Categories for the working mathematician. 2nd ed
Abstract
I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large Categories.- 8. Hom-sets.- II. Constructions on Categories.- 1. Duality.- 2. Contravariance and Opposites.- 3. Products of Categories.- 4. Functor Categories.- 5. The Category of All Categories.- 6. Comma Categories.- 7. Graphs and Free Categories.- 8. Quotient Categories.- III. Universals and Limits.- 1. Universal Arrows.- 2. The Yoneda Lemma.- 3. Coproducts and Colimits.- 4. Products and Limits.- 5. Categories with Finite Products.- 6. Groups in Categories.- IV. Adjoints.- 1. Adjunctions.- 2. Examples of Adjoints.- 3. Reflective Subcategories.- 4. Equivalence of Categories.- 5. Adjoints for Preorders.- 6. Cartesian Closed Categories.- 7. Transformations of Adjoints.- 8. Composition of Adjoints.- V. Limits.- 1. Creation of Limits.- 2. Limits by Products and Equalizers.- 3. Limits with Parameters.- 4. Preservation of Limits.- 5. Adjoints on Limits.- 6. Freyd's Adjoint Functor Theorem.- 7. Subobjects and Generators.- 8. The Special Adjoint Functor Theorem.- 9. Adjoints in Topology.- VI. Monads and Algebras.- 1. Monads in a Category.- 2. Algebras for a Monad.- 3. The Comparison with Algebras.- 4. Words and Free Semigroups.- 5. Free Algebras for a Monad.- 6. Split Coequalizers.- 7. Beck's Theorem.- 8. Algebras are T-algebras.- 9. Compact Hausdorff Spaces.- VII. Monoids.- 1. Monoidal Categories.- 2. Coherence.- 3. Monoids.- 4. Actions.- 5. The Simplicial Category.- 6. Monads and Homology.- 7. Closed Categories.- 8. Compactly Generated Spaces.- 9. Loops and Suspensions.- VIII. Abelian Categories.- 1. Kernels and Cokernels.- 2. Additive Categories.- 3. Abelian Categories.- 4. Diagram Lemmas.- IX. Special Limits.- 1. Filtered Limits.- 2. Interchange of Limits.- 3. Final Functors.- 4. Diagonal Naturality.- 5. Ends.- 6. Coends.- 7. Ends with Parameters.- 8. Iterated Ends and Limits.- X. Kan Extensions.- 1. Adjoints and Limits.- 2. Weak Universality.- 3. The Kan Extension.- 4. Kan Extensions as Coends.- 5. Pointwise Kan Extensions.- 6. Density.- 7. All Concepts are Kan Extensions.- Table of Terminology.
Chapters (1)
First we describe categories directly by means of axioms, without using any set theory, and calling them “metacategories”. Actually, we begin with a simpler notion, a (meta)graph.
... How might the categorical properties of the heteromorphisms be expressed using homomorphisms? Represent the het-bifunctors using hom-functors on the left and on the right (see any category theory text such as [17] for Alexander Grothendieck's notion of a representable functor). Any bifunctor Het : X op × A → Set is represented on the left if for each x in X there is a "universal" [3] object F(x) in A and an isomorphism Hom A (F(x), a) ∼ = Het(x, a) natural in a. ...
... where f * ←→ f ←→ f * . It only remains to drop out the middle term Het(x, a) to arrive at the "official" or usual definition of a pair of adjoint functors, which does not mention heteromorphisms [8,17]. ...
... In the rather meager attempts at interpreting adjunctions, e.g., in illustrating a "unity of opposites" [6,7], there is still no hint of directionality. But adjoints do have a direction that comes out when adjunctions are to be composed [17] (p. 104). ...
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.
... Related works. RCK has its roots in category theory [12] and sheaf theory [24], and is inspired by Grothendieck notion of relativism in the context of mathematics [14]. Grothendieck revolutionized the understanding of mathematical structures by shifting the focus from individual objects to the relationships between them, as expressed through morphisms. ...
... Category theory [12] is a branch of pure mathematics that studies abstract structures and their relationships through objects and morphisms, focusing on how they compose and interact. A category C is composed of objects having a certain structure (e.g., measurable spaces, vector spaces) and arrows (morphisms) between them preserving the structure (e.g., measurable maps, linear maps) and satisfying certain axioms (cf. ...
... GD and CB codeveloped the sheaf-theoretic characterization of RCK. [12,18]. Definition 12 (Category) A category C consists of ...
Recent advances in artificial intelligence reveal the limits of purely predictive systems and call for a shift toward causal and collaborative reasoning. Drawing inspiration from the revolution of Grothendieck in mathematics, we introduce the relativity of causal knowledge, which posits structural causal models (SCMs) are inherently imperfect, subjective representations embedded within networks of relationships. By leveraging category theory, we arrange SCMs into a functor category and show that their observational and interventional probability measures naturally form convex structures. This result allows us to encode non-intervened SCMs with convex spaces of probability measures. Next, using sheaf theory, we construct the network sheaf and cosheaf of causal knowledge. These structures enable the transfer of causal knowledge across the network while incorporating interventional consistency and the perspective of the subjects, ultimately leading to the formal, mathematical definition of relative causal knowledge.
... We hope that the benefits of this choice outweigh the price paid for it. Indeed, our formalism being formulated in the language of string diagrams (or equivalently, in the language of symmetric monoidal categories [Mac98]), we expect to be able to obtain straightforward generalizations by changing our base category (the mathematical world in which we interpret our diagrams) from relation to stochastic kernels or quantum channels. This provides solid theoretical bases to define meaningful notions of automata, transition systems, and symbolic dynamical systems in the probabilistic and quantum case, on which we could develop similar simulation-based proof techniques. ...
... All those diagrams can actually be formalized using string diagrams from category theory [Mac98]. Indeed, finite sets and relations form a strict symmetric monoidal category, called FinRel. ...
... All the diagrams of Section 2 can actually be formalized using string diagrams from category theory [Mac98]. Indeed, finite sets and relations is well known to be a strict symmetric monoidal category, called FinRel, that is: ...
Minimizing finite automata, proving trace equivalence of labelled transition systems or representing sofic subshifts involve very similar arguments, which suggests the possibility of a unified formalism. We propose finite states non-deterministic transducer as a lingua franca for automata theory, transition systems, and sofic subshifts. We introduce a compositional diagrammatical syntax for transducers in form of string diagrams interpreted as relations. This syntax comes with sound rewriting rules allowing diagrammatical reasoning. Our main result is the completeness of our equational theory, ensuring that language-equivalence, trace-equivalence, or subshift equivalence can always be proved using our rewriting rules.
... (cf. [7]) A morphism m : A → B in a category C is monic (also called monomorphism), if for any two parallel arrows ...
... (cf. [7]) An object T is terminal in a category C if to each object A in C there is exactly one arrow A → T . An object S is initial in C if to each object A there is exactly one arrow S → A. ...
... If C is a category with zero object, for any two objects A and B there is a unique arrow 0 A,B : A → 0 → B called the zero arrow from A to B. In the category Set, the empty set is an initial object and any one point set is a terminal object (cf. [7]). Definition 2.5. ...
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.
... We would like the isomorphism to coincide with (the inverse of) the natural homomorphism from Proposition 6.1. To achieve this, we refine the argument about the uniqueness of a left adjoint using the concept of conjugate natural transformations, which is introduced in the following theorem (for the proof see section I.7. in [15]). Theorem 6.2 ( [15]). ...
... To achieve this, we refine the argument about the uniqueness of a left adjoint using the concept of conjugate natural transformations, which is introduced in the following theorem (for the proof see section I.7. in [15]). Theorem 6.2 ( [15]). Let L, L ′ : C → D be two functors having right adjoints R, R ′ (respectively) and α : L ⇒ L ′ be a natural transformation. ...
We study a tensor product in the category of effect algebras and in the category of partially ordered Abelian groups with order unit. We show that the tensor product preserves all the constructions that are essentially colimits over a connected diagram. Further, we prove the construction of a universal group for an effect algebra preserves all tensor products. We establish the corresponding functor from the category of effect algebras to the category of unital Abelian po-groups as a strong monoidal functor. We note that the technique we use in establishing the result could be used in various similar situations. Finally, we show that the tensor product of effect algebras does not preserve the Riesz decomposition property, which was an open question for a while.
... The most general framework formalizing these intuitions is category theory [140]. With a category V chosen, the data containers Dpsq are just objects of V, and Dps 1 ă s 2 q : Dps 1 q Ñ Dps 2 q are morphisms of V. Every poset pS, ďq may be treated as a syntactic category catpSq, whose objects are the elements of S, and morphisms are the relations a ď b. ...
... If F is a presheaf on a topological space X and x P X, then The definition makes sense only if the category V has direct limits. From the general categorical definition of a direct limit [140] it follows that for each U Q x, U P Ω X there exists a canonical morphism FpU Q xq : FpU q Ñ F x . ...
This paper provides an overview of the applications of sheaf theory in deep learning, data science, and computer science in general. The primary text of this work serves as a friendly introduction to applied and computational sheaf theory accessible to those with modest mathematical familiarity. We describe intuitions and motivations underlying sheaf theory shared by both theoretical researchers and practitioners, bridging classical mathematical theory and its more recent implementations within signal processing and deep learning. We observe that most notions commonly considered specific to cellular sheaves translate to sheaves on arbitrary posets, providing an interesting avenue for further generalization of these methods in applications, and we present a new algorithm to compute sheaf cohomology on arbitrary finite posets in response. By integrating classical theory with recent applications, this work reveals certain blind spots in current machine learning practices. We conclude with a list of problems related to sheaf-theoretic applications that we find mathematically insightful and practically instructive to solve. To ensure the exposition of sheaf theory is self-contained, a rigorous mathematical introduction is provided in appendices which moves from an introduction of diagrams and sheaves to the definition of derived functors, higher order cohomology, sheaf Laplacians, sheaf diffusion, and interconnections of these subjects therein.
... However, there are some challenges when working with univalent categories. For instance, given a monad m on a category C, we usually define the Kleisli category K(m) of m to be the category whose objects are objects of C and whose morphisms from x to y are morphisms x → m(x) [ML13]. Even if we assume C to be univalent, the category K(m) does not have to be univalent as well, and this means that K(m) does not give the Kleisli category for a monad on a univalent category. ...
... The two coherences in Definition 7.1 are called the triangle equalities. As expected, adjunctions internal to UnivCat correspond to adjunctions of categories [ML13]. This is because the unitors and associators in UnivCat are pointwise the identity, so the triangle equalities in Definition 7.1 reduce to the usual ones. ...
We develop the formal theory of monads, as established by Street, in univalent foundations. This allows us to formally reason about various kinds of monads on the right level of abstraction. In particular, we define the bicategory of monads internal to a bicategory, and prove that it is univalent. We also define Eilenberg-Moore objects, and we show that both Eilenberg-Moore categories and Kleisli categories give rise to Eilenberg-Moore objects. Finally, we relate monads and adjunctions in arbitrary bicategories. Our work is formalized in Coq using the UniMath library.
... The minimum requirements of the relation, directed graph, and category theory for the paper include: binary relation, equivalence relation, equivalence class, quotient, category, homomorphism, isomorphism, coproduct, pullback, pushout, monic, epic, injection, initial object, functor, natural transformation, Yoneda lemma and embedding, adjunction, monad, T -algebra. For the related notions, notations, results, and a systematic introduction, the reader may consult, for instance, [13,2,6]. ...
... If F, G; η, ε : C → B is an adjunction, then GF, η, GεF is a monad on C (see [13], p.138). In fact, every monad arises this way. ...
In this paper, we study the machine learning elements which we are interested in together as a machine learning system, consisting of a collection of machine learning elements and a collection of relations between the elements. The relations we concern are algebraic operations, binary relations, and binary relations with composition that can be reasoned categorically. A machine learning system transformation between two systems is a map between the systems, which preserves the relations we concern. The system transformations given by quotient or clustering, representable functor, and Yoneda embedding are highlighted and discussed by machine learning examples. An adjunction between machine learning systems, a special machine learning system transformation loop, provides the optimal way of solving problems. Machine learning system transformations are linked and compared by their maps at 2-cell, natural transformations. New insights and structures can be obtained from universal properties and algebraic structures given by monads, which are generated from adjunctions.
... This means that we could equivalently define the compressed commuting graph of R as an undirected graph whose vertex set is the set In what follows we extend the mapping to a functor . To do this we will need some basic notions from category theory, see for example [29]. In particular, recall that a functor F between two categories A and B is a mapping that associates to each object X in A an object F(X) in B, and to each morphism f : ...
In this paper we introduce compressed commuting graph of rings. It can be seen as a compression of the standard commuting graph (with the central elements added) where we identify the vertices that generate the same subring. The compression is chosen in such a way that it induces a functor from the category of rings to the category of graphs, which means that our graph takes into account not only the commutativity relation in the ring, but also the commutativity relation in all of its homomorphic images. Furthermore, we show that this compression is best possible for matrix algebras over finite fields, i.e. it compresses as much as possible while still inducing a functor. We compute the compressed commuting graphs of finite fields and rings of 2 × 2 matrices over finite fields. ARTICLE HISTORY
... 2.A family of 1-morphisms {π j : L → D(j)} j∈Ob(J ) ,3. For each 1-morphism u : j → k in J , an invertible 2-morphismθ u : π k =⇒ D(u) • π j ,such that for any other object X equipped with a cone {f j : X → D(j)} and invertible 2-morphisms {φ u : f k =⇒ D(u) • f j }satisfying the analogous coherence conditions, there exists a unique (up to a unique invertible 2-morphism) 1-morphism u : X → L making all the corresponding diagrams commute up to the given 2-morphisms. ...
This paper develops a systematic framework for integrating local categories that model logical connectives using higher category theory. By extending these local categories into a unified two-category enriched with natural isomorphisms, the universal properties of logical operations such as negation, conjunction, disjunction, and implication are rigorously captured. Advanced techniques including pseudo-limits, pseudo-colimits, and strictification are employed to transform the resulting weak structure into a strict two-category, thereby simplifying composition rules and coherence verification without loss of semantic content. The framework is validated through detailed diagrammatic proofs and concrete examples, demonstrating its robustness and potential impact in areas such as type theory, programming language semantics, and formal verification.
... For an introduction to categories, see Mac Lane (2013); Riehl (2017). A category C is a collection of objects O(C) and morphisms ϕ : A → B between two objects. ...
Undirected graphical models are a widely used class of probabilistic models in machine learning that capture prior knowledge or putative pairwise interactions between variables. Those interactions are encoded in a graph for pairwise interactions; however, generalizations such as factor graphs account for higher-degree interactions using hypergraphs. Inference on such models, which is performed by conditioning on some observed variables, is typically done approximately by optimizing a free energy, which is an instance of variational inference. The Belief Propagation algorithm is a dynamic programming algorithm that finds critical points of that free energy. Recent efforts have been made to unify and extend inference on graphical models and factor graphs to more expressive probabilistic models. A synthesis of these works shows that inference on graphical models, factor graphs, and their generalizations relies on the introduction of presheaves and associated invariants (homol-ogy and cohomology groups).We propose to study the impact of the transformation of the presheaves onto the associated message passing algorithms. We show that natural transformations between presheaves associated with graphical models and their generalizations, which can be understood as coherent binning of the set of values of the variables, induce morphisms between associated message-passing algorithms. It is, to our knowledge, the first result on functoriality of the Loopy Belief Propagation.
... By Beck's theorem (see e.g. [18]), if A has all coequalizers and U has a left adjoint, preserves coequalizers and reflects isomorphisms, then U is monadic. If B is a category with all limits and colimits and T is a monad on B that preserves filtered colimits, then B T has all limits and colimits [2]. ...
Proto-exact and parabelian categories serve as non-additive analogues of exact and quasi-abelian categories, respectively. They give rise to algebraic K-theory and Hall algebras similarly to the additive setting. We show that every parabelian category admits a canonical proto-exact structure and we study several classes of parabelian categories, including categories of normed and Euclidean vector spaces, pointed closure spaces and pointed matroids, Hermitian vector bundles over rings of integers. We also examine finitary algebraic categories arising in Arakelov geometry and provide a criterion for determining when such a category is parabelian. In particular, we prove that the categories of pointed convex spaces and absolutely convex spaces are parabelian.
... Unlike the traditional set-theoretic approach, which characterizes mathematical objects in terms of their point-like elements and then describes relationships between them as functions (mappings from one set of points to another), category theory primarily focuses on the relationships, called "arrows" (or "morphisms") between objects. As Leinster (2014, p. 9) notes, in a category, "the objects do not live in isolation": in category theory, every object is characterized by the arrows it has to and from other objects in the category and not by its constitutive elements, which are defined independently of other objects (Mac Lane, 1978). In essence, it is "arrow-first" mathematics (Hirota et al., 2023). ...
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.
... We may regard ∆ as a (non-symmetric) strict monoidal category with monoidal product + given on objects by addition. This monoidal category ∆ has a well-known universal property that is discussed in [34,VII.5] and entails that a strict left ∆-actegory is equivalently a category C equipped with a monad T = (T, µ, η), noting that the ∆-action is then given on objects by n.A = T n A (n ∈ N = ob ∆). ...
In the well-known settings of category theory enriched in a monoidal category V, the use of V-enriched functor categories and bifunctors demands that V be equipped with a symmetry, braiding, or duoidal structure. In this paper, we establish a theory of functor categories and bifunctors that is applicable relative to an arbitrary monoidal category V and applies both to V-enriched categories and also to V-actegories. We accomplish this by working in the setting of (V-)graded categories, which generalize both V-enriched categories and V-actegories and were introduced by Wood under the name "large V-categories". We develop a general framework for graded functor categories and graded bifunctors taking values in bigraded categories, noting that V itself is canonically bigraded. We show that V-graded modules (or profunctors) are examples of graded bifunctors and that V-graded presheaf categories are examples of V-graded functor categories. In the special case where V is normal duoidal, we compare the above graded concepts with the enriched bifunctors and functor categories of Garner and L\'opez Franco. Along the way, we study several foundational aspects of graded categories, including a contravariant change of base process for graded categories and a formalism of commutative diagrams in graded categories that arises by freely embedding each V-graded category into a V-actegory.
... Definition 2.1. [38] (Mathematical category) A category C consists of a collection of objects denoted by (C) and a collection of morphisms denoted by (C). For each morphism ∈ (C) there exists an object ∈ (C) that is a domain of and an object ∈ (C) that is a target of . ...
Modern database systems face a significant challenge in effectively handling the Variety of data. The primary objective of this paper is to establish a unified data model and theoretical framework for multi-model data management. To achieve this, we present a categorical framework to unify three types of structured or semi-structured data: relation, XML, and graph-structured data. Utilizing the language of category theory, our framework offers a sound formal abstraction for representing these diverse data types. We extend the Entity-Relationship (ER) diagram with enriched semantic constraints, incorporating categorical ingredients such as pullback, pushout and limit. Furthermore, we develop a categorical normal form theory which is applied to category data to reduce redundancy and facilitate data maintenance. Those normal forms are applicable to relation, XML and graph data simultaneously, thereby eliminating the need for ad-hoc, model-specific definitions as found in separated normal form theories before. Finally, we discuss the connections between this new normal form framework and Boyce-Codd normal form, fourth normal form, and XML normal form.
... Let us be more precise. For the unexplained notions of category theory we refer the reader to [39]. ...
We provide a new foundational approach to the generalization of terms up to equational theories. We interpret generalization problems in a universal-algebraic setting making a key use of projective and exact algebras in the variety associated to the considered equational theory. We prove that the generality poset of a problem and its type (i.e., the cardinality of a complete set of least general solutions) can be studied in this algebraic setting. Moreover, we identify a class of varieties where the study of the generality poset can be fully reduced to the study of the congruence lattice of the 1-generated free algebra. We apply our results to varieties of algebras and to (algebraizable) logics. In particular we obtain several examples of unitary type: abelian groups; commutative monoids and commutative semigroups; all varieties whose 1-generated free algebra is trivial, e.g., lattices, semilattices, varieties without constants whose operations are idempotent; Boolean algebras, Kleene algebras, and G\"odel algebras, which are the equivalent algebraic semantics of, respectively, classical, 3-valued Kleene, and G\"odel-Dummett logic.
... where ⟨·|·⟩ denotes the inner product and ||·|| is the induced norm. In finite-dimensional dagger categories, this process can be implemented using dagger monic and dagger epic morphisms [34], which satisfy: The numerical implementation of these constructions requires careful consideration of computational stability and efficiency. A key aspect is the preservation of the dagger structure through what is known as dagger finiteness. ...
This paper presents a self-contained theoretical and practical framework for optimal basis selection in finite-dimensional Hilbert spaces, with applications in signal processing and numerical analysis. Based on quantum-optical state theory and dagger category frameworks, we give a rigorous mathematical development that yields a criterion for basis selection, enabling a proper trade-off between approximation accuracy and computational efficiency. The study introduces new methods to represent discrete signals via Wigner functions, along with explicit error bounds for numerical approximations. Our framework contains both theoretical foundations and practical implementations, illustrated by case studies showing the validity of the approach. Hence, it makes three key contributions: it gives a unified treatment of the basis selection criteria; it provides efficient numerical methods with provable convergence properties; and it presents practical implementation strategies for signal processing applications. Results indicate that our finite-dimensional approach attains high accuracy yet remains computationally efficient, with fidelity measures above 0.99 for optimal parameter choices. To further validate the performance of the framework, extensive numerical experiments are provided, which show the usefulness of the framework in a variety of applications from quantum state representation to digital signal processing. This work also addresses some of the theoretical challenges in dimensional scaling and error propagation, establishing a basis for future developments in the field. Our results show that finite dimensional methods offer significant benefits in mathematical tractability and computational realization at the cost of little accuracy for most practical applications.
... Category theory, from its very beginnings, was conceived as an abstraction of the notions of set and function. This intuition is clearly expressed in the first sentence of the introduction of the well-known book by Mac Lane [51] Category theory starts with the observation that many properties of mathematical systems can be unified and simplified by a presentation with diagrams of arrows. Each arrow f : X → Y represents a function; that is, a set X, a set Y , and a rule x → f (x) which assigns to each element x ∈ X an element f (x) ∈ Y . ...
The study of categories abstracting the structural properties of relations has been extensively developed over the years, resulting in a rich and diverse body of work. This paper strives to provide a modern and comprehensive presentation of these ``categories for relations'', including their enriched version, further showing how they arise as Kleisli categories of suitable symmetric monoidal monads. The resulting taxonomy aims at bringing clarity and organisation to the numerous related concepts and frameworks occurring in the literature
... Traditional process theories are commonly formulated using the mathematics of category theory Mac Lane [1998], Coecke and Paquette [2010] and in particular, as symmetric monoidal categories Mac Lane [1963], Coecke and Paquette [2010] as we briefly review in Sec. 1.1. ...
Process theories provide a powerful framework for describing compositional structures across diverse fields, from quantum mechanics to computational linguistics. Traditionally, they have been formalized using symmetric monoidal categories (SMCs). However, various generalizations, including time-neutral, higher-order, and enriched process theories, do not naturally conform to this structure. In this work, we propose an alternative formalization using operad algebras, motivated by recent results connecting SMCs to operadic structures, which captures a broader class of process theories. By leveraging the string-diagrammatic language, we provide an accessible yet rigorous formulation that unifies and extends traditional process-theoretic approaches. Our operadic framework not only recovers standard process theories as a special case but also enables new insights into quantum foundations and compositional structures. This work paves the way for further investigations into the algebraic and operational properties of generalised process theories within an operadic setting.
... Since the core idea is to move from the 1-dimensional context of monoids to the 2-dimensional context of monoidal categories, we first fix the notations and the conventions we will use for monoidal categories. For the basic theory of monoidal categories we refer to [16] or [3]. See also [15] and [14] for coherence issues. ...
In this paper we describe a homotopy torsion theory in the category of small symmetric monoidal categories. Thanks to the use of natural isomorphisms as basis for the nullhomotopy structure, this homotopy torsion theory enjoys some interesting 2-dimensional properties which may be the starting point for a definition of "2-dimensional torsion theory". As torsion objects we take symmetric 2-groups, thus generalising a known pointed torsion theory in the category of commutative monoids where abelian groups play the part of torsion objects. In the last part of the paper we carry out an analogous generalisation for the classical torsion theory in the category of abelian groups given by torsion and torsion-free groups.
... In all that follows we use standard concepts and constructions from set theory, see, e.g., [5,15]; universal algebra, see, e.g., [3,4,6,17,37]; lattice theory, see, e.g., [4,20]; semigroup theory, see, e.g., [21]; and category theory, see, e.g., [1,2,18,19,26,27]. Nevertheless, regarding set theory, we have adopted the following conventions. ...
For a plural signature and with regard to the category , of naturally preordered idempotent -algebras and surjective homomorphisms, we define a contravariant functor from to , the category of categories, that assigns to in the category -, of -semi-inductive Lallement systems of -algebras, and a covariant functor from to , that assigns to in the category , of the coverings of , i.e., the ordered pairs in which is a -algebra and a surjective homomorphism. Then, by means of the Grothendieck construction, we obtain the categories and ; define a functor from the first category to the second, which we will refer to as the Lallement functor; and prove that it is a weak right multiadjoint. Finally, we state the relationship between the Płonka functor and the Lallement functor.
... be the well-known functor[Mac98, II.7] that takes a graph to the category freely generated by it.is the subcategory of stratified strict categories defined in Definition 2.9.Proposition 2.14. ...
C-systems were defined by Cartmell as the algebraic structures that correspond exactly to generalised algebraic theories. B-systems were defined by Voevodsky in his quest to formulate and prove an initiality conjecture for type theories. They play a crucial role in Voevodsky's construction of a syntactic C-system from a term monad. In this work, we construct an equivalence between the category of C-systems and the category of B-systems, thus proving a conjecture by Voevodsky. We construct this equivalence as the restriction of an equivalence between more general structures, called CE-systems and E-systems, respectively. To this end, we identify C-systems and B-systems as "stratified" CE-systems and E-systems, respectively; that is, systems whose contexts are built iteratively via context extension, starting from the empty context.
Certain results involving "higher structures" are not currently accessible to computer formalization because the prerequisite -category theory has not been formalized. To support future work on formalizing -category theory in Lean's mathematics library, we formalize some fundamental constructions involving the 1-category of categories. Specifically, we construct the left adjoint to the nerve embedding of categories into simplicial sets, defining the homotopy category functor. We prove further that this adjunction is reflective, which allows us to conclude that has colimits. To our knowledge this is the first formalized proof that the category of categories is cocomplete.
In this work, we use the theory of quantum states over time to define joint entropy for timelike-separated quantum systems. For timelike-separated systems that admit a dual description as being spacelike-separated, our notion of entropy recovers the usual von Neumann entropy for bipartite quantum states and thus may be viewed as a spacetime generalization of von Neumann entropy. Such an entropy is then used to define dynamical extensions of quantum joint entropy, quantum conditional entropy, and quantum mutual information for systems separated by the action of a quantum channel. We provide an in-depth mathematical analysis of such information measures and the properties they satisfy. We also use such a dynamical formulation of entropy to quantify the information loss/gain associated with the dynamical evolution of quantum systems, which enables us to formulate a precise notion of information conservation for quantum processes. Finally, we show how our dynamical entropy admits an operational interpretation in terms of quantifying the amount of state disturbance associated with a positive operator- valued measurement.
This chapter presents mathematical concepts from category theory, which constitute the basis for the results discussed in the upcoming chapters. We start with the introduction of basic ideas of category theory and related definitions. After that, constructions on categories and abstract structures are defined. Next, lattice of theories concept is also important for some parts of this book, therefore it will be introduced in this chapter as well. After that, this chapter presents categorical ontology logs, or simply ologs, which combine flexibility of classical all-purpose ontologies and mathematical rigorosity of category theory. Finally, we discuss some classical and more recent applications of category theory focusing in particular on applications related to conceptual modelling.
This chapter establishes a general context for the rest of this book. We start with a general discussion on models and modelling process within the engineering context. We then discuss the model consistency and the modelling errors that may arise during the modelling process. Specifically, we emphasise that it is necessary to distinguish between the conceptual modelling errors, which appear because of violating basic modelling assumptions, and the instance modelling errors, which are related to the practical implementation of models. After that, we provide a brief overview of the conceptual modelling approaches, which could help us to evaluate models with respect to their coherence, as well as provide possibilities for a more formal description of the model creation process.
In this chapter, we explore applications of type theory and abstract algebra to conceptual modelling in engineering. This chapter begins with a presentation of ideas for automatic model generation. The key concept is to analyse the formalisation mappings that appear in categories of mathematical models from a computer science perspective. These ideas naturally lead to the application of type theory to mathematical modelling. In this concept, mathematical models are formalised by types, and the type system of a functional programming language can be utilised to verify the consistency of model derivation. A general algorithm for such verification is presented in this chapter. Finally, a relational algebra-based approach to abstract description of models is proposed. While this approach lacks some strictness of the category theory-based modelling framework, it compensates with enhanced flexibility in working with models and relations between them. Moreover, even models of physical objects, which are not necessary described in terms of mathematical expressions and equations, can also be addressed by this approach.
We introduce the monoidal Rips filtration, a filtered simplicial set for weighted directed graphs and other lattice-valued networks. Our construction generalizes the Vietoris-Rips filtration for metric spaces by replacing the maximum operator, determining the filtration values, with a more general monoidal product. We establish interleaving guarantees for the monoidal Rips persistent homology, capturing existing stability results for real-valued networks. When the lattice is a product of totally ordered sets, we are in the setting of multiparameter persistence. Here, the interleaving distance is bounded in terms of a generalized network distance. We use this to prove a novel stability result for the sublevel Rips bifiltration. Our experimental results show that our method performs better than flagser in a graph regression task, and that combining different monoidal products in point cloud classification can improve performance.
Given a compact Lagrangian L in a semipositive convex-at-infinity symplectic manifold W, we establish a cup-length estimate for the action values of L associated to a Hamiltonian isotopy whose spectral norm is smaller than some . When L is rational, this implies a cup-length estimate on the number of intersection points. This Chekanov-type result generalizes a theorem of Kislev and Shelukhin proving non-displaceability in the case when W is closed and monotone. The method of proof is to deform the pair-of-pants product on Hamiltonian Floer cohomology using the Lagrangian L.
In this survey, we provide an overview of category theory-derived machine learning from four mainstream perspectives: gradient-based learning, probability-based learning, invariance and equivalence-based learning, and topos-based learning. For the first three topics, we primarily review research in the past five years, updating and expanding on the previous survey by Shiebler et al. The fourth topic, which delves into higher category theory, particularly topos theory, is surveyed for the first time in this paper. In certain machine learning methods, the compositionality of functors plays a vital role, prompting the development of specific categorical frameworks. However, when considering how the global properties of a network reflect in local structures and how geometric properties and semantics are expressed with logic, the topos structure becomes particularly significant and profound.
Financial markets exhibit complex, multi-scale behaviors that challenge traditional mathematical modeling techniques. Existing tools, such as moving averages, momentum indicators, and statistical models, often fail to capture the continuous transformations, abrupt market shifts, and higher-order dependencies that characterize real-world financial systems. Homotopy Type Theory (HoTT) offers a novel framework that unifies continuous and discrete market structures, enabling a more adaptive and topologically robust approach to financial modeling. By leveraging homotopy spaces, fibrations, and persistent homology, HoTT allows traders, analysts, and AI-driven systems to track market dynamics, detect emerging trends, and anticipate downturns more effectively. Unlike conventional models that rely on linear correlations and pairwise dependencies, HoTT incorporates multi-scale interactions, structural market topologies, and probabilistic market shifts, making it particularly suited for algorithmic trading, risk management, and financial forecasting. This paper explores the advantages of HoTT over category theory and topos theory, outlining its applications in quantitative finance, AI-driven market prediction, and the detection of insider trading—laying the foundation for a new era of financial mathematics.
Keywords: Homotopy Type Theory, financial markets, algorithmic trading, market topology, persistent homology, fibrations, market downturns, risk management, insider trading detection, AI-driven finance, Bayesian networks, probabilistic forecasting, high-frequency trading, market structure analysis, regime shifts, quantitative finance, topos theory, category theory, multi-scale modeling, financial mathematics. 47 pages. A collaboration with GPT-4o. CC4.0.
We study the noncommutative base change of an entwining structure by a Grothendieck category , using two module like categories. These are the categories of entwined comodule objects and entwined contramodule objects in over the entwining structure . We consider criteria for maps between these noncommutative spaces, induced by generalized maps between entwining structures, known as measurings, to behave like Galois extensions. We also study conditions for extensions of these noncommutative spaces, understood as functors between module like categories, to have separability, Frobenius or Maschke type properties.
A number of model-comparison games central to (finite) model theory, such as pebble and Ehrenfeucht-Fra\"{i}ss\'{e} games, can be captured as comonads on categories of relational structures. In particular, the coalgebras for these comonads encode in a syntax-free way preservation of resource-indexed logic fragments, such as first-order logic with bounded quantifier rank or a finite number of variables. In this paper, we extend this approach to existential and positive fragments (i.e., without universal quantifiers and without negations, respectively) of first-order and modal logic. We show, both concretely and at the axiomatic level of arboreal categories, that the preservation of existential fragments is characterised by the existence of so-called pathwise embeddings, while positive fragments are captured by a newly introduced notion of positive bisimulation.
Let be a functor from a category to a homological (Borceux–Bourn) or semi-abelian (Janelidze–Márki–Tholen) category . We investigate conditions under which the homology of an object X in with coefficients in the functor F, defined via projective resolutions in , remains independent of the chosen resolution. Consequently, the left derived functors of F can be constructed analogously to the classical abelian case.
Our approach extends the concept of chain homotopy to a non-additive setting using the technique of imaginary morphisms. Specifically, we utilize the approximate subtractions of Bourn–Janelidze, originally introduced in the context of subtractive categories. This method is applicable when is a pointed regular category with finite coproducts and enough projectives, provided the class of projectives is closed under protosplit subobjects, a new condition introduced in this article and naturally satisfied in the abelian context. We further assume that the functor F meets certain exactness conditions: for instance, it may be protoadditive and preserve proper morphisms and binary coproducts—conditions that amount to additivity when and are abelian categories.
Within this framework, we develop a basic theory of derived functors, compare it with the simplicial approach, and provide several examples.
This study formulates the basic premises of materialism, which has largely lost its visibility despite being one of the fundamental philosophical approaches that have been effective in the development of modern scientific practice and the construction of philosophy of science, in an alternative way, and aims to develop a new materialist interpretation of it that is non-reductive, pluralistic and open to the use of more than one scientific discipline. This interpretation, expressed with the term relational materialism, first addresses matter with the concept of signifier and foregrounds the concept of beable as the general philosophical category of matter. Secondly, it formulates the category of beable within the irreducible integrity of the categories of relationality, nonstaticity, and finitude; and positions knownability in terms of its correspondence to these general onto-epistemological categories. Thirdly, it clarifies the conditions of existence and knownability of particular entities under general categories based on specially corresponding onto-epistemological categories (interactability, structurability, contextuality, transformability, scale-dependency, actuality, contingency). In this respect, this study offers a pluralistic philosophical framework within which different methodological positions and scientific disciplines can be formulated and criticized based on combinations of different particular categories under general categories. In the conclusion of this article, the meaning and potential of relational materialism for the development of scientific research programs are evaluated.
Quasi MV-algebras are a generalization of MV-algebras and they are motivated by the investigation of the structure of quantum logical gates. In the first part, we present relationships between ideals, weak ideals, congruences, and perfectness within MV-algebras and quasi MV-algebras, respectively. To achieve this goal, we provide a comprehensive characterization of congruence relations of a quasi MV-algebra A concerning the congruence relations of its MV-algebra of regular elements of A , along with specific equivalence relations concerning the complement of the set of regular elements. In the second part, we concentrate on perfect quasi MV-algebras. We present their representation by symmetric quasi ℓ -groups, a special kind of quasi ℓ -groups. Moreover, we establish a categorical equivalence of the category of perfect quasi MV-algebras, the category of n -perfect quasi MV-algebras, and the category of symmetric quasi ℓ -groups.
The present paper investigates information systems stemming from rough set theory against the backdrop of the Ellsberg Paradox. In the early 1960s, D. Ellsberg introduced and discussed a type of informational uncertainty/ambiguity that could not be adequately modelled by the concept of measurable risk. Twenty years later, Z. Pawlak developed rough set theory—a mathematical framework aimed at modelling and managing different types of informational uncertainty that may be found in data analysis (including various forms of imprecision, vagueness, or indecision). However, as in economics in the 1960s, by uncertainty it has commonly been meant, in default of any other alternative interpretation, a sort of measurable risk. In the present study, we would like to fill this research gap and examine D. Ellsberg’s distinction between measurable risk and unmeasurable ambiguity in the context of various forms of information systems related to rough set theory. It turns out that this distinction brings an interesting reformulation of rough sets and reveals hidden facets of this theory. Specifically, we investigate two versions of ambiguity that can be derived from the Ellsberg paradox and relate them to the row and column oriented representations of information systems.
This is a review of
Adámek, Jiří; Dostál, Matěj; Velebil, Jiří Sifted colimits, strongly finitary monads and continuous algebras. (English) £ ¢ ¡ Zbl 07977822 Theory Appl. Categ. 44, 84-131 (2025).
We introduce categorical models of spaces, which we call normed symmetric monoidal categories (NSMCs). These are ordinary symmetric monoidal categories equipped with compatible families of norm maps, and when specialized to a particular class of examples, they reveal a connection between the equivariant symmetric monoidal categories of Guillou–May–Merling–Osorno and those of Hill–Hopkins. We also give an operadic interpretation of the Mac Lane coherence theorem and generalize it to include NSMCs. Among other things, this theorem ensures that the classifying space of an NSMC is an space. We conclude by extending our coherence theorem to include NSMCs with strict relations.
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.
Topology is the branch of mathematics dedicated to the study of continuous maps. The concept of a continuous map has already proven important in calculus, for example, when studying maps between metric spaces compatible with the formation of limits. In this chapter, we will start by reviewing this context and then follow on.
Structural causal models (SCMs) allow us to investigate complex systems at multiple levels of resolution. The causal abstraction (CA) framework formalizes the mapping between high- and low-level SCMs. We address CA learning in a challenging and realistic setting, where SCMs are inaccessible, interventional data is unavailable, and sample data is misaligned. A key principle of our framework is , formalized as the high-level distribution lying on a subspace of the low-level one. This principle naturally links linear CA to the geometry of the . We present a category-theoretic approach to SCMs that enables the learning of a CA by finding a morphism between the low- and high-level probability measures, adhering to the semantic embedding principle. Consequently, we formulate a general CA learning problem. As an application, we solve the latter problem for linear CA; considering Gaussian measures and the Kullback-Leibler divergence as an objective. Given the nonconvexity of the learning task, we develop three algorithms building upon existing paradigms for Riemannian optimization. We demonstrate that the proposed methods succeed on both synthetic and real-world brain data with different degrees of prior information about the structure of CA.
In (Davydov et al. Selecta Mathematica (N.S.) 19, 237–269 2013, Rem. 3.4) the authors asked the question if any étale subalgebra of an étale algebra in a braided fusion category is also étale. We give a positive answer to this question if the braided fusion category is pseudo-unitary and non-degenerate. In the case of a pseudo-unitary fusion category we also give a new description of the lattice correspondence from (Davydov et al. J. für die reine und angewandte Mathematik. 677, 135–177 2013, Theorem 4.10). This new description enables us to describe the two binary operations on the lattice of fusion subcategories.
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