Hochschild cohomology for functors on linear symmetric monoidal categories

We consider an essentially small symmetric monoidal category, $\mathcal{X}$, enriched in $R$-Mod with $R$ a commutative ring with unit, and then $R$-linear functors from $\mathcal{X}$ to $R$-Mod. This category of functors, denoted by $\mathcal{F}$, is an abelian, symmetric monoidal, closed category, via Day’s convolution. Given a monoid $A$ in $\mathcal{F}$, we have then a category a $A$-modules, $A$-Mod. We refer the reader to [9] or [7] for the generalities on monoids and modules over them. The main example we have in mind when considering these hypothesis is $\mathcal{X}$ as the biset category and $\mathcal{F}$ as the category of biset functors (see, for example, [3]). Monoids in this case are called Green biset functors and have been extensively studied in the last years, in particular, in [4] the commutant and the center of a Green biset functor are studied. As we will see, the commutant of a Green biset functor is the Hochschild cohomology functor of degree zero. So, it is a natural question to ask for a Hochschild cohomology theory of Green biset functors. The purpose of the paper is to develop the main results and properties of this theory for monoids in $\mathcal{F}$, in order to apply them to biset functors in a forthcoming paper in collaboration with Serge Bouc.

Even though there are recent articles about Hochschild cohomology in monoidal categories (see for example [1] and [6]), none of them is suited for a direct application to our case. So, we take a different and new approach by making use of the internal hom functor in $\mathcal{F}$, defined in Section 3. The advantage of working with functors from $\mathcal{X}$ to $R$-Mod is that we can give an explicit construction of the internal hom functor, via the Yoneda-Dress construction (see chapters 2 and 3 of [7] for the abstract definition). With this, the definition of the Hochschild cochain complex of $A$ appears in a natural way in the category $\mathcal{F}$. In particular, we can give an explicit description of its arrows, in terms of these homs, which allows for a better understanding of the Hochschild cohomology functors, denoted by $\mathcal{H}H^{i}(A,\,M)$, for $M$ an $A$-bimodule and $i$ a natural number.

The article is inspired in Loday’s presentation of the classical Hochschild cohomology theory (see [8]). So, in Section 4 we introduce the bar resolution of a monoid $A$ in $\mathcal{F}$ and we define the Hochschild cochain complex of $A$. In Section 5 we deal with the important example of a separable monoid. The main result is Theorem 5.6, in which we show that if $A$ is a separable monoid in $\mathcal{F}$, then, for any $A$-bimodule $M$, the functors $\mathcal{H}H^{i}(A,\,M)$ are zero for very $i>0$ and, conversely, if $A$ is a monoid in $\mathcal{F}$ such that the first Hochschild cohomology functor $\mathcal{H}H^{1}(A,\,M)$ is zero for any $A$-bimodule $M$, then $A$ is separable. As a corollary, we obtain that the Hochschild cohomology functors, for $i>0$, of the Burnside biset functor, $RB$, as well as those of the biset functor of rational representations, $kR_{\mathbb{Q}}$, over a filed $k$ of characteristic $0$, are all zero.

Finally, in Section 6, we describe the Hochschild cohomology functors of degrees 0, 1 and 2. As in the classical case, the degree 1 can be described in terms of derivations and the degree 2 is described in terms of extensions. Also, following [5], for a monoid $A$ in $\mathcal{F}$, we endow the coproduct $\bigoplus_{i\in\mathbb{N}}\mathcal{H}H^{i}(A,\,A)$ with a structure of a graded monoid and call it the Hochschild cohomology monoid. We try too keep the proofs of the statements in this last section as brief as possible, since there are many straightforward calculations that work as in the classical case.

For an object $x$ in a category $\mathcal{C}$, we denote the identity morphism of $x$ simply as $x$.

In what follows $R$ is a commutative ring with identity and $(\mathcal{X},\,\diamond,\,\mathbf{1},\,\alpha,\,\lambda,\,\rho,\,s)$ is an essentially small symmetric monoidal category enriched in $R$-Mod, in the sense of Definition 3.1.51 of [7]. In particular the functor $\,\_\,\diamond\,\_\,:\mathcal{X}\times\mathcal{X}\rightarrow\mathcal{X}$ is $R$-bilinear. The category of $R$-linear functors from $\mathcal{X}$ to $R$-Mod is denoted by $\mathcal{F}$. Recall that $\mathcal{F}$ is an abelian, symmetric monoidal, closed category with identity given by $I=\mathcal{X}(\mathbf{1},\,\_\,)$ (see for example Section 3.3 in [7]). We denote the tensor product of two objects $M$ and $N$ in $\mathcal{F}$ as $M\otimes N$. With this, the complete notation for $\mathcal{F}$ is $(\mathcal{F},\,\otimes,\,I,\,\alpha^{\mathcal{F}},\,\lambda^{\mathcal{F}},\,\rho^{\mathcal{F}},\,S)$. As it is customary, we will avoid, whenever is possible, the parenthesis of association in tensor product of several objects in $\mathcal{F}$.

Given a monoid $A$ in $\mathcal{F}$, we denote its product by $\mu:A\otimes A\rightarrow A$, in case of confusion we may write $\mu_{A}$ instead of $\mu$. We recall that we have a morphism $e:I\rightarrow A$, or $e_{A}$, and the following commuting diagrams,

in $\mathcal{F}$.

In what follows $A$ and $C$ are monoids in $\mathcal{F}$. A morphism of monoids from $A$ to $C$ is an arrow $f:A\rightarrow C$ in $\mathcal{F}$ such that the following diagram commutes

and $f\circ e_{A}=e_{C}$.

The tensor product $A\otimes C$, of $A$ and $C$, is a monoid with product given by

where $S_{2,\,3}$ is the symmetry $C\otimes A\rightarrow A\otimes C$. When needed, as it was in this case, we will denote the symmetry with subindex.

A left $A$-module is an object $M$ in $\mathcal{F}$ together with an arrow $\nu:A\otimes M\rightarrow M$ such that the following diagrams commute

in $\mathcal{F}$. Right $A$-modules are defined in an analogous way. Morphisms of $A$-modules are defined in the obvious way and the subcategory of $\mathcal{F}$ of left $A$-modules is denoted by $A$-Mod.

We denote the opposite of $A$ in $\mathcal{F}$ as $A^{op}$. That is, $A^{op}$ denotes the object $A$ with opposite monoidal structure, $\mu\circ S:A\otimes A\rightarrow A$. The monoid $A$ is called commutative if it is equal to its opposite, i.e. $\mu\circ S=\mu$.

Recall that $C^{op}$-modules identify with right $C$-modules and thus that $(A,\,C)$-bimodules identify with $A\otimes C^{op}$-modules in the following way.

We denote the monoid $A\otimes A^{op}$ by $A^{e}$. We may indistinctly write $A^{e}$-module, $(A,\,A)$-bimodule or even $A$-bimodule.

We begin by recalling the construction of the internal hom object in $\mathcal{F}$. The general definition of this object can be found in Section 3 of [7] but, since we are working with functors on $R$-Mod, we follow the lines of [3], to give an explicit construction.

For each object $x\in\mathcal{X}$, we have an $R$-linear functor $\mathrm{p}_{x}:\mathcal{X}\rightarrow\mathcal{X}$, given by the monoidal structure of $\mathcal{X}$, that is, it sends an object $y$ to $y\diamond x$ and an arrow $\varphi$ to $\varphi\diamond x$. This in turn allows us to define the endofunctor $\mathrm{P}_{x}:\mathcal{F}\rightarrow\mathcal{F}$ called the Yoneda-Dress construction. In an object $F$, it is defined as $F_{x}:=F\circ\mathrm{p}_{x}$ and in an arrow $f:M\rightarrow N$ as $(P_{x}(f))_{y}=f_{y\diamond x}=:f_{x}(y)$, for $y$ an object of $\mathcal{X}$.

The following lemma is a generalization of Lemma 8.2.4. Its proof is straightforward.

By point 4 of the previous lemma, for every object $F$ in $\mathcal{F}$, we have an $R$-linear functor $F_{\,\_\,}:\mathcal{X}\rightarrow\mathcal{F}$.

The rest of the section is devoted to define a hom functor for modules over monoids in $\mathcal{F}$. We begin by stating the universal property of tensor products. It is easy to see that the proof of Proposition 8.4.2 in [3] holds in our case, so Remark 8.4.3 in [3] can also be generalized.

Notice that this remark can be generalized to an arrow from a finite product $\bigotimes_{i=1}^{n}M_{i}$ to $T$.

Let $A$ be a monoid in $\mathcal{F}$. The Yoneda Lemma,

together with the previous remark allows us to see the monoid $A$ as in Definition 8.5.1 of [3].

Let $x$, $y$ and $z$ be objects in $\mathcal{X}$, consider $\alpha_{x,\,y,\,z}:x\diamond(y\diamond z)\rightarrow(x\diamond y)\diamond z$, $\lambda_{x}:\mathbf{1}\diamond x\rightarrow x$ and $\rho_{x}:x\diamond\mathbf{1}\rightarrow x$. Then, $A$ is a monoid in $\mathcal{F}$ if for every pair $x,\,y$ of objects in $\mathcal{X}$, there exist bilinear maps

denoted simply by $(a,\,b)\mapsto a\times b$, and an element $\varepsilon_{A}\in A(\mathbf{1})$ satisfying the following conditions.

• •

  Associativity. For $x$, $y$ and $z$, objects in $\mathcal{X}$, and for any $a\in A(x)$, $b\in A(y)$ and $c\in A(z)$,

  $$(a\times b)\times c=A(\alpha_{x,\,y,\,z})(a\times(b\times c)).$$

• •

  Identity element. For an object $x$ in $\mathcal{X}$ and any $a\in A(x)$,

  $$a=A(\lambda_{x})(\varepsilon_{A}\times a)=A(\rho_{x})(a\times\varepsilon_{A}).$$

• •

  Functoriality. Let $\varphi:x\rightarrow x^{\prime}$ and $\psi:y\rightarrow y^{\prime}$ be arrows in $\mathcal{X}$. Then for any $a\in A(x)$ and $b\in A(y)$,

  $$A(\varphi\diamond\psi)(a\times b)=A(\varphi)(a)\times A(\psi)(b).$$

Also, a module over $A$ can be seen as an in Definition 8.5.5 of [3]. That is, a left $A$-module is an object $M$ in $\mathcal{F}$ such that for every pair $x,\,y$ of objects in $\mathcal{X}$, there exist bilinear maps

which we continue to denote by $(a,\,m)\mapsto a\times m$, satisfying corresponding conditions of associativity, identity element (on the left only) and functoriality. In a similar fashion we can define right modules and bimodules.

Morphisms of $A$-modules can also be rewritten in these terms. Let $M$ and $N$ be $A$-modules, a morphism of $A$-modules from $M$ to $N$ is an arrow $f:M\rightarrow N$ in $\mathcal{F}$ such that for any objects $x$ and $y$ in $\mathcal{X}$, we have $f_{x\diamond y}(a\times m)=a\times f_{y}(m)$ for all $a\in A(x)$ and $m\in M(y)$.

The proof of the following lemma is straightforward.

Given two $A$-modules $M$ and $N$, we have that $\textrm{Hom}_{A-\mathrm{Mod}}(M,\,N)$ is an $R$-submodule of $\textrm{Hom}_{\mathcal{F}}(M,\,N)$. Moreover, for an $A$-module $M$, we have a functor $\textrm{Hom}_{A-\mathrm{Mod}}(M,\,\_):A$-$\mathrm{Mod}\rightarrow R$-$\mathrm{Mod}$.