphism adequately captures cognitive grounding. We therefore introduce three design principles that a grounding morphism should satisfy: Axiom 1 (Monotonicity) Refinements of observational data should be appropriately reflected on the conceptual side. Axiom 2 (Sufficiency) If an observation presheaf 𝐹distinguishes certain situations, 𝑔∗should not collapse them inappropriately. Axiom 3 (Minimality) Among all concepts 𝐴capable of explaining an observation 𝐹, 𝑔∗(𝐹) should be the coarsest (most abstract) such concept. This suggests an underlying universal property. These axioms are not formal logical axioms but design principles that guide later formalization. 3.5 Summary In this section we introduced: • the observation topos 𝐸obs, • the concept topos 𝐸con, • the grounding morphism 𝑔 ∶ 𝐸obs → 𝐸con, and • the three Grounding Axioms. Together, these constitute the Two-Layer Topos–HoTT Framework. The next section states the central structural principle of this paper: the Grounding Representation Principle. 6 4 Grounding Representation Principle This section introduces the central structural principle of the two-layer topos–HoTT framework developed above: the Grounding Representation Principle. The principle asserts that observations in the observation topos are structurally embedded (represented) within suitable concept types in the concept topos. 4.1 Grounding Representation Principle Let 𝐸obs and 𝐸con be the observation and concept toposes, and let 𝑔 ∶ 𝐸obs → 𝐸con be the grounding morphism. Assume that 𝑔satisfies the Grounding Axioms (Monotonicity, Sufficiency, Minimality). We propose the following structural principle: Grounding Representation Principle. For any observation presheaf 𝐹 ∈ 𝐸obs, there exists a concept type 𝐴𝐹∈ 𝐸con such that Sh(𝐹) ↪ 𝑔∗(𝐴𝐹) as a monomorphism. Here: • Sh(𝐹) denotes the sheafification of 𝐹, representing the space of globally coherent observations; •𝑔∗(𝐴𝐹)is the inverse image of the concept 𝐴𝐹, describing all observational patterns compatible with that concept. Thus, informally: The observation 𝐹appears as a coherent substructure of the pullback of an appropriate concept type 𝐴𝐹. This expresses the idea that concepts explain observations in a structural sense. 4.2 Intuitive Interpretation This conjecture captures the relationship between observations and concepts from the perspective that “a concept explains the world.” •𝑔∗(𝐴𝐹)denotes the entire observational space permitted by the concept 𝐴𝐹; • Sh(𝐹) represents the actually obtained observation (the globally consistent data derived from 𝐹); • the monomorphism Sh(𝐹) ↪ 𝑔∗(𝐴𝐹) states that the observed configuration fits within the explanatory space determined by 𝐴𝐹. This is analogous to model selection in cognitive science: An observation lies within the range of what a concept can explain. The principle therefore provides a structural response to the Symbol Grounding Problem: it characterizes how a concept receives and organizes observational information. 7 4.3 Stronger Form: Universality (Future Work) A stronger version of the conjecture posits a universal property: Strong Form of the Principle. The concept type 𝐴𝐹satisfying Sh(𝐹) ↪ 𝑔∗(𝐴𝐹) is the coarsest (most abstract) among all such concepts. This reflects Axiom 3 (Minimality) and resembles universal constructions in categorical semantics, such as initial objects or universal solutions. We do not pursue a formal development or proof of the strong form here; its exploration is deferred to future work. 4.4 Summary In this section, we introduced the Grounding Representation Principle, the central structural claim relating observations and concepts: • observations embed into the observational space predicted by suitable concept types; • the grounding morphism ensures this alignment; • the strong form hints at a universal characterization of concept selection. The next section presents a concrete finite example—the grid world model— to visualize how the framework operates in practice. 5 Finite Grid World: A Toy Example This section illustrates how the two-layer topos–HoTT framework operates in practice by analyzing a minimal finite world. The example allows us to observe, in the simplest possible setting, the behavior of • the presheaf topos, • concept types, •𝑔∗(prediction), •𝑔∗(abstraction), and • the intuitive content of the Grounding Representation Principle. 5.1 Observation Site 𝐶: The 3×3 Grid Let the set of observation points be the 3×3grid 𝐶 = {(𝑖, 𝑗) ∣ 𝑖, 𝑗 ∈ {1, 2, 3}}. Morphisms consist of adjacency relations (up, down, left, right) together with identity morphisms. The observation topos 𝐸obs =Set𝐶𝑜𝑝 is then the category of presheaves over this grid. 8 5.2 Observation Presheaf 𝐹and Local Probability Distributions For each observation point (𝑖, 𝑗), assume that one of three colors is observed: 𝐹(𝑖, 𝑗) = {red,green,blue}. A finite probability distribution 𝑝(𝑖,𝑗) ∶ 𝐹(𝑖, 𝑗) → 𝐷(𝐹(𝑖, 𝑗)) may additionally be assigned, allowing us to express uncertainty such as “the probability that this cell is red is 0.7, green is 0.2, etc.” The sheafification Sh(𝐹) yields the set of globally coherent colorings obtained by gluing locally consistent observations. 5.3 Examples of Concept Types in 𝐸con Within the concept topos 𝐸con, we may consider concept types such as: •𝐴redRegion: the concept “there exists a connected red region,” •𝐴borderGreen: the concept “the boundary cells are mostly green.” We do not formalize the full HoTT-internal semantics of these concepts, but adopt the viewpoint that concepts are types, whose inhabitants and paths encode conceptual content and identifications. 5.4 Behavior of 𝑔∗and 𝑔∗(Prediction and Abstraction) The grounding morphism 𝑔 ∶ 𝐸obs → 𝐸con comes equipped with the adjunction 𝑔∗⊣ 𝑔∗. (1) Prediction: 𝑔∗(𝐴) For the concept 𝐴redRegion, 𝑔∗(𝐴redRegion) denotes the set of observational patterns in which a red region appears— precisely the patterns expected under that concept. (2) Abstraction: 𝑔∗(Sh(𝐹)) Given an actual observation Sh(𝐹), 𝑔∗(Sh(𝐹)) returns the concept that best explains the observed pattern. For example, if a red region appears with high probability, the observation may be classified under 𝐴redRegion. 9