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Basic Category Theory
Abstract
At the heart of this short introduction to category theory is the idea of a universal property, important throughout mathematics. After an introductory chapter giving the basic definitions, separate chapters explain three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties all three together. The book is suitable for use in courses or for independent study. Assuming relatively little mathematical background, it is ideal for beginning graduate students or advanced undergraduates learning category theory for the first time. For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations. Copious exercises are included.
... The idea of category theory is to collect together a system of mathematical objects such that there is a notion of morphism between objects which relates them to one another ( [5], p. 9). ...
... If we have jgf = ih, then we say the diagram commutes. In general, a diagram is said to commute provided that whenever there are two paths between an object A and an object B, the morphism from A to B obtained by composing along one path is equal to the morphism obtained by composing along the other ( [5], p. 11). ...
... Fundamental to category theory is the duality principle which states that every categorical definition, theorem and proof has a dual, which is simply obtained by reversing all the arrows within the given statement ( [5], p. 16). This is extremely useful, as any true statement in category theory has a dual statement which is also true; this can be a good time-saver when proving theorems as we often get two theorems for the price of one! ...
This report investigates and explores some of the introductory notions in point-free topology. Point-free topology uses the language of category theory and thus we briefly provide the necessary concepts that are needed from category theory for the rest of the report. The central objects of investigation are called frames. Frames are complete lattices which satisfy the join-infinite distribution law. The definition of a frame is motivated from the open set lattices of topological spaces. Frames essentially mimic the behaviour of the open sets in a topological space. The report proceeds to uncover the relations between spaces and frames, the functor Ω : Top → Frm is the first example of how spaces are related to frames. We define sober spaces and explain their importance in point-free topology via the full and faithful functor Ω : Sob → Frm, in this way the category of sober spaces is embedded into the category of frames and thus we may regard point-free topology as a natural extension of classical topology. Due to the contravariance of Ω, we introduce the category Loc of locales as the opposite of Frm, so that Ω : Top → Loc is covariant. We may recover a topological space from a frame via the contravariant functor Σ : Frm → Top. Thus working covariantly, we have a pair of functors Ω, Σ. We show that Σ is in fact a right adjoint to Ω. Spatiality of frames is discussed and a lattice-theoretic characterisation for spatial frames is presented. Finally we investigate the ideals in frames, the collection of all ideals in a frame constitutes a frame (called the ideal lattice). A natural problem to then investigate is the spatiality of the ideal lattice of a frame. Specifically we provide a sufficient condition, while using the Axiom of Choice, for the ideal lattice of a frame to be spatial.
... Further to this, more complex descriptions can be considered by extending graph theory with category theory (Awodey, 2010;Leinster, 2014;Spivak, 2014;Riehl, 2017). In category theory, these relationships can be visualized whereby the edges depicting relational frames represent morphisms between objects (concepts). ...
... Category theory (Awodey, 2010;Leinster, 2014;Spivak, 2014;Riehl, 2017) can be integrated into hypergraphs by defining categories where objects are different features, states or components of the data (such as a chair, the woods, or a snake), and morphisms represent transformations, relationships or dependencies between these objects (such as relational frames). Morphisms can represent simple relations or complex ToF involving observer-dependent interpretations (ToM perspective-taking). ...
... " Here coordinating "my perspective" to "your perspective, " and distinguishing between "my perspective" and "your perspective, " through time (e.g., "now" vs. "then") and space (e.g., "here" vs. "there"). These can be visualized with the use of hypergraphs as well as category theory (Awodey, 2010;Leinster, 2014;Spivak, 2014;Riehl, 2017) where these complex relational frames edges represent a relational frame with a specific label, indicating the type of relationship (e.g., "coordinates, " "distinguishes"). The models can then show how multiple relational frames combine to form more complex cognitive processes like perspective-taking and understanding others' viewpoints (ToM). ...
There have been impressive advancements in the field of natural language processing (NLP) in recent years, largely driven by innovations in the development of transformer-based large language models (LLM) that utilize “attention.” This approach employs masked self-attention to establish (via similarly) different positions of tokens (words) within an inputted sequence of tokens to compute the most appropriate response based on its training corpus. However, there is speculation as to whether this approach alone can be scaled up to develop emergent artificial general intelligence (AGI), and whether it can address the alignment of AGI values with human values (called the alignment problem). Some researchers exploring the alignment problem highlight three aspects that AGI (or AI) requires to help resolve this problem: (1) an interpretable values specification; (2) a utility function; and (3) a dynamic contextual account of behavior. Here, a neurosymbolic model is proposed to help resolve these issues of human value alignment in AI, which expands on the transformer-based model for NLP to incorporate symbolic reasoning that may allow AGI to incorporate perspective-taking reasoning (i.e., resolving the need for a dynamic contextual account of behavior through deictics) as defined by a multilevel evolutionary and neurobiological framework into a functional contextual post-Skinnerian model of human language called “Neurobiological and Natural Selection Relational Frame Theory” (N-Frame). It is argued that this approach may also help establish a comprehensible value scheme, a utility function by expanding the expected utility equation of behavioral economics to consider functional contextualism, and even an observer (or witness) centric model for consciousness. Evolution theory, subjective quantum mechanics, and neuroscience are further aimed to help explain consciousness, and possible implementation within an LLM through correspondence to an interface as suggested by N-Frame. This argument is supported by the computational level of hypergraphs, relational density clusters, a conscious quantum level defined by QBism, and real-world applied level (human user feedback). It is argued that this approach could enable AI to achieve consciousness and develop deictic perspective-taking abilities, thereby attaining human-level self-awareness, empathy, and compassion toward others. Importantly, this consciousness hypothesis can be directly tested with a significance of approximately 5-sigma significance (with a 1 in 3.5 million probability that any identified AI-conscious observations in the form of a collapsed wave form are due to chance factors) through double-slit intent-type experimentation and visualization procedures for derived perspective-taking relational frames. Ultimately, this could provide a solution to the alignment problem and contribute to the emergence of a theory of mind (ToM) within AI.
... Postulates of category theory. In general, category theory is a language that is used to express abstract concepts [8,9]. The category is defined as the combination of source ...
... Hence, Eq ( (8) and (9). Naturally, the inverse morphism may be included in Eq (8) or (9) if the appropriate definition of the inverse morphism is possible. In this EPN model, we consider instances for which the compositionality among all target objects and source objects that are necessary for the construction of the category completely satisfies Eq (4). ...
... For all manipulations described above, the identity morphism performs nothing on any token. Thereby, the morphisms associated with combination, splitting, and insertion are considered to satisfy the postulate within category theory because associativity and the existence of an identity is met [8,9]. For the above example, if the morphisms f Int , f spl , f inc , and f excl are defined at the same place p k (this may be discriminated/separated as 'place p k1 , place p k2 , place p k3 . . ...
algebraic concepts such as category are considered cornerstones on which logical consistency relies in any sophisticated study of natural phenomena. However, to the best of our knowledge, in molecular/genetic biology, their application is still severely limited because they capture neither the dynamics nor provide a visual form. The Petri net (PN) has often been used to illustrate visually parallel, asynchronous dynamic events in small data systems. A prototypal hybrid model combining both category theory and extended PNs may instead be indispensable for that purpose. This hybrid model incorporates 1) token-like elements of a group, 2) object-like places of a category, 3) square poles (rather than pentagon poles) that enable unique identifications of single-strand DNA sequences from the shape of its polygonal line, 4) creation/annihilation morphisms that generate/erase tokens, 5) Cartesian products ‘Z5×Z2×…’ that enable conversions between DNA and RNA sequences, 6) somatic recombinations (VDJ recombinations) for antibodies displayed concretely in category-theoretic form, 7) ‘identity protein Δ’ translated from a triplet of identity bases ‘EEE’ as an advanced concept from our previous display of the canonical central dogma, 8) illustrations of an incidence-matrix-like matrix A that includes operators as coordinates, and 9) basic topics concerning the canonical central dogma being displayed concretely using concepts of conventional category theory such as ‘adjoint’, ‘adjoint functor’, ‘natural transformation’, ‘Yoneda’s lemma’ and ‘Kan extension’. These ideas provide more advanced tools that expand our previous model concerning nucleic-acid-base sequences. Despite the nascent nature of our methodology, our hybrid model has potential in a variety of applications, illustrated using molecular/genetic sequences, in particular providing a simple dynamic/visual representation. With further improvements, this approach may prove effective in reducing the need for large data-storing systems.
... (cf. [5]) A category C consists of the following data: ...
... (cf. [5])A product of two object A and B in a category C is an object A Π B together with two morphisms p 1 : A Π B → A and p 2 : A Π B → B that satisfies the universal property. ie, for any object C and any two morphisms ...
A Lie groupoid is a groupoid with additional smooth manifold structures on the object set and the morphism set that makes various maps arise from the groupoid structure smooth. In this paper we describe the properties of the category Rep(G) whose objects are the classical representations of the Lie groupoid G and morphisms are the base preserving vector bundle morphisms that respects the representations. It is shown that this category is an additive category and is a monoidal category with subobjects. Also we have discussed the kernel and cokernel for certain type of mor-phisms.
... In [23], Yamazaki initiated the study of category theory [14,15] and homological algebra [10,22] in reverse mathematics; for instance, he formalized derived functors Ext n R (−, −) and Tor n R (−, −) for modules over a ring R in RCA 0 +IΣ 0 2 , he also studied projective resolutions of modules, the limits and colimits of diagrams of categories in reverse mathematics. In this paper, we follow the work of Yamazaki and continue to study category theory from the standpoint of reverse mathematics. ...
... The idea of structures (or objects) and structure-preserving maps (or morphisms) is abstracted in the concept of category. In classical texts (e.g., [10,14]), a category is defined as a structure containing a collection of objects, a collection of morphisms for each pair of objects and a composition function on morphisms subject to the unit axiom and the associativity axiom. We also have structure-preserving maps between categories, namely, functors, and structure-preserving maps between functors of categories, namely, natural transformations. ...
This paper studies categories and functors in the context of reverse and computable mathematics. In ordinary reverse mathematics, we only focuses on categories whose objects and morphisms can be represented by natural numbers. We first consider morphism sets of categories and prove several associated theorems equivalent to over the base system . The Yoneda Lemma is a basic result in category theory and homological algebra. We then develop an effective version of the Yoneda Lemma in ; as an application, we formalize an effective version of the Yoneda Embedding in . Products and coproducts are basic notions for defining special categories like semi-additive categories and additive categories. We study properties of products and coproducts of a sequence of objects of categories and provide effective characterizations of semi-additive categories and additive categories in terms of products and coproducts. Finally, we further consider the strength of theorems of category theory that are studied in this paper by methods of higher-order reverse mathematics
... As a redress to these problems towards a formally axiomatic theory of consciousness, inspiration is taken from a parallel between the meta-problem of consciousness (Chalmers, 2018(Chalmers, , 2020 and category theory (Leinster, 2014) as a kind of meta-mathematics (Eilenberg & Mac Lane, 1945) to be elaborated shortly. The meta-problem and the hard problem are supposed to be connected: "We can reasonably hope that a solution to the meta-problem will shed significant light on the hard problem." ...
... A core category theory principle is characterization of a mathematical structure by a so-called universal mapping property (UMP), i.e. a unique-existence condition satisfied by all instances of the structure (Leinster, 2014;Mac Lane, 1998). For some intuition, the largest (maximum) element in an ordered set satisfies a UMP: every element in that set is smaller than or equal to the largest element. ...
Conscious (subjective) experience permeates our daily lives, yet general consensus on a theory of consciousness remains elusive. Integrated Information Theory (IIT) is a prominent approach that asserts the existence of subjective experience (0th axiom), from an intrinsic system of causally related units, and five essential properties (axioms 1-5): intrinsicality, information, integration, exclusion and composition. However, despite empirical support for some aspects of IIT, the supposed necessity of these axioms is unclear given their informal presentation and operationalized dependence on a specific mathematical instantiation as the so-called postulates. The category theory approach presented here attempts to redress this situation. Category theory is a kind of meta-mathematics invented to make relations between formal structures formally precise and so facilitate doing "ordinary" mathematics. In this way, the five essential properties for consciousness are organized around a smaller number of meta-mathematical principles for comparison with IIT. In particular, category theory characterizes mathematical structures by their "universal mapping properties" -- a unique-existence condition for all instances of the structure. Accordingly, axioms 1-5 pertain to universal mapping properties for experience, whence the slogan, "Consciousness is a universal property."
... Categorical Framework for Formal Semantics PREPRINT however, we give a technical presentation of those models in a concise and consistent way that is (at least, to the best of the author's knowledge) otherwise absent in standard texts. In Section 3, we give a brief tour of the usual aspects of category theory, with emphasis on the components relevant here, and follow the conventions from [39,63,2]. Kripke semantics and its categorification are given in Section 4, relative to the categories of sets and relations. Much of the material here follows from [16,33,62]; here, emphasize the categorical structure of Kripke frames as they are imported into the category of intensional models, and prove various statements otherwise left alone in those sources. ...
... Working definitions of each are given in Definition 3.1, Definition 3.2, and Definition 3.3, respectively. Expositional material in this section is primarily adapted from [39,63,2], and includes additional material from [27,58]. The reader familiar with CT may skip this section without loss of context in the remainder of the paper. ...
Intensional computation derives concrete outputs from abstract function definitions; extensional computation defines functions through explicit input-output pairs. In formal semantics: intensional computation interprets expressions as context-dependent functions; extensional computation evaluates expressions based on their denotations in an otherwise fixed context. This paper reformulates typed extensional and intensional models of formal semantics within a category-theoretic framework and demonstrates their natural representation therein. We construct , the category of intensional models, building on the categories of sets, of relations, and and of Kripke frames with monotone maps and bounded morphisms, respectively. We prove that trivial intensional models are equivalent to extensional models, providing a unified categorical representation of intensionality and extensionality in formal semantics. This approach reinterprets the relationship between intensions and extensions in a categorical framework and offers a modular, order-independent method for processing intensions and recovering extensions; contextualizing the relationship between content and reference in category-theoretic terms. We discuss implications for natural language semantics and propose future directions for contextual integration and exploring 's algebraic properties.
... Given a rig groupoid, we may think of the objects as types, and the morphisms as terms [42]. The syntax of the language Π in Fig. 3 captures this idea. ...
... We say that 1 is a terminal object in the category of sets and functions. In the opposite direction, the empty set is an initial object, meaning that there is a unique function 0 → A for any set A. Similarly, the cartesian product A × B of sets can be characterized universally as a categorical product of A and B, and the disjoin union A + B as a coproduct : these are the universal objects equipped with projections A ← A × B → B respectively injections A → A + B ← B [42]. ...
Reversible computing is motivated by both pragmatic and foundational considerations arising from a variety of disciplines. We take a particular path through the development of reversible computation, emphasizing compositional reversible computation. We start from a historical perspective, by reviewing those approaches that developed reversible extensions of lambda-calculi, Turing machines, and communicating process calculi. These approaches share a common challenge: computations made reversible in this way do not naturally compose locally. We then turn our attention to computational models that eschew the detour via existing irreversible models. Building on an original analysis by Landauer, the insights of Bennett, Fredkin, and Toffoli introduced a fresh approach to reversible computing in which reversibility is elevated to the status of the main design principle. These initial models are expressed using low-level bit manipulations, however. Abstracting from the low-level of the Bennett-Fredkin-Toffoli models and pursuing more intrinsic, typed, and algebraic models, naturally leads to rig categories as the canonical model for compositional reversible programming. The categorical model reveals connections to type isomorphisms, symmetries, permutations, groups, and univalent universes. This, in turn, paves the way for extensions to reversible programming based on monads and arrows. These extensions are shown to recover conventional irreversible programming, a variety of reversible computational effects, and more interestingly both pure (measurement-free) and measurement-based quantum programming.
... A category is itself a mathematical object, and thus it too is characterized by the arrows it has to/from other categories in the "category of categories" (Leinster, 2014). A functor F : C → D between categories C and D is such an "arrow between categories," depicted as follows: ...
The concept of affordance, proposed by James J. Gibson as an opportunity for action offered by the environment to the organism, has been adopted in various fields, including psychology, neuroscience, and robotics. However, different interpretations exist as to whether it is a feature of a relation between the environment and the organism and therefore cannot exist independently of the organism, or a "resource" that exists in the environment independent of the organism's presence and is waiting to be used, or both, or neither. In this paper, we defend the position that affordances are both re-lational and resources using a category-theoretic approach. This idea is formalized by the concept of "natural transfor-mations" in category theory, which are structure-preserving transformations between "functors"-mathematical expressions representing "seeing from a particular perspective." We propose that formalizing the realism of affordance in terms of natural transformations offers a more rigorous and lucid understanding of this concept. Furthermore, our formalization enables us to relate the reality of affordances to a broader context, especially the shift in the meaning of "reality" in modern physics. Our category-theoretic approach offers a potential solution to the problems and limitations associated with existing set theory-based frameworks for affordances, paving the way for a future theory that better accounts for the open-ended interplay between organisms and their environments.
... The definitions of categories, functors, natural transforms, adjunctions, and Kan extensions are found in all of the following resources. (Riehl, 2016;Fong & Spivak, 2018;Leinster, 2016). The definition of a 2-category is adapted from its definition as an enriched category (Kelly, 2005). ...
Previous work has demonstrated that efficient algorithms exist for computing Kan extensions and that some Kan extensions have interesting similarities to various machine learning algorithms. This paper closes the gap by proving that all error minimisation algorithms may be presented as a Kan extension. This result provides a foundation for future work to investigate the optimisation of machine learning algorithms through their presentation as Kan extensions. A corollary of this representation of error-minimising algorithms is a presentation of error from the perspective of lossy and lossless transformations of data.
... Although this section should contain the definitions and notation necessary for the presentations of categorical semantics in Section 3.4 and Section 4.3, more detailed explanations can be found in the introductory textbooks by Awodey (2010); Leinster (2014) and Abramsky and Tzevelekos (2011). ...
This dissertation explores the design and implementation of programming languages that represent rounding error analysis through typing. The first part of this dissertation demonstrates that it is possible to design languages for forward error analysis, as illustrated with NumFuzz, a functional programming language whose type system expresses quantitative bounds on rounding error. This type system combines a sensitivity analysis, enforced through a linear typing discipline, with a novel graded monad to track the accumulation of rounding errors. We establish the soundness of the type system by relating the denotational semantics of the language to both an exact and floating-point operational semantics. To demonstrate the practical utility of NumFuzz as a tool for automated error analysis, we have developed a prototype implementation capable of automatically inferring error bounds. Our implementation produces bounds competitive with existing tools, while often achieving significantly faster analysis times. The second part of this dissertation explores a type-based approach to backward error analysis with Bean, a first-order programming language with a linear type system that can express quantitative bounds on backward error. Bean's type system combines a graded coeffect system with strict linearity to soundly track the flow of backward error through programs. To illustrate Bean's potential as a practical tool for automated backward error analysis, we implement a variety of standard algorithms from numerical linear algebra in Bean, establishing fine-grained backward error bounds via typing in a compositional style. We also develop a prototype implementation of Bean that infers backward error bounds automatically. Our evaluation shows that these inferred bounds match worst-case theoretical relative backward error bounds from the literature.
... For a comprehensive introduction to monads and free constructions, see Mac Lane (2013) or standard references in category theory such as Leinster (2014), which detail the colimit-based view of free objects. ...
Self-attention mechanisms have revolutionised deep learning architectures, but their mathematical foundations remain incompletely understood. We establish that these mechanisms can be formalised through categorical algebra, presenting a framework that focuses on the linear components of self-attention. We prove that the query, key, and value maps in self-attention naturally form a parametric endofunctor in the 2-category of parametric morphisms. We show that stacking multiple self-attention layers corresponds to constructing the free monad on this endofunctor. For positional encodings, we demonstrate that strictly additive position embeddings constitute monoid actions on the embedding space, while standard sinusoidal encodings, though not additive, possess a universal property among faithful position-preserving functors. We establish that the linear portions of self-attention exhibit natural equivariance properties with respect to permutations of input tokens. Finally, we prove that the ``circuits'' identified in mechanistic interpretability correspond precisely to compositions of parametric morphisms in our framework. This categorical perspective unifies geometric, algebraic, and interpretability-based approaches to transformer analysis, while making explicit the mathematical structures underlying attention mechanisms. Our treatment focuses exclusively on linear maps, setting aside nonlinearities like softmax and layer normalisation, which require more sophisticated categorical structures. Our results extend recent work on categorical foundations for deep learning while providing insights into the algebraic structure of attention mechanisms.
... Definition 2.6 (Natural Transformation). Given two functors [11] F, G : C → D, a natural transformation η : F ⇒ G assigns to each object X ∈ C a morphism η X : F (X) → G(X) in D such that for every morphism f : X → Y in C, the following diagram [12] commutes: ...
In the last decades, the theory of Categories introduced deep changes in the way math and physics are conceived, introducing new objects, theorems and laws. This sort work introduces an innovative theorem, which investigates the duality between entropy and order within the framework of a topos, a fundamental concept in category theory. The theorem establishes that, for any given family of constraints, there only exists a unique object, which I defined as the colimit, that maximizes entropy while satisfying these constraints. Achieving this result could provide a profound connection between the mathematical structure of a topos and the concept of Shannon entropy, at least in the classical world, enabling the study of constrained complex systems in an abstract and innovative manner, not kept under the chains of old symbolism. The implications of this theory extend to various fields, including machine learning, physics, and biology, thus offering new perspectives for analyzing constrained systems and optimizing resources.
... The direct product type is suitable for recursively extending from lower dimensions to higher dimensions (and the development of the integral and measure using this language is contained in [7]), but is not suitable for representations of general orders such as n-dimensional. The tuple type compensates for this disadvantage and is also used as the domain of multivariable functions [10]. However, the direct relationship (universality) between Cartesian product type and tuple type has not yet been demonstrated within the Mizar Mathematical Library (although the problem solved here is strictly connected with the Mizar choice of the formalization technique; see the outline of the encoding of corresponding topics in Isabelle/HOL [9] or Coq [5]). ...
This paper deals with the interconversion between Cartesian product types and tuple types and their integration for measures in higher dimensional spaces. We prove the universality between both types and construct a measure (and also underlying integral) based on the set of tuple types.
... Category theory, a relatively modern branch of mathematics, is the study of precisely this [7,[39][40][41] (see also Refs. [42,43] for quantum-focused introductions). ...
Harnessing the potential computational advantage of quantum computers for machine learning tasks relies on the uploading of classical data onto quantum computers through what are commonly referred to as quantum encodings. The choice of such encodings may vary substantially from one task to another, and there exist only a few cases where structure has provided insight into their design and implementation, such as symmetry in geometric quantum learning. Here, we propose the perspective that category theory offers a natural mathematical framework for analyzing encodings that respect structure inherent in datasets and learning tasks. We illustrate this with pedagogical examples, which include geometric quantum machine learning, quantum metric learning, topological data analysis, and more. Moreover, our perspective provides a language in which to ask meaningful and mathematically precise questions for the design of quantum encodings and circuits for quantum machine learning tasks.
... For basic definitions and notations of category theory, we refer the reader to [18,Chapter 1]. In this section, we will define the categories of MAT-labeled graphs and LR-vines, and construct an explicit equivalence between these two categories. ...
The present paper explores a connection between two concepts arising from different fields of mathematics. The first concept, called vine, is a graphical model for dependent random variables. This concept first appeared in a work of Joe (1994), and the formal definition was given later by Cooke (1997). Vines have nowadays become an active research area whose applications can be found in probability theory and uncertainty analysis. The second concept, called MAT-freeness, is a combinatorial property in the theory of freeness of logarithmic derivation modules of hyperplane arrangements. This concept was first studied by Abe-Barakat-Cuntz-Hoge-Terao (2016), and soon afterwards investigated further by Cuntz-Mücksch (2020).
In the particular case of graphic arrangements, the last two authors (2023) recently proved that the MAT-freeness is completely characterized by the existence of certain edge-labeled graphs, called MAT-labeled graphs. In this paper, we first introduce a poset characterization of a vine. Then we show that, interestingly, there exists an explicit equivalence between the categories of locally regular vines and MAT-labeled graphs. In particular, we obtain an equivalence between the categories of regular vines and MAT-labeled complete graphs.
Several applications will be mentioned to illustrate the interaction between the two concepts. Notably, we give an affirmative answer to a question of Cuntz-Mücksch that MAT-freeness can be characterized by a generalization of the root poset in the case of graphic arrangements.
... In this section, we assume knowledge of the definition of a category and the definition of a functor between categories. These definitions can be found in an introductory text on category theory, such as Leinster (2014); Mac Lane (1998) or Riehl (2017). ...
Group equivariant neural networks are growing in importance owing to their ability to generalise well in applications where the data has known underlying symmetries. Recent characterisations of a class of these networks that use high-order tensor power spaces as their layers suggest that they have significant potential; however, their implementation remains challenging owing to the prohibitively expensive nature of the computations that are involved. In this work, we present a fast matrix multiplication algorithm for any equivariant weight matrix that maps between tensor power layer spaces in these networks for four groups: the symmetric, orthogonal, special orthogonal, and symplectic groups. We obtain this algorithm by developing a diagrammatic framework based on category theory that enables us to not only express each weight matrix as a linear combination of diagrams but also makes it possible for us to use these diagrams to factor the original computation into a series of steps that are optimal. We show that this algorithm improves the Big-O time complexity exponentially in comparison to a na\"{i}ve matrix multiplication.
... For a general reference on terminology from category theory we suggest [5]. ...
The correspondence between definable connected groupoids in a theory T and internal generalised imaginary sorts of T, established by Hrushovski in ["Groupoids, imaginaries and internal covers," Turkish Journal of Mathematics, 2012], is here extended in two ways: First, it is shown that the correspondence is in fact an equivalence of categories, with respect to appropriate notions of morphism. Secondly, the equivalence of categories is shown to vary uniformly in definable families, with respect to an appropriate relativisation of these categories. Some elaboration on Hrushovki's original constructions are also included.
... The Yoneda lemma (see e.g. [69,Chapter 4] for a proof ) asserts that for any Set-valued presheaf on a category C, that is any functor Φ : C op Ñ Set, the set of natural transformations hom C p´, cq ñ Φ is bijective to the set Φpcq for any object c P C. ...
We develop a description of higher gauge theory with higher groupoids as gauge structure from first principles. This approach captures ordinary gauge theories and gauged sigma models as well as their categorifications on a very general class of (higher) spaces comprising presentable differentiable stacks, as e.g. orbifolds. We start off with a self-contained review on simplicial sets as models of -categories. We then discuss principal bundles in terms of simplicial maps and their homotopies. We explain in detail a differentiation procedure, suggested by Severa, that maps higher groupoids to -algebroids. Generalising this procedure, we define connections for higher groupoid bundles. As an application, we obtain six-dimensional superconformal field theories via a Penrose-Ward transform of higher groupoid bundles over a twistor space. This construction reduces the search for non-Abelian self-dual tensor field equations in six dimensions to a search for the appropriate (higher) gauge structure. The treatment aims to be accessible to theoretical physicists.
... Practically, this observation urges us to consider RH as a singularity in an enlarged field of mathematical categories and that is why the authors suggested to introduce the category theory [34][35][36][37] for handling the RH issue [22,23]. The use of this theory is justified for at least two reasons: (i) according to the work of Rota [38], the function ζ(s) can easily be expressed in the framework of partially ordered sets (which forms the basis of all standard dynamics), particular cases of categories; (ii) since the Leinster works [39,40], self-similarity as a property of a fixed point must be easily expressed by using the language of categories. Experts in algebraic geometries will consult with profit the reference [41]. ...
The novelty of the Jean Pierre Badiali last scientific works stems to a quantum approach based on both (i) a return to the notion of trajectories (Feynman paths) and (ii) an irreversibility of the quantum transitions. These iconoclastic choices find again the Hilbertian and the von Neumann algebraic point of view by dealing statistics over loops. This approach confers an external thermodynamic origin to the notion of a quantum unit of time (Rovelli Connes' thermal time). This notion, basis for quantization, appears herein as a mere criterion of parting between the quantum regime and the thermodynamic regime. The purpose of this note is to unfold the content of the last five years of scientific exchanges aiming to link in a coherent scheme the Jean Pierre's choices and works, and the works of the authors of this note based on hyperbolic geodesics and the associated role of Riemann zeta functions. While these options do not unveil any contradictions, nevertheless they give birth to an intrinsic arrow of time different from the thermal time. The question of the physical meaning of Riemann hypothesis as the basis of quantum mechanics, which was at the heart of our last exchanges, is the backbone of this note.
... There are some topics, however, whose development does appear to require it. One example often given is the definition of (Grothendieck) fibrations (and their relatives): functors p : D → C equipped with a lifting property providing (among other things) an object d of D such that pd is equal to a previously given object c of C. A similar example is the property of creating limits; see [Lei14,Remark 5.3.7] for an explicit discussion of this example. ...
We introduce and develop the notion of *displayed categories*. A displayed category over a category C is equivalent to "a category D and functor F : D --> C", but instead of having a single collection of "objects of D" with a map to the objects of C, the objects are given as a family indexed by objects of C, and similarly for the morphisms. This encapsulates a common way of building categories in practice, by starting with an existing category and adding extra data/properties to the objects and morphisms. The interest of this seemingly trivial reformulation is that various properties of functors are more naturally defined as properties of the corresponding displayed categories. Grothendieck fibrations, for example, when defined as certain functors, use equality on objects in their definition. When defined instead as certain displayed categories, no reference to equality on objects is required. Moreover, almost all examples of fibrations in nature are, in fact, categories whose standard construction can be seen as going via displayed categories. We therefore propose displayed categories as a basis for the development of fibrations in the type-theoretic setting, and similarly for various other notions whose classical definitions involve equality on objects. Besides giving a conceptual clarification of such issues, displayed categories also provide a powerful tool in computer formalisation, unifying and abstracting common constructions and proof techniques of category theory, and enabling modular reasoning about categories of multi-component structures. As such, most of the material of this article has been formalised in Coq over the UniMath library, with the aim of providing a practical library for use in further developments.
... In this paper, we assume that the readers are familiar with basic notions of category theory such as adjunctions, monads, presheaves, the Yoneda lemma, Eilenberg-Moore categories and monoidal categories as described in a textbook by Mac Lane (1971). A textbook by Leinster (2014) is also a good reference for category theory. We use advanced topics of category theory such as coends, 2-categories, bicategories and enriched categories. ...
We present an arrow calculus with operations and handlers and its operational and denotational semantics. The calculus is an extension of Lindley, Wadler and Yallop’s arrow calculus.
The denotational semantics is given using a strong (pro)monad in the bicategory of categories and profunctors. The construction of this strong monad is not trivial because of a size problem. To build denotational semantics, we investigate what -algebras are, and a handler is interpreted as an -homomorphisms between -algebras.
The syntax and operational semantics are derived from the observations on -algebras. We prove the soundness and adequacy theorem of the operational semantics for the denotational semantics.
... As a reference to basic notions from category theory, we use nicely written textbooks [28], [26], [27]. We will use the notions from category theory explicitly, often even implicitly, to formalize other concepts we employ. ...
In this paper, we investigate the concept of local homeomorphism in Esakia spaces. We introduce the notion of étale Heyting H-algebra and establish category-theoretic duality for étale Heyting H-algebra in the case of finite Heyting algebra H. Furthermore, we give an identity that axiomatizes the variety of etale Heyting H-algebras when H is finite. We also show that the category of Stone space-valued (co)presheaves over a finite Esakia space X is equivalent to the slice category of local homeomorphisms over X. The fact is used to show that, in comparison with the case of general Heyting H-algebras, it is easier to compute finite colimits in the category of étale Heyting H-algebras.
... For background on category theory and groupoids, see [ML98], [Lei14]; for groupoids in the topological and Borel contexts, see [Ram90], [Alv08], [Car11], [Bow14], [Che19b], [TDW21]. ...
We show that the category of countable Borel equivalence relations (CBERs) is dually equivalent to the category of countable theories which admit a one-sorted interpretation of a particular theory we call that witnesses embeddability into and the Lusin--Novikov uniformization theorem. This allows problems about Borel combinatorial structures on CBERs to be translated into syntactic definability problems in , modulo the extra structure provided by , thereby formalizing a folklore intuition in locally countable Borel combinatorics. We illustrate this with a catalogue of the precise interpretability relations between several standard classes of structures commonly used in Borel combinatorics, such as Feldman--Moore -colorings and the Slaman--Steel marker lemma. We also generalize this correspondence to locally countable Borel groupoids and theories interpreting , which admit a characterization analogous to that of Hjorth--Kechris for essentially countable isomorphism relations.
... Here we give informal definitions of the category-theoretic structures we use. Consult Lawvere and Schanuel [24], Leinster [25], or Mac Lane [27] for an in-depth treatment of category theory. ...
In reinforcement learning, conducting task composition by forming cohesive, executable sequences from multiple tasks remains challenging. However, the ability to (de)compose tasks is a linchpin in developing robotic systems capable of learning complex behaviors. Yet, compositional reinforcement learning is beset with difficulties, including the high dimensionality of the problem space, scarcity of rewards, and absence of system robustness after task composition. To surmount these challenges, we view task composition through the prism of category theory -- a mathematical discipline exploring structures and their compositional relationships. The categorical properties of Markov decision processes untangle complex tasks into manageable sub-tasks, allowing for strategical reduction of dimensionality, facilitating more tractable reward structures, and bolstering system robustness. Experimental results support the categorical theory of reinforcement learning by enabling skill reduction, reuse, and recycling when learning complex robotic arm tasks.
... Then we will discuss quantum computing in categorical terms, complementarity, and computational universality. For the basics of category theory, we refer to Leinster [37]. ...
We construct a computationally universal quantum programming language Quantum Π from two copies of Π, the internal language of rig groupoids. The first step constructs a pure (measurement-free) term language by interpreting each copy of Π in a generalisation of the category Unitary in which every morphism is “rotated” by a particular angle, and the two copies are amalgamated using a free categorical construction expressed as a computational effect. The amalgamated language only exhibits quantum behaviour for specific values of the rotation angles, a property which is enforced by imposing a small number of equations on the resulting category. The second step in the construction introduces measurements by layering an additional computational effect.
... These mathematical structures serve as the basis for our task planning formalism and plan transfer framework. Here, we will provide an expedient introduction to the fundamentals, however, a detailed treatment can be found in [14]. ...
This paper introduces a novel approach to ontology-based robot plan transfer using functorial data migrations from category theory. Functors provide structured maps between domain types and predicates which can be used to transfer plans from a source domain to a target domain without the need for replanning. Unlike methods that create models for transferring specific plans, our approach can be applied to any plan within a given domain. We demonstrate this approach by transferring a task plan from the canonical Blocksworld domain to one compatible with the AI2-THOR Kitchen environment. In addition, we discuss practical applications that may enhance the adaptability of robotic task planning in general.
... 10 The foregoing will take for granted some basic concepts from category theory. For background on that theory, see (for instance) Leinster (2014). and whose arrows are "structure preserving maps" between those models. ...
Teleparallel gravity shares many qualitative features with general relativity, but differs from it in the following way: whereas in general relativity, gravitation is a manifestation of space-time curvature, in teleparallel gravity, spacetime is (always) flat. Gravitational effects in this theory arise due to spacetime torsion. It is often claimed that teleparallel gravity is an equivalent reformulation of general relativity. In this paper we question that view. We argue that the theories are not equivalent, by the criterion of categorical equivalence and any stronger criterion, and that teleparallel gravity posits strictly more structure than general relativity.
... We have also filtered out documents describing books as well as meta-articles such as lists and categories. The third corpus consists of the entire text (4058 sentences) of Basic Category Theory (BCT) by Tom Leinster (Leinster, 2014). This is an introductory textbook, intended for students without advanced degrees in mathematics. ...
Mathematics is a highly specialized domain with its own unique set of challenges. Despite this, there has been relatively little research on natural language processing for mathematical texts, and there are few mathematical language resources aimed at NLP. In this paper, we aim to provide annotated corpora that can be used to study the language of mathematics in different contexts, ranging from fundamental concepts found in textbooks to advanced research mathematics. We preprocess the corpora with a neural parsing model and some manual intervention to provide part-of-speech tags, lemmas, and dependency trees. In total, we provide 182397 sentences across three corpora. We then aim to test and evaluate several noteworthy natural language processing models using these corpora, to show how well they can adapt to the domain of mathematics and provide useful tools for exploring mathematical language. We evaluate several neural and symbolic models against benchmarks that we extract from the corpus metadata to show that terminology extraction and definition extraction do not easily generalize to mathematics, and that additional work is needed to achieve good performance on these metrics. Finally, we provide a learning assistant that grants access to the content of these corpora in a context-sensitive manner, utilizing text search and entity linking. Though our corpora and benchmarks provide useful metrics for evaluating mathematical language processing, further work is necessary to adapt models to mathematics in order to provide more effective learning assistants and apply NLP methods to different mathematical domains.
... A category is itself a mathematical object, and thus it too is characterized by the arrows it has to/from other categories in the "category of categories" (Leinster, 2014). A functor F : C → D between categories C and D is such an "arrow between categories," depicted as follows: ...
The concept of affordance, proposed by James J. Gibson as an opportunity for action offered by the environment to the organism, has been adopted in various fields, including psychology, neuroscience, and robotics. However, different interpretations exist as to whether it is a feature of a relation between the environment and the organism and therefore cannot exist independently of the organism, or a "resource" that exists in the environment independent of the organism's presence and is waiting to be used, or both, or neither. In this paper, we defend the position that affordances are both relational and resources using a category-theoretic approach. This idea is formalized by the concept of "natural transformations" in category theory, which are structure-preserving transformations between "functors" -mathematical expressions representing "seeing from a particular perspective." We propose that formalizing the realism of affordance in terms of natural transformations offers a more rigorous and lucid understanding of this concept. Furthermore, our formalization enables us to relate the reality of affordances to a broader context, especially the shift in the meaning of "reality" in modern physics. Our category-theoretic approach offers a potential solution to the problems and limitations associated with existing set theory-based frameworks for affordances, paving the way for a future theory that better accounts for the open-ended interplay between organisms and their environments.
... In this section, we recall the definitions required to work with category theory as a denotational model of programming languages. The author recommends the book of Tom Leinster [Lei16], which introduces category theory with more background, details and examples. ...
This thesis revolves around an area of computer science called "semantics". We work with operational semantics, equational theories, and denotational semantics. The first contribution of this thesis is a study of the commutativity of effects through the prism of monads. Monads are the generalisation of algebraic structures such as monoids, which have a notion of centre: the centre of a monoid is made of elements which commute with all others. We provide the necessary and sufficient conditions for a monad to have a centre. We also detail the semantics of a language with effects that carry information on which effects are central. Moreover, we provide a strong link between its equational theories and its denotational semantics. The second focus of the thesis is quantum computing, seen as a reversible effect. Physically permissible quantum operations are all reversible, except measurement; however, measurement can be deferred at the end of the computation, allowing us to focus on the reversible part first. We define a simply-typed reversible programming language performing quantum operations called "unitaries". A denotational semantics and an equational theory adapted to the language are presented, and we prove that the former is complete. Furthermore, we study recursion in reversible programming, providing adequate operational and denotational semantics to a Turing-complete, reversible, functional programming language. The denotational semantics uses the dcpo enrichment of rig join inverse categories. This mathematical account of higher-order reasoning on reversible computing does not directly generalise to its quantum counterpart. In the conclusion, we detail the limitations and possible future for higher-order quantum control through guarded recursion.
The maximal proper categories of left [right] principal ideals of a monoid and their proper cones were described. Then analogous to the cross-connection representation of regular semigroups, the cross-connection representation of monoids were described.
This is a review of
Al-Rawashdeh, A.; Mesablishvili, B.
On Amitsur cohomology of monads. (English) Zbl 07960970
Cross-connectios of normal categories were introduced by K.S.S. Nambooripad to construct a regular semigroup which he call a cross-connection semigroup (see. [4]). In this paper we construct the cross-connection semigroup of a regular semigroup and hence obtained a cross-connection representation of a regular semigroup.
Convergence spaces are a generalization of topological spaces. The category of convergence spaces is well-suited for Algebraic Topology, one of the reasons is the existence of exponential objects provided by continuous convergence. In this work, we use a net-theoretic approach to convergence spaces. The goal is to simplify the description of continuous convergence and apply it to problems related to homotopy theory. We present methods to develop the basis of homotopy theory in limit spaces, define the fundamental groupoid, and prove the groupoid version of the Seifert-van Kampen Theorem for limit spaces.
In this paper, we examine the relationship between general relativity and the theory of Einstein algebras. We show that according to a formal criterion for theoretical equivalence recently proposed by Halvorson (2012, 2015) and Weatherall (2015), the two are equivalent theories.
We present colorful illustrations of particular properties of functorial diamonds, in the sense of Scholze, namely profinite reflections as categorical colors. We discuss sight as site using representable functors in the condensed formalism. We illuminate diamonds using our novel constructions of Categorical Ozma and Cinderella, the Site of Oz, and Condensed Through the Looking-Glass.
Starting without any topological assumption, we establish the existence of the universal type structure in presence of—possibly uncountably many and topologically unrestricted—conditioning events, namely, a type structure that is non-redundant, belief-complete, terminal, and unique up to measurable type isomorphism, by performing a construction in the spirit of the hierarchical one in Heifetz and Samet (J Econ Theory 82:324–341, 1998). In particular, we obtain the result by exploiting arguments from category theory and the theory of coalgebras, thus, making explicit the mathematical structure underlying all the constructions of large interactive structures and obtaining the belief-completeness of the structure [unattainable via the standard hierarchical construction à la (Heifetz and Samet in J Econ Theory 82:324–341, 1998)] as an immediate corollary of known results from these fields. Additionally, we show how our construction, with its lack of topological and cardinality assumptions on the family of conditioning events, can be employed in various game-theoretical contexts.
In this paper we investigate a ternary generalization of associativity by defining a diagrammatic calculus of hypergraphs that extends the usual notions of tensor networks, categories and relational algebras. Our key insight is to approach higher associativity as a confluence property of hypergraph rewrite systems. In doing so we rediscover the ternary structures known as heaps and are able to give a more comprehensive treatment of their emergence in the context of dagger categories and their generalizations. This approach allows us to define a notion of ternary category and heapoid, where morphisms bind three objects simultaneously, and suggests a systematic study of higher arity forms of associativity.
This is a review of
Baez, John C.; Li, Xiaoyan; Libkind, Sophie; Osgood, Nathaniel D.; Redekopp, Eric
A categorical framework for modeling with stock and flow diagrams. (English) Zbl 1533.92194
Recent research has suggested that category theory can provide useful insights into the field of machine learning (ML). One example is improving the connection between an ML problem and the design of a corresponding ML algorithm. A tool from category theory called a Kan extension is used to derive the design of an unsupervised anomaly detection algorithm for a commonly used benchmark, the Occupancy dataset. Achieving an accuracy of 93.5% and an ROCAUC of 0.98, the performance of this algorithm is compared to state-of-the-art anomaly detection algorithms tested on the Occupancy dataset. These initial results demonstrate that category theory can offer new perspectives with which to attack problems, particularly in making more direct connections between the solutions and the problem’s structure.
This chapter aims to uncover certain patterns in the considerations of Chs. 2 and 3 and provide evidence for the claim from the introduction to Part I that the presented structure theory of modules would be exemplary for algebraic structures. In particular, we address the parallels mentioned in Section 2.1 between the constructions of sub- and quotient structures for vector spaces, rings, and modules, as well as the repeatedly mentioned naturality of the constructions presented in Ch. 3. The former leads us to basic concepts of universal algebra, the latter to elementary concepts of category theory.
A comprehensive review of diamonds, in the sense of Scholze, is presented. The diamond formulations of the Fargues-Fontaine curve and BunG are reviewed. Principal results centered on the diamond formalism in the global Langlands correspondence and the geometrization of the local Langlands correspondence are given. We conclude with a discussion of future geometrizations.
Algorithms operating on real numbers are implemented as floating-point computations in practice, but floating-point operations introduce roundoff errors that can degrade the accuracy of the result. We propose Λ num , a functional programming language with a type system that can express quantitative bounds on roundoff error. Our type system combines a sensitivity analysis, enforced through a linear typing discipline, with a novel graded monad to track the accumulation of roundoff errors. We prove that our type system is sound by relating the denotational semantics of our language to the exact and floating-point operational semantics.
To demonstrate our system, we instantiate Λ num with error metrics proposed in the numerical analysis literature and we show how to incorporate rounding operations that faithfully model aspects of the IEEE 754 floating-point standard. To show that Λ num can be a useful tool for automated error analysis, we develop a prototype implementation for Λ num that infers error bounds that are competitive with existing tools, while often running significantly faster. Finally, we consider semantic extensions of our graded monad to bound error under more complex rounding behaviors, such as non-deterministic and randomized rounding.
Mathematicians manipulate sets with confidence almost every day, rarely
making mistakes. Few of us, however, could accurately quote what are often
referred to as "the" axioms of set theory. This suggests that we all carry
around with us, perhaps subconsciously, a reliable body of operating principles
for manipulating sets. What if we were to take some of those principles and
adopt them as our axioms instead? The message of this article is that this can
be done, in a simple, practical way (due to Lawvere). The resulting axioms are
ten thoroughly mundane statements about sets.
This is an expository article for a general mathematical readership.
Basic Concepts of Enriched Category Theory Also Reprints in Theory and Applications of Categories
- G M Kelly
G. M. Kelly, Basic Concepts of Enriched Category Theory. Cambridge University Press, 1982. Also Reprints in Theory and Applications of Categories 10 (2005), 1–136, available at http://www.tac.
mta.ca/tac/reprints.
Rethinking set theory Also available at https
- Tom Leinster
Tom Leinster, Rethinking set theory. American Mathematical Monthly 121 (2014), no. 5, 403–415. Also available at https://arxiv.org/
abs/1212.6543.