Representations are adjoint to endomorphisms

Abstract

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 properad. In this note, we show that more generally, given a category enriched in a monoidal category , the functor that associates to a monoid in its category of representations in is adjoint to the functor that computes the endomorphism monoid of any functor with domain . After describing the first results of the theory we give several examples of applications.

Appendices

Terminology of enriched categories

We let the reader turn to Kelly [8] for a detailed exposition on categories enriched in a monoidal category . In order to not be bothered by size issues, we fix once and for all three infinite inaccessible cardinals L < XL < XXL and use the dictionary

$$\begin{aligned} \text {small} :=\text {L-small}; \quad \text {large} :=\text {XL-small}; \quad \text {very large} :=\text {XXL-small}. \end{aligned}$$

We now assume that is large (has large sets of objects and morphisms) and has all small limits and colimits. In what follows we consider a large -enriched category

and assume that is large.

Enlargement of the universe

For convenience (when computing over large diagrams), we shall enlarge : we choose a very large monoidal category with a full monoidal embedding

The enlarged universe can be chosen to be locally large, have all large limits and colimits and the embedding can be assumed to commute with small limits and colimits. This is discussed for example by Kelly [8, §2.6] (albeit in the closed symmetric setting).

The -category can now without effort be seen as a -category

Properties of enrichments

Definition 3

(Closed monoidal category) One says that is closed when the functor $Y \mapsto Y \otimes X$ has a right adjoint $Z \mapsto X^Z$ for each object X in .

Definition 4

(Tensored) One says that is tensored over whenever is closed and for every and , the functor

$$\begin{aligned} Y \longmapsto {[X,Y]}^M \end{aligned}$$

is -representable by an object denoted . In that case, since is closed the induced functor

is naturally endowed with a monoidal structure.

Definition 5

(Faithfully tensored) We shall say that is faithfully tensored over if it is tensored and the functor

is faithful.

Definition 6

(Accessibly tensored) We shall say that is accessibly tensored over if it is tensored, both and are accessible and for every , the functor

is accessible.

Definition 7

(Accessibly enriched) When and are both accessible, we shall say that is accessibly enriched if there exists a small cardinal $\kappa$ such that for every , the functor

commutes with $\kappa$-cofiltered limits.

Remark 8

One can check that if is accessibly tensored, it is then accessibly enriched.

Examples of contexts of application

In this appendix, we give several application contexts for the adjunction

In each context, the terminology is specific, both for monoids and for their categories of representations.

Using a closed symmetric monoidal category

In the next examples, we fix a presentable closed symmetric monoidal category and denote its internal hom by $\langle - ,- \rangle$. We then consider several enrichments for .

Potential examples of such closed symmetric monoidal categories include the category of sets, vector spaces or coassociative cogebras (more generally cogebras over Hopf operads). It also includes the categories of sheaves valued in those categories.

Self enrichment

This one is the most obvious, since the monoidal structure of is closed, it is self-enriched via

$$\begin{aligned}{}[X, Y] :=\langle X, Y \rangle . \end{aligned}$$

In this context, the general idea of the adjunction was well-known to people doing reconstruction theorems à la Tannaka. It appears for example in Street’s Quantum groups: a path to current algebra [9, Ch. 16].

Operadic enrichment

Let us denote by the category of symmetric sequences: sequences of objects M(n) of endowed with right $\mathbf{S }_{n}$-actions for every natural n. The category is accessibly tensored over the category of symmetric sequences via the formula

$$\begin{aligned} M \mathbin {\triangleleft }X :=\coprod _{n \in \mathbf{N} } M(n) \otimes _{\mathbf{S }_{n}} X^{\otimes n}. \end{aligned}$$

This induces a monoidal structure on symmetric sequences

$$\begin{aligned} M \mathbin {\triangleleft }N :=\coprod _{n \in \mathbf{N} } M(n) \otimes _{\mathbf{S }_{n}} N^{\circledast n}. \end{aligned}$$

Where $\circledast$ denotes the convolution of symmetric sequences. The associated enrichment is given by

$$\begin{aligned}{}[X, Y](n) :=\langle X^{\otimes n}, Y \rangle . \end{aligned}$$

Monoids in symmetric sequences are called operads

Given an operad P, its category of representations is called the category of P-algebras. One thus gets an adjunction

The other (cogebraic) operadic enrichment

This time we let be the category of symmetric sequences with left actions of the symmetric groups. It admits a monoidal structure given by

$$\begin{aligned} M \mathbin {\triangleright }N :=\coprod _{n \in \mathbf{N} } M^{\circledast n} \otimes _{\mathbf{S }_{n}} N(n) \end{aligned}$$

and the associated enrichment is

$$\begin{aligned}{}[X, Y](n) :=\langle X, Y^{\otimes n} \rangle . \end{aligned}$$

Since left and right actions of symmetric groups are equivalent, one has an equivalence of categories

In this case, the category of representations of an operad P is its category of cogebras. Conversely, the functor associates to a functor F, seen as an object of the functor category, its coendomorphism operad.

In general, the category of P-cogebras may not be presentable, although (for example) it is presentable if the ground category is dg-vector spaces [4]. Thus, one has the adjunction

This example arises naturally in applications and appeared, for example, in unpublished work by May, who considered it well-known.

In one application, the singular chains functor from topological spaces to chain complexes factors through the category of ${\mathsf {E}}_\infty$-cogebras in chain complexes.

The following stable improvement of this example was pointed out to us by Arone: the coendomorphism operad of the suspension functor from pointed spaces to spectra can be shown to be weakly equivalent to the commutative operad [10].

Propic enrichments

Going further, one can enrich in the category of bisymmetric sequences using

$$\begin{aligned}{}[X, Y](p, q) :=\langle X^{\otimes p}, Y^{\otimes q}\rangle . \end{aligned}$$

There are several monoidal structures on bisymmetric sequences compatible with these enrichment objects, depending on the classes of graphs involved in the definition of the monoidal structure. One can allow connected graphs, in which case the monoids are properads [11, 2.1], or allow only simply connected graphs, in which case the monoids are dioperads [12, 4.2]. Similar but more exotic examples are also possible [13].

Examples with exogenous enrichments

Representations of topological monoids

The following example is taken from the duality between topological groups and their categories of representations due to Tannaka [14]. The category of finite dimensional vector spaces is canonically enriched in topological spaces. Since this category is small, one gets an adjunction

where associates to any functor $F : {\mathcal {D}}\rightarrow {\mathsf {Vect}}_{\text {fd}}$ its topological monoid of endomorphisms.

Bigebras

Let ${\varvec{K}}$ be a field. The category of associative ${\varvec{K}}$-algebras is naturally cotensored over ${\varvec{K}}$-cogebras: given a cogebra V and an algebra ${\varLambda }$, convolution gives $\text {Hom}_{{\varvec{K}}}(V, {\varLambda })$ a structure of associative algebra. This cotensorization comes with an enrichment and a tensorization [15].

Monoid objects in cogebras are bigebras. Given a bigebra H, it is an exercise to verify that the category of representations $H\text {-}{\mathsf {rep}}$ is naturally isomorphic to the category of H-module algebras studied by Hopf theorists [16, 4.1.1] equipped with the functor to algebras forgetting the H-module structure. We thus obtain an adjunction

where for an accessible functor $F :{\mathcal {D}}\rightarrow {\mathsf {Alg}}$, the endomorphism bigebra is universal among bigebras acting compatibly on the objects of ${\mathcal {D}}$.