The question "What is category theory" is approached by focusing on universal mapping properties and adjoint functors. Category theory organizes mathematics using morphisms that transmit structure and determination. Structures of mathematical interest are usually characterized by some universal mapping property so the general thesis is that category theory is about determination through universals. In recent decades, the notion of adjoint functors has moved to center-stage as category theory's primary tool to characterize what is important and universal in mathematics. Hence our focus here is to present a theory of adjoint functors, a theory which shows that all adjunctions arise from the birepresentations of "chimeras" or "heteromorphisms" between the objects of different categories. Since representations provide universal mapping properties, this theory places adjoints within the framework of determination through universals. The conclusion considers some unreasonably effective analogies between these mathematical concepts and some central philosophical themes.
Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades, the notion of adjoint functors has moved to center-stage as category theory's primary tool to characterize what is important in mathematics. Our focus here is to present a theory of adjoint functors. The basis for the theory is laid by first showing that the object-to-object "heteromorphisms" between the objects of different categories (e.g., insertion of g…
We consider analogs of Jacobson's $F$-Burnside construction and Boltje's $(-)_+$-construction for biset functors, using Mackey-functor theoretic interpretation of biset functors.
We prove an adjoint functor theorem in the setting of categories enriched in a monoidal model category $\mathcal V$ admitting certain limits. When $\mathcal V$ is equipped with the trivial model structure this recaptures the enriched version of Freyd's adjoint functor theorem. For non-trivial model structures, we obtain new adjoint functor theorems of a homotopical flavour - in particular, when $\mathcal V$ is the category of simplical sets we obtain a homotopical adjoint functor theorem appropriate to the $\infty$-cosmoi of Riehl and Verity. We also investigate accessibility in the enriched s…
There is a lot of redundancy in the usual definition of adjoint functors. We define and prove the core of what is required. First we do this in the hom-enriched context. Then we do it in the cocompletion of a bicategory with respect to Kleisli objects, which we then apply to internal categories. Finally, we describe a doctrinal setting.
We survey results on Hedetniemi's conjecture which are connected to adjoint functors in the "thin" category of graphs, and expose the obstacles to extending these results.
We survey some uses of adjoint functors in graph theory pertaining to colourings, complexity reductions, multiplicativity, circular colourings and tree duality. The exposition of these applications through adjoint functors unifies the presentation to some extent, and also raises interesting questions.
Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper we prove general adjoint functor theorems for functors between $\infty$-categories. One of our main results is an $\infty$-categorical generalization of Freyd's classical General Adjoint Functor Theorem. As an application of this result, we recover Lurie's adjoint functor theorems for presentable $\infty$-categories. We also discuss the comparison between adjunctions of $\infty$-categories and homotopy adjunctions, and give a treatment of Brown representability for $\infty$-catego…
For topological spaces $X$ and $Y$, a (not necessarily continuous) function $f:X \rightarrow Y$ naturally induces a functor from the category of closed subsets of $X$ (with morphisms given by inclusions) to the category of closed subsets of $Y$. The function $f$ also naturally induces a functor from the category of closed subsets of $Y$ to the category of closed subsets of $X$. Our aim in this expository note is to show that the function $f$ is continuous if and only if the first of the above two functors is a left adjoint to the second. We thereby obtain elementary examples of adjoint pairs (…
Diers developed a general theory of right multi-adjoint functors leading to a purely categorical, point-set construction of spectra. Situations of multiversal properties return sets of canonical solutions rather than a unique one. In the case of a right multi-adjoint, each object deploys a canonical cone of local units jointly assuming the role of the unit of an adjunction. This first part revolves around the theory of multi-adjoint and recalls or precises results that will be used later on for geometric purpose. We also study the weaker notion of local adjoint, proving Beck-Chevalley conditio…
In this paper, I characterize four particular classes of directed multigraphs, or quivers, as images under left and right adjoints to the natural vertex and edge functors. In particular, the following notions coincide: (1) independent sets of vertices with a left adjoint functor to the vertex functor, (2) independent sets of edges with a left adjoint functor to the edge functor, (3) complete digraphs with a right adjoint functor to the vertex functor, (4) bouquets with a right adjoint functor to the edge functor.
In the previous article 'A Mackey-functor theoretic interpretation of biset functors', we have constructed the 2-category $\mathbb{S}$ of finite sets with variable finite group actions, in which bicoproducts and bipullbacks exist. As shown in it, biset functors can be regarded as a special class of Mackey functors on $\mathbb{S}$. In this article, we equip $\mathbb{S}$ with a system of adjoint triplets, which satisfies properties analogous to a derivator. This system encodes the six operations for finite groups. As a corollary, we show that the associated Burnside rings satify analogous proper…
There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms (object-to-object morphisms between objects of different categories) that parses an adjunction into two separate parts (left and right representations of heteromorphisms). Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment us…
In this paper, we first construct some complete cotorson pairs on the category $\mathbb{C}_N(\mathcal{G})$ of unbounded $N$-complexes of Grothendieck category $\mathcal{G}$, from two given cotorsion pairs in $\mathcal{G}$. Next as an application, we focus on particular homotopy categories and the existence of adjoint functors between them. These are an $N$-complex version of the results were shown by Neeman in the category of ordinary complexes.
We prove how the universal enveloping algebra constructions for Lie-Rinehart algebras and anchored Lie algebras are naturally left adjoint functors. This provides a conceptual motivation for the universal properties these constructions satisfy. As a supplement, the categorical approach offers new insights into the definitions of Lie-Rinehart algebra morphisms, of modules over Lie-Rinehart algebras and of the infinitesimal gauge algebra of a module.
For our purposes, two functors Λ and Γ are said to be respectively left and right adjoints of each other if for any digraphs G and H, there exists a homomorphism of Λ(G) to H if and only if there exists a homomorphism of G to Γ(H). We investigate the right adjoints characterised by Pultr in [A. Pultr, The right adjoints into the categories of relational systems, in Reports of the Midwest Category Seminar, IV, volume 137 of Lecture Notes in Mathematics, pages 100-113, Berlin, 1970]. We find necessary conditions for these functors to admit right adjoints themselves. We give many examples where t…
Let $\mathcal{C}$ be an additive category. The nilpotent category $\mathrm{Nil} (\mathcal{C})$ of $\mathcal{C}$, consists of objects pairs $(X, x)$ with $X\in\mathcal{C}, x\in\mathrm{End}_{\mathcal{C}}(X)$ such that $x^n=0$ for some positive integer $n$, and a morphism $f:(X, x)\rightarrow (Y,y)$ is $f\in \mathrm{Hom}_{\mathcal{C}}(X, Y)$ satisfying $fx=yf$. A general theory of $\mathrm{Nil}(\mathcal{C})$ is established and it is abelian in the case that $\mathcal{C}$ is abelian. Two abelian categories are equivalent if and only if their nilpotent categories are equivalent, which generalizes a…
We give a construction of triangulated categories as quotients of exact categories where the subclass of objects sent to zero is defined by a triple of functors. This includes the cases of homotopy and stable module categories. These categories naturally fit into a framework of relative derived categories, and once we prove that there are decent resolutions of complexes, we are able to prove many familiar results in homological algebra.
Auslander and Kleiner proved in 1994 an abstract version of Green correspondence for pairs of adjoint functors between three categories. They produce additive quotients of certain subcategories giving the classical Green correspondence in the special setting of modular representation theory. Carlson, Peng and Wheeler showed in 1998 that Green correspondence in the classical setting of modular representation theory is actually an equivalence between triangulated categories with respect to a non standard triangulated structure. In the present note we first define and study a version of relative …
There exists a dispute in philosophy, going back at least to Leibniz, whether is it possible to view the world as a network of relations and relations between relations with the role of objects, between which these relations hold, entirely eliminated. Category theory seems to be the correct mathematical theory for clarifying conceptual possibilities in this respect. In this theory, objects acquire their identity either by definition, when in defining category we postulate the existence of objects, or formally by the existence of identity morphisms. We show that it is perfectly possible to get …
The functor that takes a ring to its category of modules has an adjoint if one remembers the forgetful functor to abelian groups: the endomorphism ring of linear natural transformations. This uses the self-enrichment of the category of abelian groups. If one considers enrichments into symmetric sequences or even bisymmetric sequences, one can produce an endomorphism operad or an endomorphism prop. In this note, we show that more generally, given an category C enriched in a monoidal category V, the functor that associates to a monoid in V its category of representations in C is adjoint to the f…
A family T of digraphs is a complete set of obstructions for a digraph H if for an arbitrary digraph G the existence of a homomorphism from G to H is equivalent to the non-existence of a homomorphism from any member of T to G. A digraph H is said to have tree duality if there exists a complete set of obstructions T consisting of orientations of trees. We show that if H has tree duality, then its arc graph delta H also has tree duality, and we derive a family of tree obstructions for delta H from the obstructions for H. Furthermore we generalise our result to right adjoint functors on categorie…
Representation theorems are established for fixed points of adjoint functors between categories enriched in a small quantaloid. In a very general setting these results set up a common framework for representation theorems of concept lattices in formal concept analysis (FCA) and rough set theory (RST), which not only extend the realm of formal contexts to multi-typed and multi-valued ones, but also provide a general approach to construct various kinds of representation theorems. Besides incorporating several well-known representation theorems in FCA and RST as well as formulating new ones, it i…
We prove general adjoint functor theorems for weakly (co)complete $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of (co)complete $\infty$-categories, so these $n$-categories do not admit all small (co)limits in general. We also introduce Brown representability for (homotopy) $n$-categories and prove a Brown representability theorem for localizations of compactly generated $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of presentable $\infty$-categories if $n \geq 2$, and the homotopy $n$-categories of presentable stable $\in…
The main result concerns a bicategorical factorization system on the bicategory $\mathrm{Cat}$ of categories and functors. Each functor $A\xra{f} B$ factors up to isomorphism as $A\xra{j}E\xra{p}B$ where $j$ is what we call an ultimate functor and $p$ is what we call a groupoid fibration. Every right adjoint functor is ultimate. Functors whose ultimate factor is a right adjoint are shown to have bearing on the theory of polynomial functors.
In this paper we investigate the categories of braided objects, algebras and bialgebras in a given monoidal category, some pairs of adjoint functors between them and their relations. In particular we construct a braided primitive functor and its left adjoint, the braided tensor bialgebra functor, from the category of braided objects to the one of braided bialgebras. The latter is obtained by a specific elaborated construction introducing a braided tensor algebra functor as a left adjoint of the forgetful functor from the category of braided algebras to the one of braided objects. The behaviour…
Given a symmetric monoidal stable $\infty$-category $\mathcal{C}$ and a left adjoint symmetric monoidal fiber functor to $\operatorname{Mod}_A^{\otimes}$ for some $\mathbb{E}_{\infty}$-ring $A$, one can construct a derived group scheme $G$ of monoidal automorphisms of this functor. The left adjoint fiber functor also induces a monad on $\mathcal{C}$. Under some finiteness hypothesis on the fiber functor, we show there is a comparison functor from the category of representations of $G$ to the descent category of the induced monad on $\mathcal{C}$.
There are many category-theoretic notions of algebraic theory, including Lawvere theories, monads, PROPs and operads. The first central notion of this thesis is a common generalisation of these, which we call a proto-theory. In order to define models of a proto-theory in a category, we need a way of relating the arities of the proto-theory with the objects of the category. This leads to our second central notion, that of an interpretation of arities, or aritation for short. We show that every aritation gives rise to a semantics functor sending proto-theories to models. In fact this functor alw…
Motivated by the work of of A. Zelevinsky on positive self-adjoint Hopf algebras, we define what we call a symmetric self-adjoint Hopf structure for a certain kind of semisimple abelian categories. It is known that every positive self-adjoint Hopf algebra admits a natural action of the associated Heisenberg double. We construct canonical morphisms lifting the relations that define this action on the algebra level and define an object that we call a categorical Heisenberg double that is a natural setting for considering these morphisms. As examples, we exhibit the symmetric self-adjoint Hopf st…
The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a {\em Morita context} comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is shown that in many cases equivalences between categories of algebras are induced by such Morita contexts. The Eilenberg-Moore category of representations of a Morita context is constructed. This construction allows one to associate two pairs of adjoint functors with right adjoint functors having a common domain or a {\em double adjunction} to a Morita conte…
In the context of infinity categories, we rethink the notion of derived functor in terms of correspondences. This is especially convenient for the description of a passage from an adjoint pair (F,G) of functors to a derived adjoint pair (LF,RG). In particular, canonicity of this passage becomes obvious. 2nd version: added comparison to Deligne's definition (SGA4) and a discussion of diagrams of derived functors. Introduction rewritten and references added. 3rd version: description of Kan extensions in terms of correspondences more detailed. 4th version: the final version accepted to HHA.
This dissertation is devoted to a study of adjunctions concerning categories enriched over a quantaloid Q (or Q-categories for short), with the following types of adjunctions involved: (1) adjoint functors between Q-categories; (2) adjoint distributors between Q-categories; (3) adjoint functors between categories consisting of Q-categories. For a small quantaloid Q and a distributor between Q-categories, two adjunctions between the Q-categories of contravariant and covariant presheaves are presented. These adjunctions respectively extend the fundamental construction of Isbell adjunctions and K…
This paper is a continuation of [BaSh], where we studied the behaviour of the abelianization functor under inverse limits. Our main result in [BaSh] was that if $\mathcal{T}$ is a countable directed poset and $G:\mathcal{T}\to\mathcal{G} rp$ is a diagram of groups that satisfies the Mittag-Leffler condition, then the natural map $$\mathrm{Ab}({\lim}_{t\in\mathcal{T}}G_t)\to {\lim}_{t\in\mathcal{T}}\mathrm{Ab}(G_t)$$ is surjective, and its kernel is a cotorsion group. The abelianization is an example of a left adjoint functor from groups to abelian groups. In this paper we study the behaviour u…
It is well-known that the category of presheaf functors is complete and cocomplete, and that the Yoneda embedding into the presheaf category preserves products. However, the Yoneda embedding does not preserve coproducts. It is perhaps less well-known that if we restrict the codomain of the Yoneda embedding to the full subcategory of limit-preserving functors, then this embedding preserves colimits, while still enjoying most of the other useful properties of the Yoneda embedding. We call this modified embedding the Lambek embedding. The category of limit-preserving functors is known to be a ref…
For an adjoint pair $(F, G)$ of functors, we prove that $G$ is a separable functor if and only if the defined monad is separable and the associated comparison functor is an equivalence up to retracts. In this case, under an idempotent completeness condition, the adjoint pair $(F, G)$ is monadic. This applies to the comparison between the derived category of the category of equivariant objects in an abelian category and the category of equivariant objects in the derived category of the abelian category.
If $\mathcal{M}$ is a finite abelian category and $\mathbf{T}$ is a linear right exact monad on $\mathcal{M}$, then the category $\mathbf{T}\mbox{-mod}$ of $\mathbf{T}$-modules is a finite abelian category. We give an explicit formula of the Nakayama functor of $\mathbf{T}\mbox{-mod}$ under the assumption that the underlying functor of the monad $\mathbf{T}$ has a double left adjoint and a double right adjoint. As applications, we deduce formulas of the Nakayama functor of the center of a finite bimodule category and the dual of a finite tensor category. Some examples from the Hopf algebra the…
The fundamental construction underlying descent theory, the lax descent category, comes with a functor that forgets the descent data. We prove that, in any $2$-category $\mathfrak{A} $ with lax descent objects, the forgetful morphisms create all Kan extensions that are preserved by certain morphisms. As a consequence, in the case $\mathfrak{A} = \mathsf{Cat} $, we get a monadicity theorem which says that a right adjoint functor is monadic if and only if it is, up to the composition with an equivalence, (naturally isomorphic to) a functor that forgets descent data. In particular, within the cla…
Let $\mathcal{A}$ and $\mathcal{B}$ be monoidal categories and let $R:\mathcal{A} \rightarrow \mathcal{B}$ be a lax monoidal functor. If $R$ has a left adjoint $L$, it is well-known that the two adjoints induce functors $\overline{R}={\sf Alg}(R):{\sf Alg}(\mathcal{A})\rightarrow {\sf Alg }(\mathcal{B})$ and $\underline{L}={\sf Coalg(L)}:{\sf Coalg}(\mathcal{B})\rightarrow {\sf Coalg}(\mathcal{A})$ respectively. The pair $(L,R)$ is called "liftable" if the functor $\overline{R}$ has a left adjoint and if the functor $\underline{L}$ has a right adjoint. A pleasing fact is that, when $\mathcal{A…
An adjoint pair of contravariant functors between abelian categories can be extended to the adjoint pair of their derived functors in the associated derived categories. We describe the reflexive complexes and interpret the achieved results in terms of objects of the initial abelian categories. In particular we prove that, for functors of any finite cohomological dimension, the objects of the initial abelian categories which are reflexive as stalk complexes form the largest class where a Cotilting Theorem in the sense of Colby and Fuller works.
A folklore result in category theory is that a (weakly) Cartesian closed category with finite co-products is distributive. Usually, the proof of this small result is carried on using the fact that the exponential functor is right adjoint to the product functor. And, since functors having a right adjoint preserve co-limits, the result follows immediately. But, when we try to explicitly construct the arrows, things become a bit more involved. In rare cases, it is pretty useful to have the exact arrows that make this isomorphism to hold. The purpose of this note is to develop the explicit proof w…
We give another proof of the fact that there is a dual equivalence between the $\infty$-category of monoidal $\infty$-categories with left adjoint oplax monoidal functors and that with right adjoint lax monoidal functors by constructing a perfect pairing between them.
This is a short textbook on Category Theory for Russian speaking students. It consists of three chapters: Categories and Functors, Representable Functors (including Adjoint Functors and (Co)limits) and Tensor Categories.
Monoidal functors U:C --> M with left adjoints determine, in a universal way, monoids T in the category of oplax monoidal endofunctors on M. Such monads will be called bimonads. Treating bimonads as abstract "quantum groupoids" we derive Tannaka duality between left adjointable monoidal functors and bimonads. Bialgebroids, i.e., Takeuchi's x_R-bialgebras, appear as the special case when T has also a right adjoint. Street's 2-category of monads then leads to a natural definition of the 2-category of bialgebroids.
It is shown that the multiplicative monoids of Brauer's centralizer algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself, and where, moreover, a kind of symmetry involving the self-adjoint functor is satisfied. As in a previous paper, of which this is a companion, it is shown that such a symmetric self-adjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices.
As left adjoint to the dual algebra functor, Sweedler's finite dual construction is an important tool in the theory of Hopf algebras over a field. We show in this note that the left adjoint to the dual algebra functor, which exists over arbitrary rings, shares a number of properties with the finite dual. Nonetheless the requirement that it should map Hopf algebras to Hopf algebras needs the extra assumption that this left adjoint should map an algebra into its linear dual. We identify a condition guaranteeing that Sweedler's construction works when generalized to noetherian commutative rings. …
We prove that for each locally $α$-presentable category $\mathcal K$ there exists a regular cardinal $γ$ such that any $α$-accessible functor out of $\mathcal K$ (into another locally $α$-presentable category) is continuous if and only if it preserves $γ$-small limits; as a consequence we obtain a new adjoint functor theorem specific to the $α$-accessible functors out of $\mathcal K$. Afterwards we generalize these results to the enriched setting and deduce, among other things, that a small $\mathcal V$-category is accessible if and only if it is Cauchy complete.
