Endomorphism operads of functors

TL;DR

This paper investigates the endomorphism operad of functors, especially the forgetful functor from algebras over an operad, revealing when it recovers the original operad across different categories.

Contribution

It demonstrates that the endomorphism operad of the forgetful functor recovers the original operad in vector spaces over infinite fields, but not in finite fields or sets, with several examples computed.

Findings

Recovers operad in vector spaces over infinite fields

Fails to recover operad in finite fields and sets

Provides multiple computed examples

Abstract

We consider the endomorphism operad of a functor, which is roughly the object of natural transformations from (monoidal) powers of that functor to itself. There are many examples from geometry, topology, and algebra where this object has already been implicitly studied. We ask whether the endomorphism operad of the forgetful functor from algebras over an operad to the ground category recovers that operad. The answer is positive for operads in vector spaces over an infinite field, but negative both in vector spaces over finite fields and in sets. Several examples are computed.

Equations

$$P⟼P\text{-}\mathsf{alg},$$

$${\mathcal{C}}^{\mathrm{op}}\times \mathcal{C}$$

$$\mathsf{Rep}:\mathsf{Mon}\left(\mathcal{C}\right)$$

$$\mathsf{Alg}:\mathsf{Op}\left(\mathcal{C}\right)$$

$${\mathrm{Nat}}_{\mathcal{C}}(F,G):={\int }_{\mathcal{D}}^{\ast }[F-,G-].$$

$${\mathcal{E}}_1(F):={\mathrm{Nat}}_{\mathcal{C}}(F,F)$$

$$\mathcal{E}(F)(n):={\mathrm{Nat}}_{\mathcal{C}}(F^{\otimes n},F)$$

$$\mathcal{E}(F)(n):={\mathrm{Nat}}_{\mathcal{C}}(F,F^{\otimes n}).$$

$$\mathsf{Mon}\left(\mathcal{C}\right)$$

$$\mathsf{Lie}⟶\mathcal{E}({\mathrm{T}}_{\mathrm{e}}).$$

$${\mathrm{T}}_{\mathrm{e}}:\mathsf{Lie}\mathsf{Grp}⟶{\mathsf{Vect}}_{\mathbf{R}}$$

$$\mathcal{E}({\mathrm{T}}_{\mathrm{e}})=\mathsf{Lie}.$$

$${\Omega }^{\ast }:{\mathsf{Diff}}^{\mathrm{op}}⟶{\mathsf{Vect}}_{\mathbf{R}}⟶\mathsf{Set}$$

$$\mathcal{D}$$

$${\mathrm{U}}_{\mathsf{Grp}}:\mathsf{Grp}⟶\mathsf{Set}$$

$${\mathrm{C}}^{\ast }:{\mathsf{Top}}^{\mathrm{op}}⟶{\mathsf{Ch}}_{\mathsf{Ab}}?$$

$${\mathsf{Top}}^{\mathrm{op}}⟶{\mathsf{E}}_{\infty }\text{-}\mathsf{alg}$$

$$M$$

$$P⟶\mathcal{E}({\mathrm{U}}_P)$$

$$\mathcal{C}$$

$${\mathrm{U}}_{\mathsf{C}}:\mathsf{C}\text{-}\mathsf{alg}⟶{\mathsf{Vect}}_{{\mathbf{F}}_q},$$

$$\mu \circ (P_q\otimes P_q)=P_q\circ \mu .$$

$$\prod _{s\in S}{\mathrm{Hom}}_{\mathcal{C}}(s,-):\mathcal{C}⟶\mathsf{Set}$$

$${\mathcal{E}}_1(F)⟶{\mathcal{E}}_1(F_S)$$

$$[F(d),F(d)]⟶\prod _{L(s)\to d}[FL(s),F(d)]$$

$$[F(d),F(d)]⟶\prod _{L(s)\to d}[FL(s),F(d)]⟶\prod _{L(s)\to d}[s,F(d)]$$

$$\left(\coprod _{L(s)\to d}s\right)=\left(\coprod _{s\to F(d)}s\right)⟶F(d)$$

$${\mathcal{E}}_1(F)$$

$$\mathcal{E}(F)(n)⟶\mathcal{E}(F_{S^n})(n)$$

$$[{F(d)}^{\otimes n},F(d)]⟶\prod _{\overline{s}\to F(d)}[{(\overline{s})}^{\otimes n},F(d)]$$

$$\coprod _{s\to F(d)}s⟶F(d)$$

Full text

There are many examples from geometry, topology, and algebra where this object has already been implicitly studied.

We ask whether the endomorphism operad of the forgetful functor from algebras over an operad to the ground category recovers that operad. The answer is positive for operads in vector spaces over an infinite field, but negative both in vector spaces over finite fields and in sets.

Several examples are computed.

The first and third authors were supported by IBS-R003-D1.

The second author was supported by NSF DMS-1304169

1. 1 Introduction
2. 2 A historical perspective
3. 3 Operadic approximation
4. 4 Reconstruction of operads
5. 5 Main lemmas
6. 6 Proofs of reconstruction theorems
7. 7 Computations

1. Introduction

It is commonplace in contemporary mathematics to discuss the natural endomorphisms of a functor. Cohomology operations, the center of a category, and reconstruction theorems in the style of Tannaka are examples of this line of study.

The raison d’être of this article is to discuss the natural endomorphisms in a more general sense. Specifically we study the natural endomorphism operad of a functor. As we argue below, this generalization is not at all exotic—mathematicians were studying endomorphism operads of functors decades before operads or functors were explicitly defined. And there are multiple natural questions of contemporary interest that are best described as the study of the endomorphism operad of a particular functor.

However, this point of view seems to have remained mostly implicit in the literature (one notable exception is Modules over Operads and Functors [1, § 3.4], where the subject is treated cleanly but in passing).

Our contribution here is

1. (1)

to give some outlines of the structure of the operadic theory, exploiting the fact that the endomorphism operad of a functor is adjoint to

[TABLE] 2. (2)

to discuss operadic approximations to functors, and in particular successes and failures of reconstruction in the operadic context, and 3. (3)

to calculate some interesting and illustrative examples.

Let us discuss the structure of the article.

After the introduction, in which we define our terms, we describe some historical examples of interest as motivation [§ 2]. Then we discuss the interpretation of the endomorphism operad of a functor as a universal operadic approximation to that functor [§ 3]. After that, we turn to the question of reconstruction [§ 4]. This is the question of whether and when the endomorphism construction, applied to the forgetful functor from algebras over an operad to the ground category, recovers that operad.

We defer all proofs to the following three sections, where we introduce some technical tools and apply them to justify the statements and examples of the previous sections.

For our conventions, we fix a presentable closed symmetric monoidal category $(\mathscr{C},\otimes,\mathbf{1})$ and denote by

[TABLE]

its associated self-enrichment. We let $\mathsf{Mon}\left(\mathscr{C}\right)$ denote the category of monoids in $\mathscr{C}$ and $\mathsf{Op}\left(\mathscr{C}\right)$ denote the category of operads in $\mathscr{C}$, i.e., monoids in the monoidal category of symmetric sequences in $\mathscr{C}$.

1.1. Representations of monoids and operads

Given a monoid $M$ in $\mathsf{Mon}\left(\mathscr{C}\right)$ we denote by $M\text{-}\mathsf{rep}$ the category of $M$-representations in $\mathscr{C}$. The category $M\text{-}\mathsf{rep}$ comes equipped with the forgetful functor $\mathrm{U}_{M}$ to $\mathscr{C}$.

Similarly, given an operad $P$ in $\mathsf{Op}\left(\mathscr{C}\right)$, we denote by $P\text{-}\mathsf{alg}$ the category of $P$-algebras in $\mathscr{C}$. The category $P\text{-}\mathsf{alg}$ comes equipped with the forgetful functor $\mathrm{U}_{P}$ to $\mathscr{C}$.

In both cases this assignment is functorial. A map $f:M\to N$ of monoids induces a functor $N\text{-}\mathsf{rep}\to M\text{-}\mathsf{rep}$ over $\mathscr{C}$, and a map $f:P\to Q$ of operads induces a functor $Q\text{-}\mathsf{alg}\to P\text{-}\mathsf{alg}$ over $\mathscr{C}$.

The conditions we have set on $\mathscr{C}$ ensure that the categories of representations and all induced and forgetful functors are accessible and so we get functors

[TABLE]

1.2. Endomorphisms of functors to $\mathscr{C}$

Given a functor $F$ with codomain $\mathscr{C}$, we may take the endomorphism monoid $\mathscr{E}_{1}(F)$ or the endomorphism operad $\mathscr{E}(F)$ of natural transformations from $F$ to itself. The monoidal case is classical. The operadic case occurs in more recent literature [1, 3.4].

Recall that given two functors $F,G:\mathcal{D}\to\mathscr{C}$, up to size issues which can be finessed in a variety of ways, the $\mathscr{C}$-natural transformations from $F$ to $G$ are presented by the object

[TABLE]

Here, following Yoneda’s original notation [2, § 4], $\int_{\mathcal{D}}^{\ast}$ denotes the cointegration (or end) of a functor $\mathcal{D}^{\mathrm{op}}\times\mathcal{D}\to\mathscr{C}$.

Definition 1.1**.**

Let $F:\mathcal{D}\to\mathscr{C}$ be an accessible functor. The endomorphism monoid of $F$ is

[TABLE]

the monoid of $\mathscr{C}$-natural transformations of $F$.

The endomorphism operad of $F:\mathcal{D}\to\mathscr{C}$ is, in arity $n$

[TABLE]

the operad of $\mathscr{C}$-natural transformations of $F$.

In both cases the structure operations are induced by composition in the closed category $\mathscr{C}$. In particular, it follows from the definition that the endomorphism monoid $\mathscr{E}_{1}(F)$ is the arity one component of the endomorphism operad $\mathscr{E}(F)$.

Accessibility of $F$ and presentability of $\mathscr{C}$ ensure that the endomorphism monoid and operad are small [3, 3.1].

For example, if $X:*\to\mathscr{C}$ is an object of $\mathscr{C}$, then $\mathscr{E}(X)$ is the ordinary endomorphism operad with components $[X^{\otimes n},X]$.

Remark 1.2* (Coendomorphism operad).*

Dually, one can define the coendomorphism operad of a functor $F:\mathcal{D}\to\mathscr{C}$ via

[TABLE]

This notion has also been actively studied in recent literature [4].

Remark 1.3*.*

Unfortunately, the name “endomorphism operad of a functor” has been used in the literature for at least three different notions.

1. (1)

Here the “endomorphism operad of a functor” $F:\mathcal{D}\to\mathcal{C}$ (with $\mathcal{C}$ symmetric monoidal) is the endomorphism operad of the object $F$ in the symmetric monoidal category of functors from $\mathcal{D}$ to $\mathcal{C}$; 2. (2)

in unpublished work, May used the term “endomorphism operad of a functor” $F$ for what we have called the coendomorphism operad of $F$; 3. (3)

Richter and others have defined another kind of “endomorphism operad of a functor” $F:\mathcal{D}\to\mathcal{C}$ where both $\mathcal{D}$ and $\mathcal{C}$ are monoidal in their study of transfer of algebraic structure [5, 6].

Both the endomorphism monoid and endomorphism operad are adjoint to categories of representations.

Proposition 1.4** ([3, 4.1]).**

We have the following adjunctions:

[TABLE]

In some of the motivating examples below we consider endomorphism monoids and operads of functors whose domains are not accessible. This requires a variation of the setup [3, 2.2, 2.4] but in practice does not affect the theory very much.

2. A historical perspective

We discuss some motivation and historical antecedents to this theory.

2.1. Lie groups and Lie algebras

One of the earliest examples of algebraic structures stemming from endomorphisms of functors comes from Lie groups and Lie algebras.

Consider the functor $\mathrm{T}_{\mathrm{e}}$ from real Lie groups to real vector spaces which takes the tangent space at the identity element. Since the work of Lie and Klein on “continuous groups”, it has been known that this functor factorizes through the category of Lie algebras. In other words, there is a morphism of operads

[TABLE]

Unsurprisingly (to the modern reader), this operad map is an isomorphism.

Proposition 2.1**.**

Let

[TABLE]

be the functor sending a Lie group $G$ to its tangent space at the unit. Then

[TABLE]

2.2. Cohomology operations

In the 1950s, the early category theorists applied then relatively new notions of naturality to study cohomology operations, the elements of the natural endomorphism monoid of the cohomology functor [7, 8, 9, 10, 11, 12].

Cohomology also has natural multilinear operations, most notably the cup product, and there are many examples where this multilinear structure gives an easy obstruction to the existence of some topological map. This points to the utility of a general framework with which to discuss natural multi-ary operations on a functor.

2.3. Manifolds

In the context of the geometry of manifolds, there has been a thread of research on natural differential operators on various vector bundles associated functorially to manifolds. Possibly the earliest results along these lines in the literature are due to Palais [13]. We can recast his results as a computation of the endomorphism monoid of the functor of differential forms

[TABLE]

In essence, he computes that the endomorphism monoid of $\Omega^{*}$ is spanned by scalar multiplication and exterior differentiation.

Over the years these results have been generalized, both to other tensorial constructions and to multilinear operators [14, § 34]. For example, bilinear natural maps on tensor powers of the tangent and cotangent bundle are fully classified. These contain mostly expected operations like the wedge product, Lie bracket, and variations thereof but there is an exceptional bilinear operator acting on certain spaces of densities on $1$-manifolds.

We can ask a few simple questions intended to reformulate these questions of geometric naturality in our language. Some of the answers may already be known to geometers.

Question**.**

What is the endomorphism operad of differential forms?

The natural guess is that it contains only scalar multiplication, exterior differentiation, and the wedge product, i.e., that the endomorphism operad, properly construed, is the operad governing commutative algebras equipped with a square zero derivation.

Question**.**

What is the endomorphism operad of differential forms viewed as a functor with domain a category of manifolds with some additional structure? For instance, what is the endomorphism operad in the Riemannian setting?

With the appropriate setup, this operad includes the natural operations for smooth manifolds along with the adjoint $\mathrm{d}^{*}$ of the exterior derivative and operations derived from $\mathrm{d}$, $\mathrm{d}^{*}$, and the wedge product, such as the Laplacian and the bracket. The operad governing this collection of structures with only the “obvious” relations among them is the operad governing Batalin–Vilkovisky algebras equipped with a square zero derivation of the product with no assumed compatibility with the BV operator.

A version of this question can also be asked for manifolds equipped with further structure: complex (a different point of view on this is given by Millès [15]), Kähler, and so on.

Question**.**

What is the endomorphism operad of various powers of the tangent bundle?

To give a fully satisfactory answer to this suite of questions we would want a two-colored operad dealing simultaneously with the tangent and cotangent bundle. Formulating this structure precisely is not within our purview because some of these bundles are covariant functors and some contravariant.

2.4. Hochschild-type examples

Part of the literature on Hochschild (co)homology, cyclic (co)homology, and the bar complex [8, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] (this is not exhaustive) has been devoted to finding ever-more refined natural algebraic structures on the various homology groups and chain complexes involved in the constructions, sometimes on restricted classes of algebras. This thread of work can be interpreted as providing partial or full computations of the endomorphism operad of the functor under consideration in the given context.

2.5. Homotopical versions

All of the statements in this paper are one-categorical and rigid. It is reasonable (probably more reasonable) to ask about a homotopically coherent version of this story.

We outline a speculative application. Any category $\mathscr{C}$ can be viewed as enriched in sets, and then the identity functor of $\mathscr{C}$ is the forgetful functor for representations of the trivial monoid. The endomorphism monoid in this setting is called the center of $\mathscr{C}$.

There are various settings in enriched category theory in which the notion of a derived center has been defined [32]. Perhaps there is something interesting to say about the derived central operad, defined along the lines of 4.4.

For example, it is known that the derived center of the category of simplicial algebras over a Lawvere theory, viewed as enriched in simplicial sets, is homotopy equivalent to the ordinary center of discrete algebras over the same theory [33]. But perhaps there is higher homotopical information in the derived central operads of such categories.

3. Operadic approximation

Because of the adjunction $\mathsf{Alg}\dashv\mathscr{E}$ any functor $F$ admits a universal operadic approximation $\mathscr{E}(F)$ so that the category of $\mathscr{E}(F)$-algebras is universal among those categories of operadic algebras which accept a restriction functor compatible with $F$ from its domain:

[TABLE]

Here the functor $\mathcal{D}\to\mathscr{E}(F)\text{-}\mathsf{alg}$ is the $F$ component of the counit of the adjunction between algebras and endomorphisms.

In the case that $F$ is right adjoint to a functor $L$, we can understand the endomorphism operad of $F$ as giving a universal operadic approximation to the monad $FL$. That is, there is a natural transformation from the monad associated to the endomorphism operad $\mathscr{E}(F)$ to $FL$, and moreover $\mathscr{E}(F)$ is terminal among such operads.

3.1. Loop spaces

Historically, the first explicit use of the word operad was in the context of the recognition principle [34] which, suitably interpreted, identifies an operad that naturally acts on the $n$-fold loop space functor and then characterizes the essential homotopical image of that functor (in, say, connected spaces) as the category of representations of that operad. This is a case where the universal operadic approximation recovers and thus characterizes the functor in question up to homotopy.

3.2. Groups and sets

An interesting example is the forgetful functor from groups to sets. It is well-known that this forgetful functor is monadic but not operadic (the category of groups is not a category of algebras over an operad in sets). What is the operad that describes groups the closest?

Example 3.1**.**

Let

[TABLE]

be the forgetful functor from groups to sets. Then $\mathscr{E}(\mathrm{U}_{\mathsf{Grp}})(n)$ is isomorphic to the free group on $n$ generators.

Moreover, we can explicitly specify the operadic structure. The $\mathbf{S}_{n}$-action is by permutation of generators. The operadic composition is by word substitution: given a free group element $w_{1}$ on $x_{1},\ldots,x_{m}$ and a free group element $w_{2}$ on $y_{1},\ldots,y_{n}$, then $w_{1}\circ_{i}w_{2}$ is obtained by replacing $x_{i}$ with $w_{2}$ and reindexing.

There are algebras over this operad which are not groups. For example, any unital semigroup with involution admits an action by this operad. So this furnishes an example of a non-trivial monad whose universal operadic approximation generates a different monad.

3.3. Singular chains

We have already discussed the endomorphism operads of the functors of cohomology [§ 2.2] and differential forms [§ 2.3].

Question**.**

What is the endomorphism operad of the singular cochain functor

[TABLE]

Of course, this is another context ripe for a homotopical version of our question. Various authors have described $\mathsf{E}_{\infty}$-operads which act functorially on cochains [35, 36, 37, 38]. By adjunction we get an operad map from any such $\mathsf{E}_{\infty}$-operad to the endomorphism operad $\mathscr{E}(\mathrm{C}^{*})$ of the cochain functor. Indeed one of the main results of McClure and Smith is that their $\mathsf{E}_{\infty}$-operad is a suboperad of the endomorphism operad of the normalized singular cochain functor.

But we can ask directly: is this map a weak equivalence? For some choice of $\mathsf{E}_{\infty}$-operad, is it an isomorphism? One of Mandell’s results is that the lift of $\mathrm{C}^{*}$ to a functor

[TABLE]

is not full (it is homotopically faithful when restricted to finite type nilpotent spaces). Thus if some version of $\mathsf{E}_{\infty}$ is the endomorphism operad of $\mathrm{C^{*}}$ then this is another example where the universal operadic approximation fails to yield an equivalence of categories.

4. Reconstruction of operads

Now, following the point of view of duality in the style of Tannaka, we look at the question of whether we can reconstruct an operad $P$ as the endomorphism operad of the forgetful functor $\mathrm{U}_{P}$ from $P$-algebras to $\mathscr{C}$.

The previous section considered the counit of the $\mathsf{Rep}\dashv\mathscr{E}$ adjunction. The question we are considering here is about the unit of the adjunction, and in particular whether it is an isomorphism.

4.1. Reconstruction theorems

We begin with a variation of a known result about endomorphism monoids.

Theorem 4.1** (Reconstruction for monoids).**

Suppose that $\mathscr{C}$ is strongly separated by the monoidal unit. Then for every monoid $M$, the unit

[TABLE]

of the $\mathsf{Rep}\dashv\mathscr{E}_{1}$ adjunction, is an isomorphism. In other words, $\mathsf{Rep}$ is fully faithful.

In the formalism of Tannaka, one considers a variant of the functor $\mathsf{Rep}$ where instead of considering maps $M\to[X,X]$, one uses enriched maps $[M,[X,X]]$. By doing so, the functor $\mathsf{Rep}$ sends monoids to $\mathscr{C}$-enriched categories with a $\mathscr{C}$-enriched functor to $\mathscr{C}$. In such an enriched context, the result of the theorem holds without a separation axiom for the unit because in a closed category, the monoidal unit is always a strong separator in the enriched sense [39, 40].

In our context, some hypothesis on the ground category is definitely necessary. For example, the identity functor on $\mathbf{Z}$-graded $R$-modules can also be viewed as the forgetful functor for algebras over the trivial $R$-algebra $R$. If we had reconstruction, then $\mathscr{E}_{1}$ applied to this identity functor would recover the graded $R$-algebra $R$. But the endomorphism monoid of this identity functor is $R^{\mathbf{Z}}$, not $R$ [7.3].

Remark 4.2*.*

Two rings are equivalent in the sense of Morita if their the categories of representations are equivalent. There are equivalent rings which are not isomorphic. So the variant of the functor $\mathsf{Rep}$ that only remembers the category $M\text{-}\mathsf{rep}$ and not the forgetful functor $\mathrm{U}_{M}$ to $\mathscr{C}$ is not fully faithful in general. A more refined statement is that an equivalence in the sense of Morita compatible with forgetful functors must be induced by an isomorphism of rings.

We have a similar theorem for operads but with a radically strengthened hypothesis.

Theorem 4.3** (Reconstruction for linear operads).**

Let $P$ be an operad in vector spaces over an infinite field $\boldsymbol{K}$. Then the unit

[TABLE]

is an isomorphism.

This fails both for operads in sets and for vector spaces over finite fields, as shown below.

4.2. Central operads and failures of reconstruction

As mentioned above, the endomorphism monoid of the identity functor of $\mathscr{C}$ is called the center of $\mathscr{C}$.

In line with this definition we set:

Definition 4.4** (Central operad).**

We shall call the endomorphism operad of the identity functor

[TABLE]

the central operad of $\mathscr{C}$.

We now turn our attention toward the computation of central operads of several categories. Each category whose central operad is not the identity operad is an example of failure of reconstruction.

This simple test already displays interesting behavior and often obstructs reconstruction. Here are two examples.

• •

The central operad of the category of sets is the $\mathbf{Perm}$ operad;

• •

For $R$ a commutative ring, the central operad of the category of $\mathbf{Z}$-graded $R$-modules is $R^{\mathbf{Z}}$, concentrated in arity one and acting by scalar multiplication separately in each degree.

On the other hand, for ungraded $R$-modules the central operad is the trivial operad (i.e., $R$ in arity one acting by the module action). But this is not enough to guarantee reconstruction in general for $R$-modules, as witnessed by the following example.

Proposition 4.5**.**

Let us consider

[TABLE]

the forgetful functor from $\mathbf{F}_{q}$-linear commutative algebras to $\mathbf{F}_{q}$-vector spaces.

Then the endomorphism operad $\mathscr{E}(\mathrm{U}_{\mathsf{C}})$ can be presented as

• •

generated by a multiplication operation $\mu$ of arity two and a power operation $P_{q}$ of arity one;

• •

subject to the relations that $\mu$ is commutative and associative and

[TABLE]

In particular $\mathscr{E}(\mathrm{U}_{\mathsf{C}})\neq\mathsf{C}$ so reconstruction fails for operads in vector spaces over finite fields.

Remark 4.6*.*

The natural action of $P_{q}$ on a $\mathsf{C}$-algebra is the action of taking the $q$th power. If $q=p^{n}$ it may seem as though there should be be a natural $p$-th power action, but the $p$-th power action is not $\mathbf{F}_{q}$-linear.

5. Main lemmas

Now we turn to the rigorous proofs of the statements we have made in the earlier sections. Our main tool, introduced in this section, is a reduction of the size of the cointegral computing the endomorphism operad (or monoid) which requires a datum and an assumption:

• •

we assume given a separator $S$ for the category $\mathscr{C}$, and

• •

we assume that the functor $F:\mathcal{D}\to\mathscr{C}$ is a right adjoint.

We will compute our cointegral over a category built out of the separator $S$ and the left adjoint to $F$ instead of over all of $\mathcal{D}$.

The usefulness of the reduction in computation depends directly on the size and complexity of $S$ and the objects in it. In particular, it works well for the category of sets (separated by the point) and $R$-modules (separated by $R$).

The following definition is standard and is recalled for convenience.

Definition 5.1** (Separating set).**

A set of objects $S\subset\mathscr{C}$ is separating if the functor

[TABLE]

is faithful. It is strongly separating if moreover it is conservative.

Now we build the subcategories that we will use to restrict our cointegral.

Notation 5.2**.**

Let $S\subset\mathscr{C}$ be a separating set and let $F:\mathcal{D}\to\mathscr{C}$ admit a left adjoint $L$.

We write $F_{S^{n}}$ for the restriction of $F$ to the full subcategory of $\mathcal{D}$ generated by $L(s_{1}\amalg\dots\amalg s_{n})$ with $s_{i}\in S$. We write $F_{S}$ for $F_{S^{1}}$; this is the full subcategory of $\mathcal{D}$ generated by $L(S)$.

Lemma 5.3** (Separator lemma, monoidal case).**

Let $S\subset\mathscr{C}$ be a separating set and let $F:\mathcal{D}\to\mathscr{C}$ admit a left adjoint $L$. Then the canonical map of endomorphism monoids

[TABLE]

is a monomorphism.

Proof.

For every $d\in\mathcal{D}$, we claim that the map

[TABLE]

is a monomorphism. For this, one can use the unit of the adjunction $L\dashv F$ and show that the composite

[TABLE]

is a monomorphism. Since $S$ is separating, the canonical map

[TABLE]

is an epimorphism and since the monoidal structure of $\mathscr{C}$ is symmetric closed, the functor $X\mapsto[X,F(d)]$ sends coproducts to products and epimorphisms to monomorphisms.

Now by the universal property of $\mathscr{E}_{1}(F)$ and $\mathscr{E}_{1}(F_{S})$, the following diagram commutes

[TABLE]

which implies that $\mathscr{E}_{1}(F)\to\mathscr{E}_{1}(F_{S})$ is a monomorphism. ∎

Lemma 5.4** (Separator lemma, operadic case).**

Let $S\subset\mathscr{C}$ be a separating set and let $F:\mathcal{D}\to\mathscr{C}$ admit a left adjoint $L$. Then for every natural $n$, the canonical map of components of endomorphism operads

[TABLE]

is a monomorphism.

Proof.

The proof is similar to the previous one. Explicitly, we need to show that

[TABLE]

is a monomorphism. As in the previous proof, because $S$ is separating the canonical map

[TABLE]

is an epimorphism. Since the tensor structure on $\mathscr{C}$ is closed, tensorization preserves epimorphisms, so

[TABLE]

is an epimorphism. Using the factorization

[TABLE]

we deduce that $\coprod_{\overline{s}\to F(d)}\overline{s}^{\otimes n}\to F(d)^{\otimes n}$ is again an epimorphism. We end the proof with the fact that $X\mapsto[X,F(d)]$ sends coproducts to products and epimorphisms to monomorphisms. ∎

6. Proofs of reconstruction theorems

In this section we use the separator lemmas to prove our two reconstruction theorems, for monoids in a category with a strong separator and for operads in the category of vector spaces over an infinite field.

We restate the first theorem. See 4.1

Proof.

Let $M$ be a monoid in $\mathscr{C}$. Since $\mathscr{C}$ is self enriched, the free representation generated by $\mathbf{1}$ is $M$ and we shall denote by ${M}^{\text{full}}$, the full subcategory of $M\text{-}\mathsf{rep}$ generated by this single object. Consider the following composition:

[TABLE]

We will argue that the composition is an isomorphism (which implies that the second map is a split epimorphism). Since the second map is also monic [5.3] this will suffice.

To argue that the composition is an isomorphism, first we note that the unit $M\to\mathscr{E}_{1}(\mathrm{U}_{M})$ is monic because self-enrichments are always faithfully tensored over themselves [3, 4.4].

Then the separation condition ensures that the underlying functor

[TABLE]

is conservative. Since it also preserves monomorphisms, what remains to show is that the map

[TABLE]

which we now know to be injective, is actually bijective. Since the monoidal unit is a separator, the transformation $\mathrm{Hom}_{\mathscr{C}}(M,M)\to\mathrm{Hom}_{\mathsf{Set}}(M_{0},M_{0})$ is also injective, thus we have an injection between the cointegrals

[TABLE]

Using the fact that $\mathrm{Hom}_{M\text{-}\mathsf{rep}}(M,M)=M_{0}$, one can quickly deduce that this last cointegral is canonically bijective to $M_{0}$.

Then the identity of $M_{0}$ factors as a chain of injective functions:

[TABLE]

inducing the desired bijection. ∎

In order to prove the operadic reconstruction theorem, we will need a lemma.

Lemma 6.1**.**

Let $P$ be an operad in vector spaces over an infinite field $\boldsymbol{K}$ and let $V$ be a vector space of dimension $n$. Then the natural map $P(n)\otimes V^{\otimes n}\to P\mathbin{\triangleleft}V$ induces an $\mathbf{S}_{n}$-equivariant isomorphism

[TABLE]

Proof.

One has $\mathbf{S}_{n}$-equivariant isomorphisms

[TABLE]

The first isomorphism comes by using scalar multiplication by elements of arbitrarily large order (which exist since $\boldsymbol{K}$ is infinite). The second is a consequence of $V$ being finite dimensional. For the last one, since $V$ has dimension $n$ and $\boldsymbol{K}$ is infinite one has

[TABLE]

as an $\mathbf{S}_{n}$-bimodule [41, A.2.1]. ∎

Now we are ready for the operadic reconstruction theorem. See 4.3

Proof.

The proof follows the same general logic as the monoidal case. Let $V$ be a vector space of dimension $n$ over $\boldsymbol{K}$, then the identity of $P(n)$ factors as a chain of monomorphisms as follows.

1. (1)

The $n$-th component of the unit of the adjunction $P(n)\to\mathscr{E}(\mathrm{U}_{P})(n)$ is a monomorphism because $\mathscr{C}$ is faithfully tensored over symmetric sequences in $\mathscr{C}$ ([42] or [1, 2.3.10]; [3, 4.4]); 2. (2)

Using separators, the $n$-ary component of canonical map of endomorphism operads

[TABLE]

is a monomorphism [5.4], where $\mathrm{j}_{V}$ is the inclusion

[TABLE] 3. (3)

Next, let $(v_{1},\dots,v_{n})$ be a basis of $V$. Then given any $n$-tuple $(a_{1},\dots,a_{n})$ in $P\mathbin{\triangleleft}V$, one can build a $P$-algebra map $f:\text{Free}_{P}V\to\text{Free}_{P}V$ such that the composite

[TABLE]

sends $v_{1}\otimes\dots\otimes v_{n}$ to $a_{1}\otimes\dots\otimes a_{n}$. This implies that the map induced by restriction along $V\to\mathrm{U}_{P}(\text{Free}_{P}V)$

[TABLE]

is a monomorphism. By the universal property of the cointegral, this monomorphism factors through a map

[TABLE]

which is also necessarily a monomorphism. But this last codomain is isomorphic to $P(n)$ [6.1].

By inspection, the composition is the identity map of $P(n)$, so the first map in the composition $P(n)\to\mathscr{E}(\mathrm{U}_{P})(n)$ is also an isomorphism. ∎

As a corollary, one gets the following proposition.

See 2.1

Proof.

Let $\mathsf{LieGrp}^{\circ}$ denote the full subcategory of simply connected Lie groups. It is a coreflective subcategory of $\mathsf{Lie}\mathsf{Grp}$

[TABLE]

where the coreflector $G\mapsto G^{\circ}$ sends a Lie group to the universal covering of the connected component of its unit. Since the counit $G^{\circ}\to G$ is sent to an isomorphim by $\mathrm{T}_{\mathrm{e}}$:

[TABLE]

the endormorphism operads of the functors $\mathrm{T}_{\mathrm{e}}:\mathsf{Lie}\mathsf{Grp}\to\mathsf{Vect}_{\mathbf{R}}$ and $\mathrm{T}_{\mathrm{e}}:\mathsf{LieGrp}^{\circ}\to\mathsf{Vect}_{\mathbf{R}}$ are canonically isomorphic.

Now, $\mathrm{T}_{\mathrm{e}}$ factors as

[TABLE]

Then the reconstruction theorem above shows that $\mathscr{E}(\mathrm{U}_{\mathsf{Lie}})$ is $\mathsf{Lie}$, hence $\mathscr{E}(\mathrm{T}_{\mathrm{e}})$ is also the Lie operad. ∎

7. Computations

In this section we tie up remaining loose ends, giving the details of deferred computations. In each case we shall use the same stategy: given a right adjoint $F:\mathcal{D}\to\mathscr{C}$,

1. (1)

select a separator $S\subset\mathscr{C}$; 2. (2)

use the operadic separator lemma [5.4] to get monomorphisms

[TABLE] 3. (3)

identify the biggest subobject of $\mathscr{E}(F_{S^{n}})(n)$ that acts naturally on $F$ compatibly with the action on $F_{S^{n}}$; this is $\mathscr{E}(F)(n)$.

The separators that we shall use are the typical ones: the singleton for sets, $R$ in the case of modules over a ring $R$ and the set $\{R[n]\}_{n\in\mathbf{Z}}$ in the case of graded $R$-modules.

Lemma 7.1**.**

Let

[TABLE]

be the forgetful functor from groups to sets. Then $\mathscr{E}(\mathrm{U}_{\mathsf{Grp}})(n)$ is naturally isomorphic to the set underling a free group on $n$ generators, with action as specified below.

Proof.

Let $\operatorname{F}_{X}$ be the free group on $X=\{x_{1},\ldots,x_{n}\}$. We shall show that

[TABLE]

is naturally isomorphic to $\operatorname{F}_{X}$.

Let $\phi$ be a map from $\operatorname{F}_{X}^{n}$ to $\operatorname{F}_{X}$ commuting with every group homomorphism of $\operatorname{F}_{X}$, let $g_{1},\ldots,g_{n}$ be a tuple in $\operatorname{F}_{X}^{n}$, and let $f$ be the group homomorphism taking $x_{i}$ to $g_{i}$. Then

[TABLE]

so $\phi$ is determined by its value on $(x_{1},\ldots,x_{n})$, which is an element of $\operatorname{F}_{X}$.

These are all distinct because they take different values on the tuple $(x_{1},\ldots,x_{n})$, so it remains only to argue that all of these in fact yield natural maps for arbitrary groups.

For any group $G$, an element in $\operatorname{F}_{X}$ yields a map $G^{\times n}\to G$ which takes $(g_{1},\ldots,g_{n})$ to the word in $G$ obtained from $w$ by replacing $x_{i}$ with $g_{i}$; this is clearly natural. ∎

For computations of endomorphisms of identity functors, recall that the identity functor is isomorphic to the forgetful functor from algebras over the trivial operad.

Proposition 7.2**.**

The central operad of the category $R\text{-}\mathsf{mod}$ of modules over a commutative ring $R$ is the trivial operad (i.e., $R$ in arity one).

Proof.

We shall compute the cointegrals

[TABLE]

i.e., the linear maps $\phi$ from $(R^{n})^{\otimes n}$ to $R^{n}$ commuting with every map $R^{n}\to R^{n}$.

• •

For $n=0$, we have $R^{n}=R^{0}=0$, and there is only one map $R\to 0$;

• •

For $n=1$, the center of $\operatorname{GL}_{1}(R)$ is $R$ itself;

• •

For $n>1$, commuting with the map $\textnormal{p}_{i}:R^{n}\to R^{n}$ killing the basis element $e_{i}$ and acting as the identity on the other variables means that any such $\phi$ must take $e_{1}\otimes\cdots\otimes e_{n}$ to the kernel of $\textnormal{p}_{i}$. For distinct $i$ and $j$, the kernel of $\textnormal{p}_{i}$ and the kernel of $\textnormal{p}_{j}$ have null intersection so any such transformation must take the generic primitive tensor to zero.

Thus this is the trivial operad $R$, which clearly acts naturally by scalar multiplication. ∎

Proposition 7.3**.**

The central operad of the category of $\mathbf{Z}$-graded $R$-modules is $R^{\mathbf{Z}}$, concentrated in arity one (it acts by scalar multiplication separately in each degree).

Proof.

We shall compute a cointegral over modules of the form

[TABLE]

Then we need to determine the maps $(R^{\vec{\jmath}})^{\otimes n}\to R^{\vec{\jmath}}$ which commute with every module map from $R^{\vec{\jmath}}$ to itself. The computation in the ungraded case goes through as before for each $\vec{\jmath}$. In particular, this shows that $\mathscr{E}(\operatorname{id})$ is concentrated in arity one. For the calculation in arity one, the full subcategory spanned by the modules $R[j]$ has no maps between $R[j_{1}]$ and $R[j_{2}]$ for distinct $j_{1}$ and $j_{2}$. Thus

[TABLE]

On the other hand, we can realize such a product $(r_{j})_{j\in\mathbf{Z}}$ as the natural transformation which multiplies degree $j$ by $r_{j}$. ∎

The situation is more interesting in sets. Let us recall that $\mathbf{Perm}$ is the operad whose algebras are sets endowed with a binary operation $(x,y)\mapsto xy$ satisfying $(xy)z=x(yz)=x(zy)$.

Proposition 7.4**.**

The central operad of the category of sets is the $\mathbf{Perm}$ operad.

Proof.

We shall compute

[TABLE]

i.e., the maps $\phi$ from $[n]^{n}$ to $[n]$ commuting with every endomorphism of $[n]$, where $[n]$ denotes the set $\{1,\dots,n\}$. For $n=0$ this set is empty. For $n$ strictly larger than zero, let $\phi$ be such a function and suppose $\phi(1,\ldots,n)=i$. For any tuple $(a_{1},\ldots,a_{n})$ be a tuple in $[n]^{n}$ let $f$ be the function $[n]\to[n]$ which takes $j$ to $a_{j}$. Then

[TABLE]

which shows that $\phi$ is the projection in the $i$th coordinate.

Thus this $n$-th cointegral is the set $\{\textnormal{p}_{1},\ldots,\textnormal{p}_{n}\}$, with symmetric group action on the subscript. There are evident relations between the projections for $n=2$:

[TABLE]

as all of these are the projection $\textnormal{p}_{1}$ in arity $3$. This is a presentation for the $\mathbf{Perm}$ operad which acts naturally on sets in the obvious way. ∎

Finally we conduct our computation of the endomorphism operad of the forgetful functor from commutative algebras to vector spaces over a finite field.

See 4.5

Proof.

Let us consider the cointegral

[TABLE]

where $\mathrm{S}(-)$ denotes the symmetric algebra and $V_{n}$ is an $n$-dimensional vector space with a choice of basis.

Let $(e_{1},\dots,e_{n})$ be a basis of $V_{n}$ and let $(f_{1},\ldots,f_{n})$ be an ordered set of polynomials in the variables $\mathrm{X}_{1},\ldots,\mathrm{X}_{n}$. There is an endomorphism of $\mathrm{S}(V_{n})$ which takes $e_{i}$ to $f_{i}(e_{1},\ldots,e_{n})$. Equalizing over this algebra map implies that any function in the cointegral is fully determined by its value $f$ on $(e_{1},\dots,e_{n})$, so that there is a monomorphism

[TABLE]

We are now left to find the biggest subobject of $\mathrm{S}(V_{n})$ that acts naturally on $\mathrm{U}_{\mathsf{C}}$.

Thus consider an element of $\mathrm{S}(V_{n})$, i.e., a polynomial of the form

[TABLE]

Such a polynomial acts set-theoretically on commutative algebras $A$ via evaluation

[TABLE]

This is already natural as a map of sets. But we must restrict to those $f$ for which $\mathrm{Ev}_{f}$ is a $\mathbf{F}_{q}$-multilinear, i.e., to induce a map

[TABLE]

For this, let us see what multilinearity means in the case where $A=\mathrm{S}(V_{n+1})$ is the free commutative algebra on $n+1$ generators $e_{1},\dots,e_{n+1}$: consider the equation in $\mathrm{S}(V_{n+1})^{\times n}$

[TABLE]

Via direct computation, a first condition for the function $\mathrm{Ev}_{f}(\mathrm{S}(V_{n+1}))$ to be linear with respect to the above equation is that each of the monomials of $f$ act linearly in its $i$-th variable.

Then for the linearity of each monomial, the above equation yields also

[TABLE]

This is only possible if $m_{i}$ is a power of $p$, the characteristic of $\mathbf{F}_{q}$. This already implies, e.g., that $m_{i}\neq 0$.

For multilinearity we also need $(\alpha e_{i})^{m_{i}}$ to equal $\alpha e_{i}^{m_{i}}$ for arbitrary $\alpha\in\mathbf{F}_{q}$. This implies that $q-1$ must divide $m_{i}-1$. This fact then further implies that $m_{i}$ is not only a power of $p$ but also a power of $q$.

So far, we have argued that the $n$-ary operations in $\mathscr{E}(\mathrm{U}_{\mathsf{C}})$ inject into the $\mathbf{F}_{q}$ polynomials in $\mathrm{X}_{1},\ldots,\mathrm{X}_{n}$ such that the exponent of each $\mathrm{X}_{i}$ is a power of $q$. Conversely any such polynomial clearly acts linearly. This presentation corresponds to the presentation in the hypotheses of the proposition as follows. The binary product $\mu$ corresponds to the two-variable monomial $\mathrm{X}_{1}\mathrm{X}_{2}$ and the power operation $P_{q}$ corresponds to the one-variable monomial $\mathrm{X}_{1}^{q}$. Operadic composition is via substitution of variables. ∎

Acknowledgments

The authors would like to thank Rune Haugseng, Theo Johnson-Freyd, Claudia Scheimbauer, and John Terilla for useful discussions, Greg Arone, Michael Batanin and Birgit Richter for pointing us to pertinent references in the literature, and Benoit Fresse for a remark that partially inspired this paper.