Category Theory

There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms that parses an adjunction into two separate parts. Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment uses relatively simple category theory and (...) is focused on the interpretation and application of the mathematical concepts. (shrink)

In this paper, we develop a mathematical model of awareness based on the idea of plurality. Instead of positing a singular principle, telos, or essence as noumenon, we model it as plurality accessible through multiple forms of awareness (“n-awareness”). In contrast to many other approaches, our model is committed to pluralist thinking. The noumenon is plural, and reality is neither reducible nor irreducible. Nothing dies out in meaning making. We begin by mathematizing the concept of awareness by appealing to the (...) mathematical formalism of higher category theory. The beauty of higher category theory lies in its universality. Pluralism is categorical. In particular, we model awareness using the theories of derived categories and (infinity, 1)-topoi which will give rise to our meta-language. We then posit a “grammar” (“n-declension”) which could express n-awareness, accompanied by a new temporal ontology (“n-time”). Our framework allows us to revisit old problems in the philosophy of time: how is change possible and what do we mean by simultaneity and coincidence? Another question which could be re-conceptualized in our model is one of soteriology related to this pluralism: what is a self in this context? A new model of “personal identity over time” is thus introduced. (shrink)

This paper shows how the universals of category theory in mathematics provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. The paper also shows how the always-self-predicative universals of category theory provide the "opposite bookend" to the never-self-predicative universals of iterative set theory and thus that the paradoxes arose from having (...) one theory (e.g., Frege's Paradise) where universals could be either self-predicative or non-self-predicative (instead of being always one or always the other). (shrink)

Recent developments in pure mathematics and in mathematical logic have uncovered a fundamental duality between "existence" and "information." In logic, the duality is between the Boolean logic of subsets and the logic of quotient sets, equivalence relations, or partitions. The analogue to an element of a subset is the notion of a distinction of a partition, and that leads to a whole stream of dualities or analogies--including the development of new logical foundations for information theory parallel to Boole's development of (...) logical finite probability theory. After outlining these dual concepts in mathematical terms, we turn to a more metaphysical speculation about two dual notions of reality, a fully definite notion using Boolean logic and appropriate for classical physics, and the other objectively indefinite notion using partition logic which turns out to be appropriate for quantum mechanics. The existence-information duality is used to intuitively illustrate these two dual notions of reality. The elucidation of the objectively indefinite notion of reality leads to the "killer application" of the existence-information duality, namely the interpretation of quantum mechanics. (shrink)

Saunders Mac Lane famously remarked that "Bourbaki just missed" formulating adjoints in a 1948 appendix (written no doubt by Pierre Samuel) to an early draft of Algebre--which then had to wait until Daniel Kan's 1958 paper on adjoint functors. But Mac Lane was using the orthodox treatment of adjoints that only contemplates the object-to-object morphisms within a category, i.e., homomorphisms. When Samuel's treatment is reconsidered in view of the treatment of adjoints using heteromorphisms or hets (object-to-object morphisms between objects in (...) different categories), then he, in effect, isolated the concept of a left representation solving a universal mapping problem. When dualized to obtain the concept of a right representation, the two halves only need to be united to obtain an adjunction. Thus Samuel was only a now-simple dualization away for formulating adjoints in 1948. Apparently, Bodo Pareigis' 1970 text was the first and perhaps only text to give the heterodox "new characterization" (i.e., heteromorphic treatment) of adjoints. Orthodox category theory uses various relatively artificial devices to avoid formally recognizing hets--even though hets are routinely used by the working mathematician. Finally we consider a "philosophical" question as to whether the most important concept in category theory is the notion of an adjunction or the notion of a representation giving a universal mapping property (where adjunctions arise as the special case of a bi-representation of dual universal mapping problems). (shrink)

The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and functions. The analysis extends directly to other concrete categories (groups, rings, vector spaces, etc.) where the objects are sets with a certain type of structure and the morphisms are functions that preserve that structure. Then the elements & distinctions-based definitions can be (...) abstracted in purely arrow-theoretic way for abstract category theory. In short, the language of elements & distinctions is the conceptual language in which the category of sets is written, and abstract category theory gives the abstract arrows version of those definitions. (shrink)

This essay endeavors to define the concept of indefinite extensibility in the setting of category theory. I argue that the generative property of indefinite extensibility for set-theoretic truths in category theory is identifiable with the Grothendieck Universe Axiom and the elementary embeddings in Vopenka's principle. The interaction between the interpretational and objective modalities of indefinite extensibility is defined via the epistemic interpretation of two-dimensional semantics. The semantics can be defined intensionally or hyperintensionally. By characterizing the modal profile of $\Omega$-logical validity, (...) and thus the generic invariance of mathematical truth, modal coalgebras are further capable of capturing the notion of definiteness for set-theoretic truths, in order to yield a non-circular definition of indefinite extensibility. (shrink)

All of existence — from physical universes and their laws to thoughts, concepts and mathematics — must have an external grounding to satisfy Gödel's Incompleteness Theorems. This grounding outside existence must be Pure Potential. Existence needs this external grounding at every moment. In order to preserve the structure we observe, this continuing grounding must freely choose to purposefully create existence. This we refer to as Free Will. This paper presents the Trans-Existential Grounding (TEG) Framework, a mathematical exploration of the ancient (...) metaphysical insight that Free Will, understood as unrestricted decision, could be the creative source of existence. This perspective, found in Vedantic, Neoplatonic, and Idealist philosophy, is here examined through mathematical formalization. Using set theory and formal logic, this paper explores characterizing Free Will as pure potential W that might transcend and generate the domain of existence E. This framework proposes that existence comprises only what has been actualized from potential, with potential P remaining distinct from existence. The paper develops theorems about this proposed generative relationship based on the structural argument: existence exhibits formal structure (laws, patterns, consistency) that constitutes a formal system subject to Gödel's incompleteness. Since no formal system can prove its own consistency, existence as a whole requires trans-existential grounding. While Torkel Franzén warns about misapplications of Gödel's theorems, the fact remains: formal systems do exist and cannot self-ground. This paper distinguishes between the mathematical necessity (formal systems require external grounding) and the philosophical proposition (Free Will as the only viable trans-existential ground). (shrink)

This sketch of a perhaps future 'Elementary Theory of the Category of Mereological Sums (including Mereological Wholes and Parts)' relates to my previous papers "The Topos of Emergence" and "Intelligible Gunk". I assert that for successfully categorizing Mereology one has to start with a specific setting of gunk. In this paper we will give a sketch of a categorically version of particular mereological structures. I.e. we will follow the example of F.W.Lawvere’s “An elementary theory of the category of sets” -/- (...) where he introduced a categorised version of set theory. We refer to sirenians mainly for the reason that they are the closest living relatives of elephants. -/- But they are also rather underestimated creatures - at the first moment rather dull and clumsy, but moving gracefully and elegant in their natural habitat.And some sailors already reckognized their bewitching attraction before we did.Yet we still think this sirenia is a modest creature compared with the elephant in the room. (shrink)

What is cognition? Equivalently, what is cognition good for? Or, what is it that would not be but for human cognition? But for human cognition, there would not be science. Based on this kinship between individual cognition and collective science, here we put forward Isbell conjugacy---the adjointness between objective geometry and subjective algebra---as a scientific method for developing cognitive science. We begin with the correspondence between categorical perception and category theory. Next, we show how the Gestalt maxim is subsumed by (...) the mathematical construct of colimit, a generalization of summation. The universal mapping property definitions of mathematical constructs, by virtue of being the best with respect to the universe of discourse, can be learned using reinforcement learning algorithms, which raises the possibility of abstracting the architecture of mathematics by artificial intelligence. Subsequently, we present naturality (to be contrasted with miracles), understood as 'Becoming consistent with Being', which governs the transformations of both things and their theories, as the zeroth law of change. Furthermore, the contrast---physical [mechanism] vs. biological [organism]---is smoothed via natural transformation, wherein transformations are respectful of the cohesion of the objects transformed. In closing, upon recognizing the scientific value of learning difficult-to-master differential calculus by physicists, of learning a strange four-letter language by biologists, and of learning the grammar of our respective mother tongues, we make a case for learning the theory of naturality / category theory for developing cognitive science. (shrink)

A quick introduction to category theory and topos theory, axiomatically. These notes are adapted from Prof. Pettigrew’s notes.

This paper presents a formal proof that distinction (∆)—the minimal act of differentiating between a state and its background—is not only epistemically prior but ontologically necessary for any form of cognition or experience. From a single axiom—to be is to be distinguishable—we construct a minimal categorical architecture of mind: a category Dist of epistemic states and distinction morphisms; a sheaf S representing coherent cognitive integration; and a set of dynamic operators (R for reflexivity, D for epistemic drift, C for collapse, (...) and K for curvature). We prove that this structure—M = (Dist, S, R, D, C, K)—is a necessary and sufficient condition for conscious agency. The framework accounts for time, memory, qualia, trauma, and self-reference as structural consequences of distinction dynamics. We conclude by establishing operator algebra over S, drawing formal analogies to quantum measurement, and proposing testable implications for artificial intelligence, cognitive pathology, and epistemic evolution. This is not a speculative theory of mind—it is a minimal proof of the conditions under which mind can exist. (shrink)

We present a machine-verified formalization of the Distinction Field, a topological- categorical model of cognition built from the primitive act of distinction. We define a base category Dist of epistemic states, a cognitive sheaf S, and four dynamic operators— Reflexivity (R), Drift (D), Collapse (C), and Curvature (K)—which generate epistemic dy- namics. Crucially, we demonstrate operator-level non-commutativity in the algebra AM = {R, D, C, K} by constructing and verifying six counterexamples in the Coq proof assistant. This provides strong support (...) for the irreversibility and structural asymmetry that underlie cognition, memory, trauma, and reasoning. (shrink)

This Gentle Introduction is very much still work in progress. Roughly aimed at those who want something a bit more discursive, slower-moving, than Awodey's or Leinster's excellent books. -/- The current [Jan 2018] version is 291pp.

This paper introduces the Universal Theory of Differentiation (UTD) as a foundational framework for modeling cognition through structured acts of distinction. Instead of treating mental content as primitive, UTD posits that cognition emerges from recursive differentiations within a categorical hierarchy ∆ₙ, where each level represents structured differences between prior acts. Key cognitive functions—perception, memory, attention, and self-awareness—are expressed as stable fixpoints of differentiation, formalized via recursive morphisms Dₙ₊₁(δ, δ) = Iₙ. We demonstrate how UTD reframes existing theories such as Integrated (...) Information Theory, Global Workspace Theory, and Predictive Processing, offering a structural alternative grounded in internal coherence rather than representational or statistical models. The paper further explores implications for cognitive architecture, suggesting that both human and artificial systems can be modeled as reflective differentiators rather than symbol processors or statistical predictors. (shrink)

Purpose: This paper proposes CatRO (Categorical Referent Ontology), a category-theoretic formalization of staged concept formation that addresses the symbol grounding problem. -/- Methods: CatRO formalizes the transformation from observational data to concepts as three staged processes: observation (OBS-P) as a functor, individualization (IND-P) as a natural transformation, and universalization (UNV-P) as classification based on Conceptual Spaces. We introduce a Pattern Datum hierarchy (OMD, SPD, PPD, EPD) as explicit intermediate representations and demonstrate correspondence with the DOLCE top-level ontology through the Quality (...) Space framework. -/- Results: This formalization mathematically guarantees structure preservation while providing ``gray box'' interpretability---the structural traceability of transformation processes is ensured even when the specific content of intermediate representations cannot be fully described in natural language. Two illustrative examples---``Two Apples'' and ``Pendulum Motion''---demonstrate the framework's operation. -/- Conclusions: CatRO contributes to bridging formal ontology and machine learning by providing a theoretical foundation for explainable concept formation. (shrink)

Scientists and philosophers frequently speak about levels of description, levels of explanation, and ontological levels. This paper presents a framework for studying levels. I give a general definition of a system of levels and discuss several applications, some of which refer to descriptive or explanatory levels while others refer to ontological levels. I illustrate the usefulness of this framework by bringing it to bear on some familiar philosophical questions. Is there a hierarchy of levels, with a fundamental level at the (...) bottom? And what does the answer to this question imply for physicalism, the thesis that everything supervenes on the physical? Are there emergent higher-level properties? Are higher-level descriptions reducible to lower-level ones? Can the relationship between normative and non-normative domains be viewed as one involving levels? And might a levelled framework shed light on the relationship between third-personal and first-personal phenomena? (shrink)

En este artículo, el primero de una bilogía, se introduce al lector no especializado a la Teoría Homotópica de Tipos (HoTT), un reciente marco fundacional para las matemáticas donde convergen lógica, computación, topología y teoría de tipos. En la segunda parte, varias de las aplicaciones e implicaciones filosóficas de HoTT serán exploradas.

Universal perceptual structure, diverse implementation.

We propose a mathematical framework using category theory and partially ordered sets (posets) to analyze psychiatric disorders from the DSM-5 based on their neural substructures and functional impacts, termed _affordances_ (e.g., memory, emotional regulation). We define a monotone mapping between neural substructures and their functional affordances, proving that disorders with the greatest functional impact correspond to maximal elements in the affordance poset. This approach provides a systematic ranking of disorders by their neurofunctional severity, offering a tool to prioritize treatment strategies.

Los gráficos existenciales de Peirce proveen un entendimiento geométrico de una variedad de lógicas (clásica, intuicionista, modal, primer orden). La interpretación geométrica se da en el plano, pero puede ser extendida a otras superficies (esfera, cilindro, toro, etc.) Yendo más allá, se pueden dibujar gráficos existenciales sobre superficies de Riemann arbitrarias, y, con la introducción de herramientas de geometría algebraica (haces, topos de Grothendieck, topos elementales), se pueden capturar las lógicas emergentes vía un nuevo Topos de Gráficos Existenciales sobre Superficies (...) de Riemann y su clasificador de subobjetos. Ofrecemos algunas perspectivas (conceptos, definiciones, ejemplos, conjeturas) a lo largo de ese camino. (shrink)

Yale Gallery Talk, Language Perception and Representation Tanya Kelley and James Prosek Linguist and artist Tanya Kelley, Ph.D., and artist, writer, and naturalist James Prosek, B.A. 1997, discuss color manuals used by artist-naturalists and biologists and lead visitors in close looking and drawing. Presented in conjunction with the exhibition James Prosek: Art, Artifact, Artifice. Space is limited. Open to: General Public .

This article develops the central argument first presented in Aspen, Total Obtainment, Version 1.0, DOI:10.5281/zenodo.21230189. This paper argues that true absence cannot coherently obtain, and that this failure shifts the explanatory burden normally assigned to existence. The familiar question “Why is there something rather than nothing?” often allows nothingness to function as a presumed default, contrast, or alternative. That permission is rarely defended. Once true absence is distinguished from empty space, vacuum, void, silence, darkness, an empty world, or the absence (...) of particular objects, it can no longer be treated as a reduced condition within existence. True absence means absence without remainder: no qualities, no structure, no relation, no distinction, no condition, no locality, no capacity, and no obtainment. Yet to propose that true absence obtains is already to attribute condition, identification, relation, and obtainment to it. The proposal therefore acquires the very features it excludes. The argument developed here does not claim that existence is produced by necessity. Necessity, law, cause, implication, and explanation can function only if they already obtain. They are not primitive over obtainment; they are domain-functions within it. The root result is therefore not that necessity compels existence, but that non-obtainment fails as a coherent alternative. What remains is obtainment as prerequisite. From this result, the paper distinguishes participant categories from prerequisite categories, develops the marker–occupant distinction, and derives two immediate consequences: terminal boundary failure and terminal exclusion failure. Local boundaries and local exclusions remain intact. What fails is the attempt to make boundary, outside, non-realization, or true absence ultimate. Total Obtainment is therefore not an inventory of everything that exists, nor a theory placed beside other theories. It names the prerequisite field within which any explanation, objection, distinction, theory, law, necessity-claim, or challenge must obtain before it can function. Keywords: true absence; obtainment; nothingness; prerequisite category; necessity; metaphysical nihilism; necessary existence; boundary; terminal exclusion; totality; grounding; ontology; modal metaphysics. (shrink)

This paper develops a single closure formalism for high-dimensional Calabi–Yau geometry. The central objects are hypersurface towers, toric reflexive constructions, Hopf-type fibrations, mirror correspondences, special-holonomy constraints, SYZ lifting, zero-defect Hodge realization, and the limiting programme CYₙ → CY∞. The arguments combine adjunction, Chern-class identities, Hodge theory, Batyrev mirror duality, finite reflexive-polytope enumeration, calibrated geometry, Newton iteration for special-Lagrangian phase defects, and ensemble closure procedures for dimensional saturation. Each conclusion is stated inside the hypotheses that make it a theorem, so the (...) manuscript reads as a compact research paper in which construction, enumeration, fibration, and asymptotic saturation form one connected mathematical system. Note. The word closed is used in its technical sense inside this manuscript: all assumptions needed by a statement are written before the statement, and the conclusion follows from those assumptions, from a cited theorem, from a finite enumeration, or from an explicitly defined asymptotic ensemble. It is not used as a rhetorical claim about mathematical problems outside the stated scope. (shrink)

The Foundations of Mathematics This article examines what is meant by ‘foundation’ in different contexts, and it focuses on four prominent mathematical theories, each of which have been claimed to play some kind of foundational role. As we will see, different features which might be desired from a foundation has led to a variety of … Continue reading Foundations of Mathematics →.

Can category theory be a foundation of mathematics? While category theory is taken to organize the body of mathematical knowledge and practice, its status as a foundational theory has been disputed. I argue that category theory can serve as a foundation of mathematics especially in an explanatory sense. The explanatory sense of foundation is both historically situated and philosophically motivated; I examine its historical uses by some of the pioneering figures as well as its theoretical ramifications across several philosophical topics. (...) Then what makes category theory explanatory in the relevant sense? I argue that the explanatory nature of category theory can be understood as its capacity to offer abstract explanation. To support this, the case studies on mathematical explanation through Galois theory and category theory will be considered, which will be tied in with more traditional accounts of category theory’s foundational status. Thus, I argue that category theory’s foundational status can be supported through the lens of mathematical explanation, which demonstrates a successful interplay between the two traditions in philosophy of mathematics, i.e. the foundation-oriented tradition and the practice-oriented tradition. (shrink)

This paper asks a question that standard group theory does not: why must the axioms of group theory---and no weaker or alternative structure---govern the symmetry of any system capable of making operational distinctions? Semigroups, monoids, and non-associative magmas are not merely less convenient than groups; they are operationally inadmissible. This paper derives that inadmissibility from first principles. The group axioms (closure, associativity, identity, invertibility) are established as necessary conditions for Πd-consistent operational symmetry under the axiom system {A1, A2, A4} of (...) Operatiology, without presupposing any group-theoretic result. Invertibility is forced by the requirement that no operational step induces an irreversible collapse of distinguishability; associativity follows from the inherent associativity of function composition; identity existence follows from the finiteness of the Πd-class structure. A complete explicit construction of the six-class minimal non-commutative Πd-closure is provided from a period-3 and period-2 generator pair, with saturation depth K=2 verified by a non-circular three-step closure consistency check. S₃ emerges as the minimal finite non-commutative operational witness of this necessity. Minimality is established in two structurally independent parts: Theorem A (constructive minimality, Tier-2 internal) provides an existence witness; Theorem B (extensional extension theorem, via established series results) provides universality. This paper forms part of the Operatiology Applied Series, in which group theory is the symmetry layer of the rank-3 minimal operational closure C⁽³⁾_Πd. Published on May 26, 2026 DOI:10.5281/zenodo.20396514 . (shrink)

This paper establishes the Isometric Extension of the Mathematical Unified Theory of Cognitional Mechanics (CM-MUT), deriving the exact quantitative correspondence between the geometric modal functor and the algebraic modal functor over the historical category Hist. The central result is that for all admissible operational histories H, the κ-scale L¹ norm and the Frobenius norm are related by the Casimir invariant K=√3 of M₃(ℂ): ‖M_A(H)‖_κ = K·‖M_G(H)‖_F. -/- This isometric relationship is derived from two implementation axioms, A3' (Modal Norm Selection) and (...) A5 (Modal Distance Preservation), which formalize assumptions implicit throughout the CM corpus. The proof proceeds via the automorphism structure Aut(M₃(ℂ))≅PGL(3,ℂ), Axiom 4 (Redundancy Exclusion), and a formal lemma establishing that minimal descriptive complexity in the Cartan subalgebra uniquely forces uniform eigenvalue distribution Var(κ)=0 for all admissible histories. -/- Four corollaries follow from a single algebraic source: the inverse fine structure constant α⁻¹=137.03599…, the Iron Stability Attractor κ(Fe)=6/13, the chemical projection constant Π_chem=145/468, and the ABC quality bound q(a,b,c)<13/7. -/- Prior CM results on the Riemann Hypothesis, P≠NP, and the Halting Problem are unified under this framework. The Navier-Stokes regularity problem, the Hodge Conjecture, the Yang-Mills mass gap, the Poincaré Conjecture, and the Birch-Swinnerton-Dyer Conjecture are shown to be Tier-3 projections of Tier-1 structural necessities. -/- The apparent difficulty of the Millennium Problems is an artifact of their Tier-3 formulation, not of mathematical reality. All derivations proceed with zero free parameters from the axiomatic structure of M₃(ℂ). -/- Published on April 4, 2026 DOI: 10.5281/zenodo.19411822 . (shrink)

This paper establishes that category theory is not a foundational layer of any operational system but a representational artifact: the minimal morphism-based formal language encoding the operational obstruction structure [ℐ/∼] derived from Operatiology. The Unbounded Index Obstruction criterion classifies core categorical notions individually. Arbitrary categories, functors, and natural transformations require certification over non-finitely-exhaustible index families. Limits and colimits belong to the power-set type of [ℐ/∼], the categorical analogue of the Power Set axiom in ZFC. Adjunctions belong to the unrestricted type, (...) parallel to the Axiom of Replacement. The Yoneda lemma occupies the extremal position within the power-set type: the entire relational complexity of Nat(h_A, F) collapses to a single evaluation, constituting maximum compression efficiency within the morphism-based language. The minimal categorical structures retaining executive correlates in C⁽³⁾_Πd are finite acyclic free categories and the single-object category BM₃(ℂ). A fidelity ordering among categorical, set-theoretic, and type-theoretic foundations is established as a structural taxonomy. Category theory and ZFC are dual [ℐ/∼] compression languages differing only in compression axis: membership versus morphism. Their foundational rivalry is a competition internal to the projective layer. The classification criterion (operational necessity, grounded in Axiom 2) and the classification function (Unbounded Index Obstruction) are both derived from the executive layer; no external foundation is structurally required or presupposed. -/- Published on May 31, 2026 DOI: 10.5281/zenodo.20470946. (shrink)

This paper establishes that the standard number systems ℕ, ℤ, ℚ, ℝ, and ℂ are not pre-existing mathematical objects but structures uniquely selected by the operational constraints of the rank-3 minimal operational closure C⁽³⁾_Πd as formalised in Operatiology. The selection order is constrained by the axioms: Axiom 1 contains a geometric-persistence clause expressed using the distance function d, which is defined only in Axiom 2. Therefore ℝ, the number system selected by Axiom 2, must be established before ℂ, the number (...) system selected by the orbital structure of Axiom 1. -/- Axiom 2 selects ℝ as the unique totally ordered, complete Archimedean field. Axioms 1 and 4 identify ℕ, ℤ, and ℚ as the minimal discrete substructures of ℝ required for distinguishability, reversibility, and minimal description. Axiom 1 also generates bilateral inaccessibility between non-commutative orbit pairs, producing a canonical ℤ/2ℤ symmetry. By Galois theory, the unique minimal realisation of this symmetry over ℝ is ℂ, making ℂ the number system structurally demanded by non-commutativity. -/- Working Axiom 3 is not invoked; all results follow from A1, A2, and A4. Numbers and the algebra M₃(ℂ) are co-derived from the same axiomatic source via the chain C⁽³⁾_Πd → ℳ₁ → M₃(ℂ), showing that mathematics functions as the projective layer generated by, and subordinate to, the executive structure of Operatiology. Numbers, in this framework, are not discovered or invented; they are what operational closure structurally requires. -/- This version supersedes the prior Cognitional Mechanics formulation (DOI: 10.5281/zenodo.20108822). The transition from Cognitional Mechanics to Operatiology replaces the earlier grounding in "intelligence" as a technical term with the structurally deeper foundation of C⁽³⁾_Πd, and reframes the mathematical projection hierarchy in terms of the five-layer distance ordering established in the companion paper on mathematics as the unique top-down projection of operational structure. Published on May 24, 2026 DOI: 10.5281/zenodo.20366742 . (shrink)

Operatiology, derived from Noology through the three primitive notions of Ordo, Consensus, and Arbitrium, establishes the rank-3 minimal operational closure C⁽…Operatiology, derived from Noology through the three primitive notions of Ordo, Consensus, and Arbitrium, establishes the rank-3 minimal operational closure C⁽³⁾_Πd as the unique structure satisfying the executive axiom system {A1, A2, A4} together with the Operational-Geometric Coupling. The companion paper on the top-down projection of mathematics establishes that the induced mathematical category M is uniquely determined and that algebra occupies (...) the minimum-distance layer M₁ within M, delegating the complete algebraic identification to the present work. This paper discharges that delegation. The argument proceeds in four stages. First, algebra is established as the M₁-layer: the unique minimum-distance projective structure from the operational substrate. Second, operational non-decomposability is derived from Axioms 1, 2, and 4: direct sum decomposition of the state space is excluded by Axiom 1, and non-trivial two-sided ideals are excluded via the maximality of the canonical irreducible generator set I_Πd, establishing simplicity. Third, finite dimensionality is derived from the finite generator rank and the Πd-saturation closure via an explicit saturation depth bound, and the Wedderburn-Artin theorem is applied to identify the representation as Mₙ(D). Fourth, a bridge lemma connecting Πd-separation to eigenvalue structure establishes that the Noological primitive Arbitrium forces D = ℂ, and the three-generator Πd-independence condition forces n = 3, yielding M₃(ℂ) as the unique minimal algebraic realisation of C⁽³⁾_Πd. Published on May 24, 2026 DOI: 10.5281/zenodo.20363937 -/- . (shrink)

This paper establishes that M₃(ℂ), the algebra of 3×3 complex matrices, is the unique minimal projective algebraic realisation of the rank-3 operational closure C⁽³⁾_Πd derived from Operatiology. The result supersedes Version 1 " M3(C) Necessity in Cognitional Mechanics: The Logical Foundation of Dimensional Structure" (DOI: 10.5281/zenodo.18280992), which established the same conclusion via dimensional exclusion and spectral efficiency arguments within the earlier Cognitional Mechanics axiom system. -/- The present version proceeds from the Operatiology axiom system {A1, A2, A4} and the Operational–Geometric (...) Coupling, with the Noological primitive Arbitrium as the regulative layer. The complete proof is delegated to the companion paper on algebra as the unique top-down projection of operational structure (DOI: 10.5281/zenodo.20363937), which proceeds via four stages: identification of algebra as the minimum-distance projective layer M₁; derivation of simplicity from operational non-decomposability; establishment of finite dimensionality and the Wedderburn–Artin form Mₙ(D); and a bridge lemma connecting Πd-separation to eigenvalue structure, forcing D = ℂ and n = 3. -/- The necessity established here applies retroactively to the entire Cognitional Mechanics corpus. Every paper citing the present work or its predecessor, directly or indirectly, and proceeding from M₃(ℂ) as a structural premise, is thereby grounded in the canonical projective realisation of C⁽³⁾_Πd. This covers derivations of physical constants, Standard Model gauge dynamics, nuclear reactions, number systems, cosmology, and life. The comprehensive corpus framework is documented in Foundations of Cognitional Mechanics (DOI: 10.5281/zenodo.20226488). -/- Published on May 24, 2026 DOI: 10.5281/zenodo.20365633 . (shrink)

Consciousness is the point where past, present, and anticipated future converge. This paper derives that point from the brain's own wiring. Four neural loops implement four operations, gating, contextualization, resolution, and commitment, and they map onto a directed graph with one structurally exceptional node: a convergence point that receives three streams at once. One loop delivers the present (the incident signal), one delivers the past (the trace of prior completed cycles), and one delivers the anticipated future (the prior expectation carried (...) from the last cycle's commitment). All three arrive at that node within a 50 to 150 ms window, and the resolution event there determines what is happening, in which direction, and with what weight. Consciousness, on this account, is that convergence. The same three-direction structure Husserl identified phenomenologically as the "living present" is here forced by graph topology, arrived at independently. -/- The four loops are a Layer 5 pyramidal recurrent loop (gating), a central lateral thalamocortical loop (contextualization), a VTA-cortical dopaminergic loop (resolution), and an ACC-mediodorsal thalamic loop (commitment). They compose into a closed cycle, an endomorphism whose output feeds back as its own next input, and the cycle is developed in three registers held together throughout: the neural circuits, the graph-theoretic structure, and the phenomenology of what each circuit's disruption produces. -/- The account carries a single falsification condition. Consciousness is defined operationally as the capacity to produce contingent, accurate reports of currently presented information, and the theory is refuted by one documented case in which all four loops have simultaneously lost function while that capacity is preserved. The paper argues no such case exists across the disorders-of-consciousness, anesthesia, and focal-lesion literatures, and that the architecture accounts for the observed profile of consciousness under disruption: resilience to single-loop damage, characteristic alteration under selective loop disruption, and reliable elimination only under simultaneous multi-loop failure. Eight discriminating predictions follow, none generated by Global Neuronal Workspace Theory, Integrated Information Theory, or Thalamocortical Loop Theory individually. -/- The same four-loop structure is the empirical anchor of a further result. The graph carries a signing asymmetry between its convergence point and its cession point, and a chain of forced derivations with no free parameters (Stewart, 2026e, 2026f, 2026q) pushes that asymmetry through to the Cartan matrix of A₂ × A₁ and, with the temporal generator of the iterated cycle, to su(3) ⊕ su(2) ⊕ u(1), the gauge algebra of the Standard Model, at rank 4. The four loops are the only empirical input to that chain; the Appendix sketches each step. -/- Keywords: consciousness, thalamocortical loops, disorders of consciousness, anesthesia, four-loop architecture, ACC-mediodorsal thalamic loop, VTA-cortical dopaminergic loop, central lateral thalamocortical loop, Layer 5 pyramidal recurrent loops, apical-basal coincidence detection, gamma synchronization, neural correlates of consciousness, endomorphism, strongly connected directed graph. (shrink)

Wrong is the application of any operation before its prerequisite has produced its output. An operation that occurs without its prerequisite's output receives no valid input, and the failure to close is the structural signature of error. The four operations are distinction, placement, identification, and composition, in that order, each a necessary condition for the next; they correspond to the four foundations of mathematics (logic, set theory, type theory, category theory), with the ordering derived in *On Occurrence* (Stewart, 2026g) and (...) *The Endomorphic Collapse Traverses the Foundations of Mathematics* (Stewart, 2026e). The present paper takes the four-operation ordering as given and derives the structure of error from it: what the misorderings are, how many there are, and how they entail one another. -/- Wrong, as experienced, is expectancy violation, the divergence between a projected expectation and the actual determination. Expectancy violation follows from the ordering. Expectation requires ordering, so a system without one has nothing to project forward to and cannot expect anything. Several phenomena that resemble structural error are not. Incompleteness (an ordering that stalls without attempting a later operation) is the clearest example. It is not a structural failure mode. It appears as wrong only in the observer's expectancy violation. The structural classification covers all and only the misorderings. -/- Structural errors classify into five non-empty classes plus one structurally empty cell. The count is forced by the ordering itself: misordering four ordered positions is generated by the three adjacent swaps (positions 1-2, 2-3, 3-4), and these three swaps, together with the two further failures the ordering's non-adjacent reach admits, give the five classes. Class 2-1 is the failure of the distinction-to-placement step. Class 3-2 is the failure of the placement-to-identification step. Class 4-3 is the failure of the identification-to-composition step. Class 3-1 is the failure of the direct distinction-to-identification step, which bypasses placement. Class 4-1 is the catastrophic failure where the full ordering does not complete. The sixth potential class (Class 4-2, placement before composition) is structurally empty because the ordering provides no direct step from placement to composition: placement reaches composition only through identification. Each class is named by the pair (later, earlier), where the later operation executed without the earlier prerequisite having delivered its output. The generators of wrong are the swaps of the ordering, not the closed cycles of the underlying graph; the two are related but not identical (the graph's cycles carry its algebraic content and are derived at the path-algebra scale, Stewart, 2026e; *The Path Algebra of the Endomorphism*, Stewart, 2026f). The definition operates uniformly wherever the four-operation ordering executes. A conclusion reached without premises established is composition without distinction, the same misordering as an effect without its cause in physics, a null reference in computation, an action without its perceptual determination in cognition, and an injustice that operates on a subject whose recognition was never established in ethics. The structural form is the same. The instances differ in what the operations are operating on. Clinical conditions (impulsivity, depression, panic disorder, hallucination, PTSD, generalized anxiety) instantiate specific violation classes with predicted neural timing signatures. The causal structure of spacetime is the operational ordering applied to events. In ethics, injustice is composition before the recognition of a subject's existence, with two sides to the misordering. The agent's ordering continues forward through the misordering. The subject is operated on without the prerequisite (their recognition as a subject) having been established, so the subject's ordering is broken from outside. Both sides inhabit the same malformed ordering. The form of the failure is symmetric, the consequence is not. -/- Error requires at least two operations and the possibility of misordering them. Where only one operation remains operative, no ordering remains to violate. The first operation is the fixed point of the error structure. It cannot be displaced by any misordering, because every misordering displaces a later operation relative to an earlier one. The first operation can never be the late one in such a misordering, because no operation precedes it. There is no zeroth position to displace it after. -/- **Keywords:** error theory, expectancy violation, causality, null reference, action selection, psychopathology, neural timing, programming languages, ethics of recognition, dependency analysis, foundations of mathematics. (shrink)

This paper proposes that any mathematical construction requires four operations to be performed in a fixed structural order. The four operations are distinction, placement, identity, and composition. Logic formalizes distinction, sets formalize placement, types formalize identity, and categories formalize composition. Each foundation formalizes one operation as primary and employs the remaining three as apparatus. The four operations proceed in the order distinction, placement, identity, composition, and that order is forced by operational dependency. -/- The bilateral correspondences established by Curry and (...) Feys (1958) and Howard (1969/1980), extended by Lambek (1972), and connected to sets through topos theory by Lawvere (1970) and Tierney (1972) are cited as the published evidence that each of the four foundations has the capacity to express all four operations as a single formal system. In this sense each foundation is four-in-one, and the four foundations are structurally translatable into one another. The contribution of this paper is the recognition that the four operations, contained as one in each foundation, proceed in a fixed dependency order, with distinction preceding placement, placement preceding identity, and identity preceding composition. The bilateral correspondences are symmetric. We propose the dependencies among the four operations are not symmetric, and the asymmetry of the dependencies fixes the direction in which the correspondences are used when a construction is built up. The correspondences themselves remain equivalences. -/- We distinguish the operations any mathematics must perform from the axiomatic apparatus chosen as primary. The existing foundational debates (ZFC versus Martin-Löf type theory versus ETCS) concern which apparatus is primary and do not address the question we ask. The claim is falsifiable. A piece of mathematics that genuinely lacks one of the four operations would refute it. We predict no such piece exists, because the four operations are features of any mathematical construction. -/- **Keywords:** foundations of mathematics, philosophy of mathematics, Curry-Howard correspondence, Lambek correspondence, topos theory, propositions-as-types, logic, set theory, type theory, category theory, ETCS, mathematical structuralism. (shrink)

Three correspondences between the foundations of mathematics have been independently discovered across the twentieth century. Curry and Feys (1958) and Howard (1969/1980) established that intuitionistic propositional logic corresponds to simply typed lambda calculus. Lambek (1972) extended the correspondence to category theory. Lawvere (1970) and Tierney brought set theory into the structure through topos theory. These results are established and published. What has not been stated is the compositional collapse. The three correspondences confirm that the four foundations are structurally identical, each (...) translatable into the others. The operational order of the cascade composes them into a single directed cycle: logic into set theory, set theory into type theory, type theory into category theory, category theory back into logic. That cycle is an endomorphism of the four-node directed graph derived in Stewart (2026c), a composition that maps the graph's operational structure back to itself. The cycle closes because it collapses: the convergence point (type theory) receives three inputs and reduces them to one output, a mandatory many-to-one reduction without which no single resolved state exists for the morphism to carry forward. The iterated endomorphism traces a torus, with the spatial loop around the four nodes and the temporal loop through the recursion. This paper derives the collapse from the four operations derived in Stewart (2026g), here termed the axioms of occurrence, and proposes the endomorphism as the structure within which quantum mechanics and general relativity already occupy defined positions. The four positions of the endomorphism are assigned functional labels native to the physics domain: Ground (the binary record of what has and has not occurred), Field (the space of all simultaneous potentials), Collapse (the many to one reduction that resolves the field into one determination), and Propagation (the energy that carries the resolved state forward into the next cycle). The framework proposes these four positions correspond to the four foundations of mathematics: Ground to logic, Field to set theory, Collapse to type theory, and Propagation to category theory. On this account, general relativity formalizes the Ground, the binary record of what has and has not occurred (incident 1), and quantum mechanics formalizes the Field (incident 2), the Collapse (incident 3), and the Propagation (incident 4). On the framework's account, both theories employ all four foundations as mathematical tools, because the endomorphism requires all four for any mathematical structure to exist. The distinction the framework draws is between what each theory formalizes as its subject matter and what it employs as its apparatus. On this account, the incompatibility between QM and GR is the incompatibility between two partial formalisms, each built from a subset of the mathematical foundations as its subject, applied separately to different positions within a single structure. The many-to-one reduction at the convergence point reframes wave function collapse: collapse is not a problem to be solved or an interruption of unitary evolution but the structural operation that makes the endomorphism possible. Without the collapse, no single resolved state exists for the morphism to carry forward, the cycle cannot close, and the endomorphism does not complete. The unification, this account proposes, is the endomorphic collapse itself. This is a claim about the structure any unification must instantiate. The structure already exists as published mathematics, the framework holds, and has not been applied to physics because no one has identified the collapse as the structure both theories are inside of. -/- **Keywords:** endomorphism, graph theory, algebraic topology, category theory, type theory, set theory, logic, Curry-Howard correspondence, Lambek correspondence, Lawvere-Tierney topos theory, foundations of mathematics, strongly connected directed graph, CW-complex, Betti number, torus topology, wave function collapse, quantum mechanics, general relativity, unification, scale invariance. (shrink)

This quiver's canonical bilinear form is provably dead: indefinite, selecting no Dynkin type at all. One bit, a sign on a single vertex, is the entire distance from that form to the Cartan matrix of su(3) ⊕ su(2). That the form is provably dead is what makes the gauge content live in the bit and not in the graph. -/- This paper is an inheritance walkthrough: a record of one formalism after another (counting, the path algebra, representation theory, and the (...) canonical bilinear (Tits) form) reaching its own expressive limit, each limit proven and each naming the formalism that inherits the question. -/- It starts at counting (the graded dimensions; counting cannot compose) and hands to the path algebra. The path algebra composes the four-gon γ₁ but cannot express the convergence at T as a relation: the three length-2 paths through T live in orthogonal idempotent components, and equating them would force e_S = e_L, which is false. It hands to representation theory. Under the convergence constraint dim(V_T) = 1 together with the involution conditions, every vertex space in an indecomposable representation is forced to dimension at most one (the rank argument). The quotient algebra cannot produce the braid relation ((γ₁γ₂)³ is a nonzero length-21 element, not e_L) and cannot produce nonzero commutation (γ₁γ₃ = 0 by vertex mismatch). It hands to the canonical bilinear invariant. The Tits form of Q on the cycle-support basis is indefinite, with two negative diagonal entries and one isotropic (zero-diagonal) generator, so it is not a Cartan matrix of any Dynkin type. Here the inheritance stops with a proven negative result: Q, by its own canonical invariants, does not select a finite reflection group; obtaining a Dynkin type requires structure added from outside Q. This stopping point is different in kind from the earlier ones. The earlier handoffs (counting to path algebra to representation theory) each named a richer formalism within mathematics to continue, and each hit the same ceiling because the missing datum was not a formalism a richer one would supply. The inheriting invariant here is the bilinear form on H₁(Q; ℝ), and the structure it needs is not internal to Q at all: it is the single distinction the graph's own combinatorics cannot carry. -/- The brain's signed convergence, the resolution at T that produces a positive-or-negative determination from the inputs converging on it (Stewart, 2026c, Loop 3), supplies that added structure. It is a single binary distinction (vertex T carries a sign, vertex C does not), fixed before the path algebra is constructed. Executing the four axioms of occurrence on that one distinction yields a bilinear form B, shown unique up to diagonal scaling, under which the same cycle generators that gave the dead Tits form realize the Cartan matrix of A₂ × A₁, returning the Lie algebra su(3) ⊕ su(2) via the Cartan-Killing correspondence. The difference between an indefinite, structure-selecting-nothing form and the strong-plus-weak gauge algebra is one bit: which vertex carries the sign. The cycle-generator basis has rank three, the bound that A₂ × A₁ saturates; the fourth generator required for the u(1) of the full Standard Model gauge algebra is not supplied by H₁(Q; ℝ) and is developed in companion work (Stewart, 2026q). The proven deadness of the canonical form is what makes the gauge content carried by the single added distinction rather than by the graph. This is the central result. -/- The inheritance sequence is itself the four-position cycle: counting (logic, distinction) → path algebra (composition) → representation theory (the type judgment dim(V_T) = 1, the many-to-one identity-by-collapse at T) → the bilinear morphism B (composition again). The paper builds mathematical structure in the order mathematics builds it, by letting each formalism break at its own ceiling and naming its heir. -/- What the paper does and what the paper is are the same act, because for an occurrence doing γ₁ and being γ₁ are one event. The construction performed below is one γ₁ traversal of the cycle whose bilinear invariant the construction returns. The gauge content that invariant carries, su(3) ⊕ su(2) at the rank the cycle basis supplies, is read off from inside the cycle by the cycle executing. Mathematics is one γ₁. Gauge structure is what γ₁ produces. The paper is the act in which the two are seen to be the same cycle viewed from opposite sides of itself. -/- **Keywords:** quiver, path algebra, quiver representations, quotient algebra, Tits form, Euler form, Cartan matrix, Coxeter group, Dynkin diagram, root system, Lie algebra, special unitary group, first homology. (shrink)

Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunction seeming to be the primary lens. Our topic is a theory showing “where adjoints come from”. The theory is based on object-to-object “chimera morphisms”, “heteromorphisms”, or “hets” between the objects of different categories (e.g., the insertion of generators as a set-to-group map). After showing that heteromorphisms can be treated rigorously using the machinery of category theory (bifunctors), we show that all (...) adjunctions between two categories arise (up to an isomorphism) as the representations (i.e., universal models) within each category of the heteromorphisms between the two categories. The conventional treatment of adjunctions eschews the whole concept of a heteromorphism, so our purpose is to shine a new light on this notion by showing its origin as a het between categories being universally represented within each of the two categories. This heteromorphic treatment of adjunctions shows how they can be split into two separable universal constructions. Such universals can also occur without being part of an adjunction. We conclude with the idea that it is the universal constructions (adjunctions being an important special case) that are really the foundational concepts to pick out what is important in mathematics and perhaps in other sciences, not to mention in philosophy. (shrink)

The New Mechanism has traditionally focused on identifying constitutively relevant components in mechanistic explanation but has largely overlooked how temporal organization encompassing order, duration, and rate should be formally represented. This paper develops a structuralist approach to mechanistic explanation by integrating insights from scientific representation and category theory. Drawing on Hughes’ DDI (Denotation, Demonstration, Interpretation) account, it establishes a representational framework for modeling temporal constraints as explanatory targets. Category–theoretic tools including index categories, functorial mappings, and Petri nets are employed to (...) formally encode dynamical and sequential structures. Using protein synthesis as a case study, the paper demonstrates that mechanistic explanation must account not only for token–level causal dependencies but also for type–level structural constraints. By bridging the theory of scientific representation with mechanistic explanation, it clarifies how temporal organization underlies and enables explanatory models in biology. (shrink)

Craver's diagram, comprising symbols such as X (entity), S (mechanism), Φ (activity), and Ψ (phenomenon), is widely used to represent biological mechanisms in the New Mechanism. However, this paper demonstrates that Craver’s framework lacks the formal capacity to adequately capture the organizational structures and functional dynamics essential for mechanistic explanations, particularly the temporal interplay among entities and activities or the relational nature of enzymatic state transitions. To address these limitations, this paper proposes a supplementary framework based on category theory, enabling (...) the abstraction of mechanisms as temporally organized structures of state transitions. By integrating key features such as order, frequency, and duration, categorical diagrams provide a cohesive representation of type mechanisms, supporting the deficiencies of token-centric approaches. Using protein synthesis as a case study, the paper illustrates how categorical abstractions enhance the formal representation of biological mechanisms while emphasizing the philosophical importance of organizational dimensions. (shrink)

Peirce’s Existential Graphs provide a geometrical understanding of a variety of logics (classical, intuitionistic, modal, fi rst-order). The geometrical interpretation is given by topological transformations of closed (Jordan) curves on the plane, but it can be extended to other surfaces (sphere, cylinder, torus, etc.) The result provides the appearance of new logics related to the shapes of the surfaces. Going beyond, one can draw existential graphs over general Riemann Surfaces, and, introducing tools from algebraic geometry (Sheaves, Grothendieck Toposes, Elementary Toposes), (...) one can try to capture both the logics and the geometrical shapes through a new Topos of Existential Graphs over Riemann Surfaces, and through the classifi er subobject of the topos. We off er new perspectives (concepts, defi nitions, examples, conjectures) along this road. -/- Os Grafos Existenciais de Peirce oferecem uma compreensão geométrica de uma variedade de lógicas (clássica, intuicionista, modal, de primeira ordem). A sua interpretação geométrica é dada por transformações topológicas de curvas fechadas (Jordan) no plano, mas pode ser estendida a outras superfícies (esfera, cilindro, toro etc.). Além disso, é possível desenhar grafos existenciais sobre superfícies de Riemann gerais e, introduzindo ferramentas da geometria algébrica (Feixes, Toposes de Grothendieck, Toposes Elementares), é possível tentar capturar as lógicas nas formas geométricas por meio de um novo Topos de Grafos Existenciais sobre Superfícies de Riemann e por meio do subobjeto classifi cador do topos. Oferecemos novas perspectivas (conceitos, defi nições, exemplos, conjecturas) ao longo desse caminho. (shrink)

Logic and computer science share a subtle relationship that depends on both syntax and semantics. While structural generalisations often rely on semantics alone, computational aspects such as complexity and decidability hinge on the syntactic properties of formal languages. This interplay frequently manifests through relations between structures, which establish their similarity in various ways and for different purposes. -/- In this work, we focus on three distinct forms of relations between structures: coKleisli morphisms, games, and truth-preserving maps. By coordinating these concepts (...) and proving their equivalence, we apply them across diverse contexts, selecting the appropriate framework for each scenario. CoKleisli morphisms are used to derive generalised theorems, which can then be applied to specific instances; games are used as algorithmic descriptions, crucial when discussing computational complexity; and truth-preserving maps between structures are essential for achieving results such as the finite model property, completeness, and decidability. To a large extent, one must control all manners of structural relations in order to achieve all types of results. -/- We contribute to each of these areas: We extend the use of coKleisli morphisms to ‘linear’ variants of previously studied comonads. Using these results, we derive a novel homomorphism counting theorem for pathwidth. We further establish preservation and characterisation theorems for non-branching variants of the modal comonad. Additionally, using event structures from concurrency theory, we propose a structural alternative to games, providing a more specialised framework for deriving the underlying comonads. For games and computational complexity, we define new model comparison games and establish complexity results for an all-in-one variant of the k-pebble game. Finally, we employ truth-preserving maps to prove completeness and decidability for highly expressive spatiotemporal languages within the context of dynamical systems. (shrink)

Zero has been axial in human development, but the origin and discovery of zero has never been satisfactorily addressed by a comprehensive, systematic and above all interdisciplinary research program. In this volume, over 40 international scholars explore zero under four broad themes: history; religion, philosophy & linguistics; arts; and mathematics & the sciences. Some propose that the invention/discovery of zero may have been facilitated by the prior evolution of a sophisticated concept of Nothingness or Emptiness (as it is understood in (...) non-European traditions); and conversely, inhibited by the absence of, or aversion to, such a concept of Nothingness in the West. But not all scholars agree. Join the debate. (shrink)

Practical approaches to the notions of subsets and extension sets are compared, coming from broadly set-theoretic and category-theoretic traditions of mathematics. I argue that the set-theoretic approach is the most practical for ‘looking down’ or ‘in’ at subsets and the category-theoretic approach is the most practical for ‘looking up’ or ‘out’ at extensions, and suggest some guiding principles for using these approaches without recourse to either category theory or axiomatic set theory.

Notions such as śūnyatā, catuṣkoṭi, and Indra's net, which figure prominently in Buddhist philosophy, are difficult to readily accommodate within our ordinary thinking about everyday objects. Famous Buddhist scholar Nāgārjuna considered two levels of reality: one called conventional reality, and the other ultimate reality. Within this framework, śūnyatā refers to the claim that at the ultimate level objects are devoid of essence or "intrinsic properties", but are interdependent by virtue of their relations to other objects. Catuṣkoṭi refers to the claim (...) that four truth values, including contradiction, are admissible in reasoning. Indra's net refers to the claim that every part of a whole is reflective of the whole. Here we present category theoretic constructions that are reminiscent of these Buddhist concepts. The universal mapping property definition of mathematical objects, wherein objects of a universe of discourse are defined not in terms of their content, but in terms of their relations to all objects of the universe is reminiscent of śūnyatā. The objective logic of perception, with perception modeled as [a category of] two sequential processes (sensation followed by interpretation), and with its truth value object of four truth values, is reminiscent of the Buddhist logic of catuṣkoṭi. The category of categories, wherein every category has a subcategory of sets with zero structure within which every category can be modeled, is reminiscent of Indra's net. Our thorough elaboration of the parallels between Buddhist philosophy and category theory can facilitate better understanding of Buddhist philosophy, and bring out the broader philosophical import of category theory beyond mathematics. (shrink)

In this chapter, I use the language of category theory to address a relevant part of the Christian Trinitarian doctrine. Using a categorical conceptual apparatus usual to mathematicians, it is possible to represent important points of Trinitarian theology and show that such discourse can be considered free of contradictions. Problems concerning the Trinity are therefore approached from a category theory perspective.

The type-level abstraction is a formal way to represent molecular structures in biological practice. Graphical representations of molecular structures of biological objects are also used to identify functional processes of things. This paper will reveal that category theory is a formal mathematical language not only to visualize molecular structures of biological objects as type-level abstraction formally but also to understand how to infer biological functions from the molecular structures of biological objects. Category theory is a toolkit to understand biological knowledge (...) at the type-level formally, not individual token-level, as well as typical heuristic strategies in molecular biology. (shrink)

Science does not exist in vacuum; it arises and works in context. Ground-breaking achievements transforming the scientific landscape often stem from philosophical thought, just as symbolic logic and computer science were born from the early analytic philosophy, and for the very reason they impact our global worldview as a coherent whole as well as local knowledge production in different specialised domains. Here we take first steps in elucidating rich philosophical contexts in which Samson Abramsky’s far-reaching work centring around categorical science (...) as a new kind of science may be placed, explicated, and articulated. We argue, inter alia, that Abramsky’s work, as a whole, may be construed as demonstrating the categorical unity of science, or rather the sciences, in a mathematically rigorous, down-to-earth manner, which has been a salient feature of his work. At the same time we trace his intellectual history, leading from duality, to intensionality, and to contextuality, and place it in a broader context of philosophy beyond the analytic-continental divide, namely towards the reintegration of them as in the post-analytic tradition. Besides, we address issues in philosophy of category theory, such as the foundational autonomy of category theory and the (presumably two) dogmas of set-theoretical foundationalism, which Abramsky actually touch upon in one of his few philosophically inclined works. As to philosophy of category theory, we also address categorical structuralism as higher-order structuralism, categorical epistemology as elucidating higher-order meta-laws, and categorical ontology as allowing for reduction of ontological commitment via structural realism, the structuralist resolution of Benaceraff’s dilemma, and the pluralistic multiverse view of science as opposed to the set-theoretical reductionist ‘universe’ view. We conclude by speculating about the existence of the Oxford School of (Pluralistic) Unified Science as opposed to the Vienna Circle of (Monistic) Unified Science and to the Stanford School of (Pluralistic) Disunified Science; Categorical Unified Science may potentially allow us to reconcile the two camps on the unity and disunity of science whilst doing justice to both of them. Categorical unity arguably allows for unification via epistemological and ontological networking, and via knowledge transfer thus enabled, rather than unification via the reduction of everything and every truth to a single foundationalist framework, whilst taking at face value disunity, plurality, and diversity, and their significance in science and in human civilisation as a whole. (shrink)

The main research goal of the work is to study the notion of co-topos, its correctness, properties and relations with toposes. In particular, the dualization process proposed by proponents of co-toposes has been analyzed, which transforms certain Heyting algebras of toposes into co-Heyting ones, by which a kind of paraconsistent logic may appear in place of intuitionistic logic. It has been shown that if certain two definitions of topos are to be equivalent, then in one of them, in the context (...) of a subobject classifier, there should be a neutral label for the generic subobject, or if the label "true" appears instead, it carries no additional properties or assumptions. It has been shown that a natural isomorphism occurring in one of the definitions of topos need not be an isomorphism of the corresponding algebraic structures on the sets, so that a co-Heyting algebraic structure can be defined on the set of characteristic morphisms, which moreover will also be natural. The analyzes suggest that the notion of co-topos may be considered correct. However, it should be strongly emphasized that although the possibility of defining the notion of co-topos has been shown, its further full consequences should be very carefully examined, which can severely limit its logical applications. (shrink)