# Generalized polynomial functors

Jonathan D. Axtell  Sungkyunkwan University, Suwon 16419, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

We define Schur categories, $\Gamma^{d}\mathcal{C}$, associated to a
$\Bbbk$-linear category $\mathcal{C}$, over a commutative ring $\Bbbk$. The
corresponding representation categories,
$\mathbf{rep}\,\Gamma^{d}\mathcal{C}$, generalize categories of strict
polynomial functors. Given a $\Bbbk$-superalgebra $A$, we show that for
certain categories $\mathcal{V}=\boldsymbol{\mathcal{V}}_{A}$,
$\boldsymbol{\mathcal{E}}_{A}$ of $A$-supermodules, there is a Morita
equivalence between $\mathbf{rep}\,\Gamma^{d}\mathcal{V}$ and the category of
supermodules over generalized Schur superalgebras of the form $S^{A}(m|n,d)$
and $S^{A}(n,d)$, respectively. We also describe a formulation of generalized
Schur-Weyl duality from the viewpoint of the category
$\mathbf{rep}\,\Gamma^{d}\boldsymbol{\mathcal{E}}_{A}$.

##  1\. Introduction

Let $\Bbbk$ be a commutative ring and suppose $A=A_{\bar{0}}\oplus
A_{\bar{1}}$ is a $\Bbbk$-superalgebra such that each $A_{\epsilon}$
($\epsilon=\bar{0},\bar{1}$) is a finitely generated, free $\Bbbk$-module. The
generalized Schur superalgebras $S^{A}(n,d)$ were defined by Evseev and
Kleshchev [6, 7] in order to prove the Turner double conjecture. These
superalgebras are related to wreath product algebras by a generalized Schur-
Weyl duality established in [6].

The category $\mathcal{P}_{d}$ of (degree $d$, homogeneous) strict polynomial
functors was defined by Friedlander and Suslin in [8] in order to prove
results about the cohomology of finite group schemes. In the case where
$\Bbbk$ is a field of characteristic $p\neq 2$, certain super-analogues of
these categories were defined in [1], denoted
$\mathsf{Pol}^{(\mathrm{I})}_{d}$ and $\mathsf{Pol}^{(\mathrm{II})}_{d}$,
which are isomorphic to the categories of finitely generated supermodules over
the Schur superalgebras, $S(m|n,d)$ and $Q(n,d)$, respectively, provided that
$m,n\geq d$.

In this paper, we describe a generalization of the categories
$\mathcal{P}_{d}$ and $\mathsf{Pol}_{d}^{(\dagger)}$
($\dagger=\mathrm{I},\mathrm{II}$). Suppose first that $\mathcal{C}$ is a
$\Bbbk$-linear category whose Hom sets belong to the category
$\boldsymbol{\mathcal{V}}_{\Bbbk}$ of finitely generated, projective
$\Bbbk$-supermodules, and such that composition induces an even $\Bbbk$-linear
map (see Section 2.7 for more details). In Section 3, we define the Schur
categories, $\Gamma^{d}\mathcal{C}$, whose morphisms consist of divided powers
of morphisms in the category $\mathcal{C}$. Then in Section 4, we define the
category of generalized polynomial functors
$\boldsymbol{\mathcal{P}}_{d,\mathcal{C}}$ to be the representation category

$\boldsymbol{\mathcal{P}}_{d,\mathcal{C}}:=\mathbf{rep}\,\Gamma^{d}\mathcal{C}$

consisting of all even, $\Bbbk$-linear functors from $\Gamma^{d}\mathcal{C}$
to $\boldsymbol{\mathcal{V}}_{\Bbbk}$.

If the Hom-sets of $\mathcal{C}$ belong to the subcategory
$\mathcal{V}_{\Bbbk}\subset\boldsymbol{\mathcal{V}}_{\Bbbk}$ of ordinary (non-
graded) projective $\Bbbk$-modules, then the morphisms of
$\Gamma^{d}\mathcal{C}$ also belong to $\mathcal{V}_{\Bbbk}$. In this case, we
also define the category of generalized polynomial functors
$\mathcal{P}_{d,\mathcal{C}}$ to be the representation category

$\mathcal{P}_{d,\mathcal{C}}:=\mathrm{rep}\,\Gamma^{d}\mathcal{C},$

consisting of all $\Bbbk$-linear functors from $\Gamma^{d}\mathcal{C}$ to
$\mathcal{V}_{\Bbbk}$.

Now suppose that $A\in\boldsymbol{\mathcal{V}}_{\Bbbk}$ is a
$\Bbbk$-superalgebra as described above. Let $\boldsymbol{\mathcal{V}}_{A}$
denote the category of all finitely-generated, projective right
$A$-supermodules. We then introduce evenly-generated $A$-supermodules in
Section 2.5, and we let
$\boldsymbol{\mathcal{E}}_{A}\subset\boldsymbol{\mathcal{V}}_{A}$ denote the
full subcategory of evenly-generated, projective right $A$-supermodules.

Suppose that $m,n\geq d$ are integers. Then we also describe a generalized
Schur superalgebra of the form $S^{A}(m|n,d)$ in Section 3.4. In our main
result Theorem 4.3, we show that there are equivalences of categories

$\boldsymbol{\mathcal{P}}_{d,\boldsymbol{\mathcal{V}}_{A}}\ \simeq\
S^{A}(m|n,d)\kern 1.0pt\mathchar
45\relax\mathbf{mod},\qquad\boldsymbol{\mathcal{P}}_{d,\boldsymbol{\mathcal{E}}_{A}}\
\simeq\ S^{A}(n,d)\kern 1.0pt\mathchar 45\relax\mathbf{mod},$

between a category of generalized polynomial functors and a corresponding
category of left supermodules over a Schur superalgebra.

Considering the special cases where $A=\Bbbk$ or $A=\mathcal{C}(1)$, a rank
one Clifford algebra, gives the following equivalences of categories:

$\boldsymbol{\mathcal{P}}_{d,\boldsymbol{\mathcal{V}}_{\Bbbk}}\simeq\mathsf{Pol}^{(\mathrm{I})}_{d},\qquad\boldsymbol{\mathcal{P}}_{d,\boldsymbol{\mathcal{E}}_{\mathcal{C}(1)}}\simeq\mathsf{Pol}^{(\mathrm{II})}_{d}$

respectively. Similarly, we have isomorphisms of superalgebras:

$S^{C(1)}(n,d)\cong Q(n,d),\qquad S^{\Bbbk}(m|n,d)\cong S(m|n,d).$

It follows that Theorem 4.3 generalizes Theorem 4.2 of [1].

Now consider the case of an ordinary ungraded algebra $A$. Then we may regard
$S^{A}(n,d)$ as an ordinary algebra. In Theorem 4.4, we describe an
equivalence

$\mathcal{P}_{d,\mathcal{V}_{A}}\ \simeq\ S^{A}(n,d)\kern 1.0pt\mathchar
45\relax\mathrm{mod}$

between $\mathcal{P}_{d,\mathcal{V}_{A}}$ and the category of left
$S^{A}(n,d)$-modules. In the case $A=\Bbbk$, we have an equivalence of
categories

$\mathcal{P}_{d,\mathcal{V}_{\Bbbk}}\simeq\mathcal{P}_{d,\Bbbk},$

and there is algebra isomorphism between $S^{\Bbbk}(n,d)$ and the classical
Schur algebra $S(n,d)$ defined in [9]. It follows that Theorem 4.4 generalizes
Theorem 3.2 of [8].

Finally, in Section 5, we describe an exact functor from the category
$\boldsymbol{\mathcal{P}}_{d,\boldsymbol{\mathcal{E}}_{A}}$ to the category of
finite dimensional left supermodules over the wreath product superalgebra
$A\wr\mathfrak{S}_{d}$ (Theorem 5.1). This functor may be viewed as a
categorical analogue of the generalized Schur-Weyl duality described by
Kleshchev and Etseev in [7].

##  2\. Preliminaries

We assume throughout that $\Bbbk$ is a commutative ring with unit. Let
$\mathbb{N}$ and $\mathbb{N}_{0}$ denote the positive and nonnegative
integers, respectively.

###  2.1. Definitions

Let $\mathcal{M}_{\Bbbk}$ denote the category of all $\Bbbk$-modules and
$\Bbbk$-linear maps. Given $M,N\in\mathcal{M}_{\Bbbk}$, we write $M\otimes
N:=M\otimes_{\Bbbk}N$, $\mathrm{Hom}(M,N):=\mathrm{Hom}_{\Bbbk}(M,N)$ and
$\mathrm{End}(M):=\mathrm{Hom}(M,M)$.

A $\Bbbk$-supermodule is a $\mathbb{Z}/2$-graded $\Bbbk$-module,
$M=M_{\bar{0}}\oplus M_{\bar{1}}$. We write $|M|\in\mathcal{M}_{\Bbbk}$ to
denote the ordinary $\Bbbk$-module obtained by forgetting the
$\mathbb{Z}/2$-grading. Given $\varepsilon\in\mathbb{Z}/2$ and homogeneous
$v\in M_{\varepsilon}$, we also write $\overline{v}=\varepsilon$. Elements of
$M_{\bar{0}}$ (resp. $M_{\bar{1}}$) are called even (resp. odd).

Let us write $\boldsymbol{\mathcal{M}}_{\Bbbk}$ to denote the category of all
$\Bbbk$-supermodules and $\Bbbk$-linear maps. Given objects
$M,N\in\boldsymbol{\mathcal{M}}_{\Bbbk}$, both the tensor product $M\otimes N$
and the hom-set $\mathrm{Hom}(M,N)$ are again objects of
$\boldsymbol{\mathcal{M}}_{\Bbbk}$, with $\mathbb{Z}/2$-gradings defined by

$(M\otimes N)_{\epsilon}:=M_{\bar{0}}\otimes N_{\epsilon}\oplus
M_{\bar{1}}\otimes N_{\bar{1}+\epsilon}$

and

$\mathrm{Hom}(M,N)_{\epsilon}:=\mathrm{Hom}(M_{\bar{0}},N_{\epsilon})\oplus\mathrm{Hom}(M_{\bar{1}},N_{\bar{1}+\epsilon}),$

respectively, for each $\epsilon\in\mathbb{Z}/2$.

We will consider $\mathcal{M}_{\Bbbk}$ as a subcategory of
$\boldsymbol{\mathcal{M}}_{\Bbbk}$ by identifying an ordinary $\Bbbk$-module
$M\in\mathcal{M}_{\Bbbk}$ as a $\Bbbk$-supermodule concentrated in degree
$\bar{0}$.

###  2.2. Superalgebras

A superalgebra is a $\Bbbk$-supermodule $A=A_{\bar{0}}\oplus A_{\bar{1}}$
which is a unital associative $\Bbbk$-algebra, such that multiplication
$m_{A}:A\otimes A\to A$ is an even $\Bbbk$-linear map. A superalgebra
homomorphism $\vartheta:A\to B$ is an even $\Bbbk$-linear map that is an
algebra homomorphism in the usual sense.

Given a pair of superalgebras $A,B$, the tensor product $A\otimes B$ is again
a superalgebra in a natural way. The multiplication is defined by the usual
rule of signs convention

(1) $(a_{1}\otimes b_{1})(a_{2}\otimes
b_{2})=(-1)^{\overline{b_{1}}\,\overline{a_{2}}}(a_{1}a_{2})\otimes(b_{1}b_{2})$

for all $a_{1},a_{2}\in A$, $b_{1},b_{2}\in B$, and the unit is given by
$1_{A\otimes B}:=1_{A}\otimes 1_{B}$. Expressions of the form (1) are assumed
to hold for homogeneous elements and are then extended linearly to the general
case.

###  2.3. Supermodules

Let $A$ be a superalgebra. A left $A$-supermodule is a $\Bbbk$-supermodule
$M\in\boldsymbol{\mathcal{M}}_{\Bbbk}$ which is a left $A$-module in the usual
sense, such that $A_{\epsilon}M_{\epsilon^{\prime}}\subseteq
M_{\epsilon+\epsilon^{\prime}}$ for
$\epsilon,\epsilon^{\prime}\in\mathbb{Z}/2$. One may define right
$A$-supermodules similarly. A subsupermodule of a left (resp. right)
$A$-supermodule $M$ is an $A$-submodule $N\subset M$ such that $N=(N\cap
M_{\bar{0}})\oplus(N\cap M_{\bar{1}})$.

A homomorphism $\varphi:M\rightarrow N$ of left (resp. right) $A$-supermodules
$M,N$ is a (not necessarily homogeneous) $\Bbbk$-linear map such that

$\varphi(av)=(-1)^{\overline{\varphi}\,\overline{a}}a\varphi(v),\quad\text{resp.
}\ \varphi(va)=\varphi(v)a,$

for all $a\in A,v\in V$.

Considering $A$ as a right supermodule over itself, notice that the map

(2) $A\,\xrightarrow{\sim}\,\mathrm{End}_{A}(A):\,a\mapsto m_{A}(a\otimes-)$

is an isomorphism of superalgebras.

Let ${}_{A}\boldsymbol{\mathcal{M}}$ (resp. $\boldsymbol{\mathcal{M}}_{A}$)
denote the category of all left (right) $A$-supermodules and homomorphisms,
with composition of morphisms defined by the rule of signs, as in (1). Given a
pair $M,N$ of left (right) $A$-supermodules, let $\mathrm{Hom}_{A}(M,N)$
denote the set of homomorphisms, $\varphi:M\to N$. We may also consider the
superalgebra $\mathrm{End}_{A}(M):=\mathrm{Hom}_{A}(M,M)$.

We write $A\kern 1.0pt\mathchar 45\relax\mathbf{mod}$ (resp.
$\mathbf{mod}\kern 1.0pt\mathchar 45\relax A$) to denote the full subcategory
of $A$-supermodules which are finitely generated as ordinary $A$-modules. The
underlying $\Bbbk$-module $|A|\in\mathcal{M}_{\Bbbk}$ is an ordinary
$\Bbbk$-algebra. In this case, we respectively write

${}_{|A|}\mathcal{M},\quad\mathcal{M}_{|A|},\quad|A|\kern 1.0pt\mathchar
45\relax\mathrm{mod},\quad\mathrm{mod}\kern 1.0pt\mathchar 45\relax|A|,$

to denote the corresponding categories of ordinary left $|A|$-modules, etc.

###  2.4. Free $A$-supermodules

Let $A$ be a superalgebra. Given any right $A$-supermodule
$M\in\boldsymbol{\mathcal{M}}_{A}$, define the parity change $\Pi M$ to be the
same $A$-module, but with opposite $\mathbb{Z}/2$-grading. For example, given
$m,n\in\mathbb{N}$ and considering $A$ as a right $A$-supermodule, we write
$A^{n|m}:=A^{n}\oplus(\Pi A)^{m}$. We will also write

$\mathit{M}_{n}(A):=\mathrm{End}_{A}(A^{n}),\quad\qquad\mathit{M}_{n|m}(A):=\mathrm{End}_{A}(A^{n|m})$

to denote corresponding superalgebras.

We define the (right) parity change functor $\Pi:\mathbf{mod}\kern
1.0pt\mathchar 45\relax A\to\mathbf{mod}\kern 1.0pt\mathchar 45\relax A$ which
sends $M\to\Pi M$. On a right A-supermodule homomorphism $\phi:M\to N$, we set
$\Pi(\phi)=(-1)^{\bar{\phi}}\phi$ as a linear map.

Suppose the free right $A$-supermodule $A^{n|m}$ (resp.
$A^{n^{\prime}|m^{\prime}}$) has homogeneous $A$-linear basis
$v_{1},\dots,v_{m+n}$ (resp. $w_{1},\dots,w_{m^{\prime}+n^{\prime}}$), such
that $v_{1},\dots,v_{n}$ (resp. $w_{1},\dots,w_{n^{\prime}}$) are even and
$v_{n+1},\dots,v_{m+n}$ (resp.
$w_{n^{\prime}+1},\dots,w_{m^{\prime}+n^{\prime}}$) are odd. Then for a given
$a\in A$, we write $e(j,i)$ (resp. $e^{(a)}(j,i)$) to denote the homomorphism
in $\mathrm{Hom}_{A}(A^{n|m},A^{n^{\prime}|m^{\prime}})$ which sends
$v_{k}\mapsto 0$ for all $k\neq i$ and sends $v_{i}\mapsto w_{j}$ (resp.
$v_{i}\mapsto w_{j}a$).

Notice that the parity of the element $e(j,i)$ is given by:

$\overline{e(j,i)}=\begin{cases}\bar{0},&\text{if }\,1\leq i\leq n\,\text{ and
}\,1\leq j\leq n^{\prime},\\\ &\text{or if }\,n<i\leq m+n\,\text{ and
}\,n^{\prime}<j\leq m^{\prime}+n^{\prime};\\\
\bar{1},&\text{else.}\end{cases}$

Similarly, given a homogeneous element $a\in A$, we have

(3) $\overline{e^{(a)}(j,i)}=\bar{a}+\overline{e(j,i)}.$

Now suppose $A\in\boldsymbol{\mathcal{V}}_{\Bbbk}$. If $A$ is free as a
$\Bbbk$-supermodule, then we may identify
$\mathrm{Hom}(\Bbbk^{n|m},\Bbbk^{n^{\prime}|m^{\prime}})$ as a
$\Bbbk$-subsupermodule of
$\mathrm{Hom}_{A}(A^{n|m},A^{n^{\prime}|m^{\prime}})$ which is generated by
the elements $e(j,i)$. The isomorphism (2) may then be generalized as follows.

######  Lemma 2.1.

Let $A$ be a superalgebra which is finitely-generated and free as a
$\Bbbk$-module. Considering $A$ as a right $A$-supermodule, there exists for
any $m,n,m^{\prime},n^{\prime}\in\mathbb{N}$ an even isomorphism of
$\Bbbk$-supermodules

$A\otimes\mathrm{Hom}(\Bbbk^{n|m},\Bbbk^{n^{\prime}|m^{\prime}})\simeq\mathrm{Hom}_{A}(A^{n|m},A^{n^{\prime}|m^{\prime}}):\quad
a\otimes e(j,i)\mapsto e^{(a)}(j,i).$

In particular, we have the following superalgebra isomorphisms:

  * (i)

$A\otimes\mathrm{End}(\Bbbk^{n|m})\simeq\mathit{M}_{n|m}(A)$

  * (ii)

$A\otimes\mathrm{End}(\Bbbk^{n})\simeq\mathit{M}_{n}(A)$

###### Proof.

Since $A$ is a free $\Bbbk$-supermodule, we have $A\simeq\Bbbk^{p|q}$, for
some $p,q\in\mathbb{N}_{0}$. Let $a_{1},\dots,a_{p}$ (resp.
$a_{p+1},\dots,a_{p+q}$) be a $\Bbbk$-basis of $A_{\bar{0}}$ (resp.
$A_{\bar{1}}$). Then the set

$\\{e^{(r)}(j,i):=e^{(a_{r})}(j,i)\mid 1\leq i\leq m+n,\,1\leq j\leq
m^{\prime}+n^{\prime},\,1\leq r\leq p+q\\}$

is a homogeneous $\Bbbk$-basis of
$\mathrm{Hom}_{A}(A^{n|m},A^{n^{\prime}|m^{\prime}})$. It follows by (3) that
$\overline{e^{(r)}(j,i)}=\overline{a}_{r}+\overline{e(j,i)}$, for any $r$. The
mapping

$a\otimes e(j,i)\mapsto e^{(a)}(j,i)$

thus yields an even isomorphism. For the case $m=m^{\prime}$ and
$n=n^{\prime}$, it is easy to check that the isomorphism

$A\otimes\mathrm{End}(\Bbbk^{n|m})\simeq\mathrm{End}_{A}(A^{n|m})$

is a superalgebra homomorphism. This shows part (i) of the lemma. Part (ii)
follows by setting $m=0$. ∎

###  2.5. Evenly generated supermodules

Notice that $\boldsymbol{\mathcal{M}}_{A}$ and
${}_{A}\boldsymbol{\mathcal{M}}$ are not abelian categories. It will thus be
convenient to consider the abelian subcategory
${}_{A}\boldsymbol{\mathcal{M}}^{\mathsf{ev}}\subset{}_{A}\boldsymbol{\mathcal{M}}$
(resp.
$\boldsymbol{\mathcal{M}}_{A}^{\mathsf{ev}}\subset\boldsymbol{\mathcal{M}}_{A}$),
consisting of the same objects but only even morphisms. This allows us to make
use of the basic notions of homological algebra by restricting our attention
to only even morphisms. For example, by a short exact sequence in
${}_{A}\boldsymbol{\mathcal{M}}$ (resp. $\boldsymbol{\mathcal{M}}_{A}$), we
mean a sequence

$0\to L\to M\to N\to 0$

with all the maps being even.

Now suppose that $A$ is a superalgebra and that $M$ belongs to $A\kern
1.0pt\mathchar 45\relax\mathbf{mod}$ (resp. $\mathbf{mod}\kern 1.0pt\mathchar
45\relax A$). Then $M$ is automatically finitely generated as an $A$-module.
It follows that there is a short exact sequence of the form

$A^{n|m}\to M\to 0.$

We will say that $M$ is evenly generated if there exists an even surjective
homomorphism of the form

$A^{n}\twoheadrightarrow M.$

###  2.6. Projective supermodules

We say that a right $A$-supermodule $M$ is projective if $M$ is a projective
object of $\boldsymbol{\mathcal{M}}_{A}^{\mathsf{ev}}$. Let us write
$\boldsymbol{\mathcal{V}}_{A}$ to denote the full subcategory of
$\mathbf{mod}\kern 1.0pt\mathchar 45\relax A$ consisting of all finitely
generated, projective right $A$-supermodules. Let us also write

$\boldsymbol{\mathcal{E}}_{A}\subset\boldsymbol{\mathcal{V}}_{A}$

to denote the full subcategory consisting of the evenly-generated, projective
right $A$-supermodules.

######  Remark 2.2.

The categories $\boldsymbol{\mathcal{E}}_{A}$ and
$\boldsymbol{\mathcal{V}}_{A}$ are not distinct in general. For example, let
$C(1)$ denote the rank 1 Clifford algebra (see [1, Example 2.6]). Then one may
check that every $C(1)$-supermodule is evenly generated, and thus
$\boldsymbol{\mathcal{E}}_{C(1)}=\boldsymbol{\mathcal{V}}_{C(1)}$.

We further write $\mathcal{V}_{|A|}$ to denote the category of ordinary (non-
graded) finitely generated projective $|A|$-modules. Note for example that
$\boldsymbol{\mathcal{V}}_{\Bbbk}$ (resp. $\mathcal{V}_{\Bbbk}$) denotes the
category of all finitely generated, projective $\Bbbk$-supermodules (resp.
$\Bbbk$-modules). In this case, we may identify
$\boldsymbol{\mathcal{E}}_{\Bbbk}$ with $\mathcal{V}_{\Bbbk}$, considered as a
subcategory of $\boldsymbol{\mathcal{M}}_{\Bbbk}$.

###  2.7. Enriched categories

We recall the definition of a specific type of $\mathcal{M}$-enriched
category, where we assume that
$\mathcal{M}\subset\boldsymbol{\mathcal{M}}_{\Bbbk}^{\mathsf{ev}}$. See [11]
for more details about $\mathcal{M}$-enriched categories in general.

First notice that
$\boldsymbol{\mathcal{M}}_{\Bbbk}=(\boldsymbol{\mathcal{M}}_{\Bbbk},\otimes,\Bbbk)$
is a symmetric monoidal category with symmetry isomorphism given by the so-
called supertwist map

$\tau:M\otimes N\xrightarrow{\sim}N\otimes M.$

This is the $\Bbbk$-linear map sending

(4) $v\otimes w\mapsto(-1)^{\overline{v}\,\overline{w}}w\otimes v\quad(v\in
M,w\in N)$

for any $M,N\in\boldsymbol{\mathcal{M}}_{\Bbbk}$. It follows by (4) that the
supertwist $\tau$ is even. It is thus natural to view each of the
subcategories

$\mathcal{M}_{\Bbbk},\
\boldsymbol{\mathcal{M}}_{\Bbbk}^{\mathsf{ev}}\,\subset\
\boldsymbol{\mathcal{M}}_{\Bbbk}$

as a monoidal subcategory by restriction.

In the remainder, we assume that $\mathcal{M}$ denotes a full subcategory of
$\boldsymbol{\mathcal{M}}_{\Bbbk}^{\mathsf{ev}}$ such that:

(5) $\Bbbk\in\mathcal{M}\ \text{ and }\ M\otimes N\in\mathcal{M},\,\text{
whenever }M,N\in\mathcal{M}.$

It follows that $\mathcal{M}$ has the structure of a symmetric monoidal
category given by restriction.

We now recall the corresponding definition of enriched categories.

######  Definition 2.3 ([11]).

An $\mathcal{M}$-enriched category (or $\mathcal{M}$-category) is a category
$\mathcal{C}$ such that the hom-set $\mathrm{Hom}_{\mathcal{C}}(X,Y)$ is an
object of $\mathcal{M}$ for all $X,Y\in\mathcal{C}$, the composition is
$\Bbbk$-bilinear and the induced map

$\circ^{\mathcal{C}}_{X,Y,Z}:\,\mathrm{Hom}_{\mathcal{C}}(Y,Z)\otimes\mathrm{Hom}_{\mathcal{C}}(X,Y)\
\rightarrow\ \mathrm{Hom}_{\mathcal{C}}(X,Z)$

is an even $\Bbbk$-linear map (that is, a morphism in $\mathcal{M}$) for all
$X,Y,Z\in\mathcal{C}$.

If $\mathcal{M}\subset\mathcal{M}^{\prime}$ are both full subcategories of
$\boldsymbol{\mathcal{M}}_{\Bbbk}^{\mathsf{ev}}$ satisfying (5), then it is
clear that any $\mathcal{M}$-category is automatically an
$\mathcal{M}^{\prime}$-category. Notice for example that an
$\mathcal{M}_{\Bbbk}$-enriched category is just a $\Bbbk$-linear category.

Let us fix a category
$\mathcal{M}\subset\boldsymbol{\mathcal{M}}_{\Bbbk}^{\mathsf{ev}}$ as above.
We next recall the definition of enriched functors and natural
transformations.

######  Definition 2.4.

Suppose $\mathcal{C},\mathcal{D}$ are $\mathcal{M}$-enriched categories. An
$\mathcal{M}$-enriched functor $F:\mathcal{C}\to\mathcal{D}$ is a functor such
that the map

$F_{X,Y}:\mathrm{Hom}_{\mathcal{C}}(X,Y)\to\mathrm{Hom}_{\mathcal{D}}(F(X),F(Y))$

is an even $\Bbbk$-linear map for any pair of objects $X,Y\in\mathcal{C}$.

######  Definition 2.5.

Next suppose that $F,G:\mathcal{C}\to\mathcal{D}$ is a pair of
$\mathcal{M}$-enriched functors. We first define an even (resp. odd)
$\mathcal{M}$-enriched natural transformation, $\eta:F\to G$, to be a
collection of even (odd) linear maps

$\eta(X)\in\mathrm{Hom}_{\mathcal{M}}(F(X),G(X))\qquad(^{\forall}X\in\mathcal{C})$

such that for a given $\varphi\in\mathrm{Hom}_{\mathcal{M}}(X,Y)$ we have

$G(\varphi)\circ\eta(X)=(-1)^{\overline{\varphi}\,\overline{\eta(X)}}\eta(Y)\circ
F(\varphi).$

In general, an $\mathcal{M}$-enriched natural transformation, $\eta:F\to G$,
is then defined to be a collection of linear maps,
$\eta(X)=\eta_{\bar{0}}(X)\oplus\eta_{\bar{1}}(X)$ $(X\in\mathcal{C}),$ such
that

$\eta_{\epsilon}(X)\in\mathrm{Hom}_{\mathcal{M}}(F(X),G(X))_{\epsilon}\qquad(\epsilon\in\mathbb{Z}/2)$

and $\eta_{\bar{0}}$ (resp. $\eta_{\bar{1}}$): $F\to G$ is an even (resp. odd)
$\mathcal{M}$-enriched natural transformation.

We also recall the corresponding category of enriched functors and natural
transformations.

######  Definition 2.6.

Given a pair $\mathcal{C},\mathcal{D}$ of $\mathcal{M}$-enriched categories,
we write $\mathrm{Fun}_{\Bbbk}(\mathcal{C},\mathcal{D})$ to denote the
$\mathcal{M}$-enriched category consisting of all even $\Bbbk$-linear functors

$X:\mathcal{C}\to\mathcal{D}$

and whose morphisms are $\mathcal{M}$-enriched natural transformations.

Let $\mathcal{C}$ be an $\mathcal{M}$-enriched category. We write $X\cong Y$,
if $X,Y$ are isomorphic in $\mathcal{C}$. If there is an even isomorphism
$\varphi:X\cong Y$ (i.e.,
$\varphi\in\mathrm{Hom}_{\mathcal{C}}(X,Y)_{\bar{0}}$), then we use the
notation $X\simeq Y$.

Let $\mathcal{C}^{\mathrm{ev}}$ denote the subcategory of $\mathcal{C}$
consisting of the same objects but only even morphisms. Since
$\mathcal{M}$-enriched functors send even morphisms to even morphisms, they
give rise to the corresponding functors between the underlying even
subcategories.

Notice that $\mathcal{V}_{\Bbbk}$ and
$\boldsymbol{\mathcal{V}}_{\Bbbk}^{\mathsf{ev}}$ are both full subcategories
of $\boldsymbol{\mathcal{M}}_{\Bbbk}^{\mathsf{ev}}$ which satisfy (5). In the
remainder, we will mostly consider enriched categories where
$\mathcal{M}=\mathcal{V}_{\Bbbk}$ or
$\boldsymbol{\mathcal{V}}_{\Bbbk}^{\mathsf{ev}}$.

Given a superalgebra $A$, we may view $A\kern 1.0pt\mathchar
45\relax\mathbf{mod}$ and $\mathbf{mod}\kern 1.0pt\mathchar 45\relax A$ as
$\boldsymbol{\mathcal{V}}_{\Bbbk}^{\mathsf{ev}}$-enriched categories, while
${}_{A}\boldsymbol{\mathcal{M}}$ and $\boldsymbol{\mathcal{M}}_{A}$ are
$\boldsymbol{\mathcal{M}}_{\Bbbk}^{\mathsf{ev}}$-enriched categories.
Furthermore, the categories ${}_{A}\boldsymbol{\mathcal{M}}^{\mathsf{ev}}$ and
$\boldsymbol{\mathcal{M}}_{A}^{\mathsf{ev}}$ are abelian, while the respective
subcategories $(A\kern 1.0pt\mathchar 45\relax\mathbf{mod})^{\mathsf{ev}}$,
$(\mathbf{mod}\kern 1.0pt\mathchar 45\relax A)^{\mathsf{ev}}$,
${}_{A}\boldsymbol{\mathcal{V}}^{\mathsf{ev}}$, and
$\boldsymbol{\mathcal{V}}_{A}^{\mathsf{ev}}$ are exact categories in the sense
of Quillen (see [5, 10]).

###  2.8. Tensor Products of $\mathcal{M}$-categories

Suppose $\mathcal{C},\mathcal{D}$ are $\mathcal{M}$-enriched categories. Then
define $\mathcal{C}\otimes\mathcal{D}$ to be the $\mathcal{M}$-enriched
category whose objects are pairs $(X,X^{\prime})$ with $X\in\mathcal{C}$,
$X^{\prime}\in\mathcal{D}$ and with morphisms defined by:

$\mathrm{Hom}_{\mathcal{C}\otimes\mathcal{D}}((X,X^{\prime}),(Y,Y^{\prime})):=\mathrm{Hom}_{\mathcal{C}}(X,Y)\otimes\mathrm{Hom}_{\mathcal{D}}(X^{\prime},Y^{\prime})$

for all $X,Y\in\mathcal{C}$ and $X^{\prime},Y^{\prime}\in\mathcal{D}$. The
composition of morphisms is defined similarly to (1).

##  3\. Categories of Divided Powers

After recalling some basic properties of divided powers, we define the
categories $\Gamma^{d}\mathcal{C}$ associated to any
$\boldsymbol{\mathcal{V}}_{\Bbbk}^{\mathsf{ev}}$-category $\mathcal{C}$.

###  3.1. Divided powers

Suppose $M\in\boldsymbol{\mathcal{V}}_{\Bbbk}$ and $d\in\mathbb{Z}_{\geq 1}$.
There is a unique (even) right action of the symmetric group
$\mathfrak{S}_{d}$ on the tensor power $M^{\otimes d}$ such that each
transposition $(i\ i+1)$ for $1\leq i\leq d-1$ acts by the supertwist:

$(v_{1}\otimes\cdots\otimes v_{d}).(i\ i+1)\ :=\
(-1)^{\overline{v}_{i}\overline{v}_{i+1}}v_{1}\otimes\cdots\otimes
v_{i+1}\otimes v_{i}\otimes\cdots\otimes v_{d},$

for any $v_{1},\dots,v_{d}\in M$, with $v_{i},v_{i+1}$ being
$\mathbb{Z}/2$-homogeneous.

We define the $d$-th divided power of $M$ to be the invariant subsupermodule

$\Gamma^{d}M:=(M^{\otimes d})^{\mathfrak{S}_{d}}$

and set $\Gamma^{0}M:=\Bbbk$.

######  Remark 3.1.

We have used an equivalent definition to the one given in [1, 2]. See [13,
Example 2.1.1] for an analogous statement regarding divided powers of ordinary
$\Bbbk$-modules.

For any $M,M^{\prime},N,N^{\prime}\in\boldsymbol{\mathcal{V}}_{\Bbbk}$, there
is an even isomorphism

(6)
$\mathrm{Hom}(M,M^{\prime})\otimes\mathrm{Hom}(N,N^{\prime})\simeq\mathrm{Hom}(M\otimes
N,M^{\prime}\otimes N^{\prime})$

sending $\phi\otimes\psi\mapsto\phi\boxtimes\psi$, where

$(\phi\boxtimes\psi)(v\otimes
w):=(-1)^{\overline{\psi}\,\overline{v}}\phi(v)\otimes\psi(w).$

Notice that the isomorphism (6) is natural with respect to composition.

Next, recall that the functor
$\otimes^{d}:\boldsymbol{\mathcal{V}}_{\Bbbk}\to\boldsymbol{\mathcal{V}}_{\Bbbk}$
sending $M\mapsto M^{\otimes d}$, whose action on morphisms is defined by

$\otimes^{d}_{M,N}(\varphi):=\,\varphi\boxtimes\cdots\boxtimes\varphi:M^{\otimes
d}\to N^{\otimes d}$

for any $\varphi\in\mathrm{Hom}(M,N)$.

Given $d,e\in\mathbb{N}$, there is an embedding

(7) $\Gamma^{d}M\otimes\Gamma^{e}N\,\hookrightarrow\,\Gamma^{d+e}(M\oplus
N):\,x\otimes y\mapsto x\cdot y,$

where

$x\cdot y\,:=\sum_{\sigma\in\mathfrak{S}_{d+e}/\mathfrak{S}_{d,e}}(x\otimes
y).\sigma\,\in\,M^{\otimes(d+e)}$

for $x\in M^{\otimes d}$, $y\in M^{\otimes e}$, where $\mathfrak{S}_{d,e}$
denotes the image of
$\mathfrak{S}_{d}\times\mathfrak{S}_{e}\hookrightarrow\mathfrak{S}_{d+e}$.
Then as in [1, Eq.(26)], there is a corresponding decomposition

(8) $\Gamma^{d}(M\oplus N)\cong\bigoplus_{0\leq c\leq
d}\Gamma^{c}M\otimes\Gamma^{d-c}N$

where we identify $\Gamma^{c}M\otimes\Gamma^{d-c}N$ as a subsupermodule of
$\Gamma^{d}(M\oplus N)$ under the embedding (7).

It follows easily from (8) that the divided power $\Gamma^{d}M$ of a
projective $\Bbbk$-supermodule $M\in\boldsymbol{\mathcal{V}}_{\Bbbk}$ is again
finitely-generated and projective. This yields a functor
$\Gamma^{d}:\boldsymbol{\mathcal{V}}_{\Bbbk}\to\boldsymbol{\mathcal{V}}_{\Bbbk}$
which is a subfunctor of $\otimes^{d}$. In particular, the action of
$\Gamma^{d}$ on morphisms is defined by restriction

$\Gamma^{d}_{M,N}(\varphi):=(\varphi^{\otimes
d})|_{\Gamma^{d}M}:\Gamma^{d}M\to\Gamma^{d}N$

for any $\varphi\in\mathrm{Hom}(M,N)$. It is then clear that the (non-linear)
maps $\Gamma^{d}_{M,N}$ are even for all
$M,N\in\boldsymbol{\mathcal{V}}_{\Bbbk}$, so that $\Gamma^{d}$ restricts to a
functor
$\Gamma^{d}:\boldsymbol{\mathcal{V}}_{\Bbbk}^{\mathsf{ev}}\to\boldsymbol{\mathcal{V}}_{\Bbbk}^{\mathsf{ev}}$.

Suppose that $I=(d_{1},\dots,d_{s})$ is a tuple of positive integers, and let
$\mathfrak{S}_{I}$ denote the subgroup
$\mathfrak{S}_{d_{1}}\times\cdots\times\mathfrak{S}_{d_{s}}\subseteq\mathfrak{S}_{|I|}$,
where $|I|:=\sum d_{i}$. Given $M\in\boldsymbol{\mathcal{V}}_{\Bbbk}$ and
distinct nonzero elements $v_{1},\dots,v_{s}\in M_{\bar{0}}$, we define the
new element

$(v_{1},\dots,v_{s};I)_{0}:=\sum_{\sigma\in\mathfrak{S}_{|I|}/\mathfrak{S}_{I}}(v_{1}^{\otimes
d_{1}}\otimes\cdots\otimes v_{s}^{\otimes d_{s}}).\sigma,$

which belongs to $(M_{\bar{0}}^{\otimes|I|})^{\mathfrak{S}_{|I|}}$.
Similiarly, if $v_{1}^{\prime},\dots,v_{t}^{\prime}\in M_{\bar{1}}$, we define
the (possibly zero) element

$(v_{1}^{\prime},\dots,v_{t}^{\prime})_{1}:=\sum_{\sigma\in\mathfrak{S}_{t}}(v_{1}^{\prime}\otimes\cdots\otimes
v_{t}^{\prime}).\sigma,$

which belongs to $(M_{\bar{1}}^{\otimes t})^{\mathfrak{S}_{t}}$.

######  Lemma 3.2.

Suppose that $M\in\boldsymbol{\mathcal{V}}_{\Bbbk}$ is a free
$\Bbbk$-supermodule, and that $\\{v_{1},\dots,v_{\mu}\\}$ $($resp.
$\\{v_{\mu+1},\dots,v_{\mu+\nu}\\})$ is an ordered basis of $M_{\bar{0}}$
$($resp. $M_{\bar{1}})$. Then a basis for $\Gamma^{d}M$ is given by the set of
all elements of the form

$(v_{i_{1}},\dots,v_{i_{s}};I)_{0}\cdot(v_{j_{1}},\dots,v_{j_{t}})_{1},$

such that $i_{1},\dots,i_{s}$ $($resp. $j_{1},\dots,j_{t})$ are distinct
elements of the set $\\{1,\dots,\mu\\}$ $($resp. $\\{\mu+1,\dots,\mu+\nu\\})$
and $|I|+t=d$.

###### Proof.

For any $\Bbbk$-module $V\in\mathcal{V}_{\Bbbk}$, let $D^{d}V$ and
$\Lambda^{d}V$ denote the ordinary $d$-th divided and exterior powers of $V$,
respectively. Then it follows, as in [1, (8)], that we have the following
isomorphisms of $\Bbbk$-modules

(9) $\Gamma^{d}M\ \simeq\
\bigoplus_{k+l=d}D^{k}|M_{\bar{0}}|\otimes\Lambda^{l}|M_{\bar{1}}|\
\simeq\bigoplus_{k+l=d}\Gamma^{k}M_{\bar{0}}\otimes\Gamma^{l}M_{\bar{1}}$

for each $d\geq 0$.

One may check by comparison with Proposition 4 of [4, Ch.IV, §5] that the set

$\\{(v_{i_{1}},\dots,v_{i_{s}};I)_{0};\ |I|=k\mbox{ and }1\leq
i_{1}<\cdots<i_{s}\leq\mu\\}$

is a basis of $\Gamma^{k}M_{\bar{0}}$. It is also not difficult to verify that

$\\{(v_{j_{1}},\dots,v_{j_{l}})_{1};\ 1\leq j_{1}<\cdots<j_{l}\leq\nu\\}$

is a basis of $\Gamma^{l}M_{\bar{1}}$. The lemma then follows from (9). ∎

###  3.2. The category $\Gamma^{d}\mathcal{C}$

Suppose $M,N\in\boldsymbol{\mathcal{V}}_{\Bbbk}$. Then define
$\psi^{d}=\psi^{d}(M,N)$ to be the unique map which makes the following square
commute:

(10) ${\Gamma^{d}M\otimes\Gamma^{d}N}$${\Gamma^{d}(M\otimes N)}$${M^{\otimes
d}\otimes N^{\otimes d}}$${(M\otimes N)^{\otimes
d}}$$\scriptstyle{\psi^{d}}$$\scriptstyle{\sim}$

We are now ready to define divided powers of any given
$\boldsymbol{\mathcal{V}}_{\Bbbk}^{\mathsf{ev}}$-enriched category.

######  Definition 3.3.

Suppose that $\mathcal{C}$ is a
$\boldsymbol{\mathcal{V}}_{\Bbbk}^{\mathsf{ev}}$-enriched category. Then we
define a new $\boldsymbol{\mathcal{V}}_{\Bbbk}^{\mathsf{ev}}$-category,
$\Gamma^{d}\mathcal{C}$, which has the same objects as $\mathcal{C}$, with
morphisms given by

$\mathrm{Hom}_{\Gamma^{d}\mathcal{C}}(X,Y):=\Gamma^{d}\mathrm{Hom}_{\mathcal{C}}(X,Y)$

for all $X,Y\in\mathcal{C}$. The composition of morphisms

$\circ^{\Gamma^{d}\mathcal{C}}_{X,Y,Z}:\
\mathrm{Hom}_{\Gamma^{d}\mathcal{C}}(Y,Z)\otimes\mathrm{Hom}_{\Gamma^{d}(\mathcal{C})}(X,Y)\,\rightarrow\,\mathrm{Hom}_{\Gamma^{d}\mathcal{C}}(X,Z)$

is defined by the following composite

$\Gamma^{d}\mathrm{Hom}_{\mathcal{C}}(Y,Z)\otimes\Gamma^{d}\mathrm{Hom}_{\mathcal{C}}(X,Y)\
\xrightarrow{\,\psi^{d}\,}\
\Gamma^{d}\big{(}\mathrm{Hom}_{\mathcal{C}}(Y,Z)\otimes\mathrm{Hom}_{\mathcal{C}}(X,Y)\big{)}$
$\hskip 71.13188pt\xrightarrow{\,\Gamma^{d}(\circ^{\mathcal{C}}_{X,Y,Z})\,}\
\Gamma^{d}\mathrm{Hom}_{\mathcal{V}}(X,Z)$

for all $X,Y,Z\in\mathcal{C}$.

###  3.3. The algebra $\Gamma^{d}A$

Suppose that $A\in\boldsymbol{\mathcal{V}}_{\Bbbk}$ is a superalgebra. Then we
may identify $A$ as a
$\boldsymbol{\mathcal{V}}_{\Bbbk}^{\mathsf{ev}}$-enriched category with one
object, $o$, and hom-set $\mathrm{Hom}_{A}(o,o):=A$. Hence, the divided power
$\Gamma^{d}A$ of a superalgebra $A$ is also a superalgebra. The multiplication
$m_{\Gamma^{d}A}$ is induced by the composition

$\Gamma^{d}A\otimes\Gamma^{d}A\xrightarrow{\psi^{d}}\Gamma^{d}(A\otimes
A)\xrightarrow{\Gamma^{d}(m_{A})}\Gamma^{d}A.$

In particular, $\Gamma^{d}A$ is a subsuperalgebra of $A^{\otimes d}$.

###  3.4. Generalized Schur algebras

Fix $m,n,d\in\mathbb{N}$, and let $A\in\boldsymbol{\mathcal{V}}_{\Bbbk}$ be a
superalgebra. Then we may identify $\mathit{M}_{n}(A)$ with the superalgebra
of $n\times n$ matrices over $A$. We may also identify $\mathit{M}_{n}(A)$
with the tensor product $A\otimes\mathit{M}_{n}(\Bbbk)$ of superalgebras. The
generalized Schur algebra $S^{A}(n,d)$ defined in [6] may then be identified
as the superalgebra

$S^{A}(n,d)=\Gamma^{d}\mathit{M}_{n}(A).$

The classical Schur algebra, $S(n,d)$, is obtained as the special case:
$S^{\Bbbk}(n,d)=\Gamma^{d}\mathit{M}_{n}(\Bbbk)$. As mentioned in [13, Example
2.1.1], this definition is equivalent to the original definition given by
Green in [9].

We next introduce another type of generalized Schur superalgebra which will be
needed later. Given nonnegative integers $m,n$, let

$S^{A}(m|n,d):=\Gamma^{d}\mathit{M}_{m|n}(A).$

Notice that $S^{\Bbbk}(m|n,d)$ is isomorphic to the Schur superalgebra
$S(m|n,d)$ introduced by Muir in [12].

##  4\. Generalized Polynomial Functors

We now define a generalization of both the ordinary categories of strict
polynomial functors, defined by Friedlander and Suslin in [8], and the super-
analogues of these categories defined in [1].

###  4.1. Representations of a category

Suppose that $\mathcal{C}$ is a
$\boldsymbol{\mathcal{V}}_{\Bbbk}^{\mathsf{ev}}$-enriched category. Then write

$\mathbf{rep}\,\mathcal{C}:=\mathrm{Fun}_{\Bbbk}(\mathcal{C},\boldsymbol{\mathcal{V}}_{\Bbbk}).$

to denote the category of all even $\Bbbk$-linear functors,
$F:\mathcal{C}\to\boldsymbol{\mathcal{V}}_{\Bbbk}$.

Next suppose that $X\in\mathcal{C}$, and consider the superalgebra
$E:=\mathrm{End}_{\mathcal{C}}(X)$. The categories $\mathbf{rep}\,\mathcal{C}$
and $E\kern 1.0pt\mathchar 45\relax\mathbf{mod}$ are both
$\boldsymbol{\mathcal{V}}_{\Bbbk}^{\mathsf{ev}}$-enriched categories. Recall
that $(E\kern 1.0pt\mathchar 45\relax\mathbf{mod})^{\mathsf{ev}}$ and
$\boldsymbol{\mathcal{V}}_{\Bbbk}^{\mathsf{ev}}$ are both exact categories.
Now since direct sums, products, kernels and cokernels can be computed
objectwise in the target category $\boldsymbol{\mathcal{V}}_{\Bbbk}$, we see
that $(\mathbf{rep}\,\mathcal{C})^{\mathsf{ev}}$ is also an exact category.

The relationship between $\mathbf{rep}\,\mathcal{C}$ and $E\kern
1.0pt\mathchar 45\relax\mathbf{mod}$ is given by evaluation on $X$. If
$F\in\mathbf{rep}\,\mathcal{C}$, the (even) functoriality of $F$ makes the
$\Bbbk$-supermodule $F(X)$ into a left $E$-supermodule. We thus have the
evaluation functor:

$\mathbf{rep}\,\mathcal{C}\to\ E\kern 1.0pt\mathchar
45\relax\mathbf{mod}:\,F\mapsto F(X).$

This evaluation functor also has the following interpretation. Since the
covariant hom-functor $h^{X}:=\mathrm{Hom}_{\mathcal{C}}(X,-)$ is an even
$\Bbbk$-linear functor, it must belong to $\mathbf{rep}\,\mathcal{C}$. In this
situation, Yoneda’s lemma takes the form of an even isomorphism

(11) $\mathrm{Hom}_{\mathbf{rep}\,\mathcal{C}}(h^{X},F)\ \simeq\ F(X)$

for any $F\in\mathbf{rep}\,\mathcal{C}$. In particular,

$E\ =\ h^{X}(X)\ \simeq\ \mathrm{End}_{\mathbf{rep}\,\mathcal{C}}(h^{X}).$

Hence, Yoneda’s lemma allows us to interpret “evaluation at $X$” as the
functor
$\mathrm{Hom}_{\mathbf{rep}\,\mathcal{C}}(h^{X},-):\mathbf{rep}\,\mathcal{C}\to{}E\kern
1.0pt\mathchar 45\relax\mathbf{mod}$.

Notice that the parity change functor,
$\Pi:\boldsymbol{\mathcal{V}}_{\Bbbk}\rightarrow\boldsymbol{\mathcal{V}}_{\Bbbk}$,
induces by composition a functor
$\Pi\circ-:\mathbf{rep}\,\mathcal{C}\to\mathbf{rep}\,\mathcal{C}$.

We now describe a condition on $X$ which ensures that evaluation is in fact an
equivalence of categories. The following lemma is a generalization of [14,
Proposition A.1]. It was stated in [1, Proposition A.1] assuming that $\Bbbk$
is a field. However, the generalization to the case of a commutative ring
$\Bbbk$ is easily obtained.

######  Lemma 4.1 ([14, 1]).

Let $\mathcal{C}$ be a
$\boldsymbol{\mathcal{V}}_{\Bbbk}^{\mathsf{ev}}$-enriched category. Assume
that there exists an object $P\in\mathcal{C}$ such that for all
$X,Y\in\mathcal{C}$, the composition induces a surjective map

$\mathrm{Hom}_{\mathcal{C}}(P,Y)\otimes\mathrm{Hom}_{\mathcal{C}}(X,P)\twoheadrightarrow\mathrm{Hom}_{\mathcal{C}}(X,Y).$

Then the following hold.

  1. (i)

For all $F\in\mathbf{rep}\,\mathcal{C}$ and all $Y\in\mathcal{C}$, the
canonical map $\mathrm{Hom}_{\mathcal{C}}(P,Y)\otimes F(P)\to F(Y)$ is
surjective.

  2. (ii)

The set $\\{h^{P},\Pi h^{P}\\}$ is a projective generator of
$(\mathbf{rep}\,\mathcal{C})^{\mathsf{ev}}$, where
$h^{P}=\mathrm{Hom}_{\mathcal{C}}(P,-)$ as above.

  3. (iii)

If $E=\mathrm{End}_{\mathcal{C}}(P)$, then evaluation on $P$ induces an
equivalence of categories, $\mathbf{rep}\,\mathcal{C}\simeq{}E\kern
1.0pt\mathchar 45\relax\mathbf{mod}$.

We note that for a given a $\mathcal{V}_{\Bbbk}$-category $\mathcal{C}$, we
may also consider the category

$\mathrm{rep}\,\mathcal{C}:=\mathrm{Fun}_{\Bbbk}(\mathcal{C},\mathcal{V}_{\Bbbk})$

consisting of ordinary $\Bbbk$-linear functors and natural transformations.

###  4.2. Generalized polynomial functors

Given a $\boldsymbol{\mathcal{V}}_{\Bbbk}^{\mathsf{ev}}$-category
$\mathcal{C}$ and $d\in\mathbb{N}_{0}$, we associate the corresponding
category

$\boldsymbol{\mathcal{P}}_{d,\mathcal{C}}:=\mathbf{rep}\,\Gamma^{d}\mathcal{C}$

which we call the category of (homogeneous, degree $d$) generalized polynomial
functors associated to the category $\mathcal{C}$.

Suppose now that $\mathcal{C}$ is a $\mathcal{V}_{\Bbbk}$-enriched category.
Then we may also consider the category

$\mathcal{P}_{d,\mathcal{C}}:=\mathrm{rep}\,\Gamma^{d}\mathcal{C}$

whose objects we again call generalized polynomial functors.

######  Example 4.2.

Suppose that $\Bbbk$ is a field. Then we have the following special cases:

  * (a)

$\mathcal{P}_{d,\mathcal{V}_{\Bbbk}}$ is equivalent to the category
$\mathcal{P}_{d,\Bbbk}$ of homogenous strict polynomial functors, defined by
Friedlander and Suslin in [8].

  * (b)

$\boldsymbol{\mathcal{P}}_{d,\boldsymbol{\mathcal{V}}_{\Bbbk}}$ is equivalent
to the category $\mathsf{Pol}^{\mathrm{I}}_{d}$ defined in [1, Section 3].

  * (c)

Recall from Remark 2.2 that $C(1)$ denotes the rank 1 Clifford algebra. Then
$\boldsymbol{\mathcal{P}}_{d,\boldsymbol{\mathcal{E}}_{C(1)}}=\boldsymbol{\mathcal{P}}_{d,\boldsymbol{\mathcal{V}}_{C(1)}}$
is equivalent to the category $\mathsf{Pol}^{\mathrm{II}}_{d}$ of spin
polynomial functors defined in [1, Section 3].

We are now ready to state our main result.

######  Theorem 4.3.

Suppose $A\in\boldsymbol{\mathcal{V}}_{\Bbbk}$ is a superalgebra which is free
as a $\Bbbk$-module. Then for $m,n,d\in\mathbb{N}$, with $m,n\geq d$, we have
the following equivalences of categories:

  1. (i)

$\displaystyle\boldsymbol{\mathcal{P}}_{d,\boldsymbol{\mathcal{V}}_{A}}\
\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\ S^{A}(m|n,d)\kern
1.0pt\mathchar 45\relax\mathbf{mod};$

  2. (ii)

$\displaystyle\boldsymbol{\mathcal{P}}_{d,\boldsymbol{\mathcal{E}}_{A}}\
\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\ S^{A}(n,d)\kern
1.0pt\mathchar 45\relax\mathbf{mod}$,

given by evaluation on $A^{m|n}$ and $A^{n}$, respectively.

###### Proof.

We prove part (i). Since
$S^{A}(m|n,d)=\mathrm{End}_{\Gamma^{d}\boldsymbol{\mathcal{V}}_{A}}(A^{m|n})$,
it suffices by Lemma 4.1 to show that the composition map

(12)
$\mathrm{Hom}_{\Gamma^{d}\boldsymbol{\mathcal{V}}_{A}}(A^{m|n},Y)\otimes\mathrm{Hom}_{\Gamma^{d}\boldsymbol{\mathcal{V}}_{A}}(X,A^{m|n})\rightarrow\mathrm{Hom}_{\Gamma^{d}\boldsymbol{\mathcal{V}}_{A}}(X,Y)$

is surjective, for all $X,Y\in\boldsymbol{\mathcal{V}}_{A}$.

By naturality, one may reduce to the case where $X$ is a free (right)
$A$-supermodule. Since $Y$ is projective and thus a direct summand of a free
$A$-supermodule, we may also assume that $Y$ is free. We may thus assume that
$X=A^{p|q}$, $Y=A^{s|t}$, for some $p,q,s,t\in\mathbb{N}_{0}$.

Let us identify $\mathrm{Hom}(\Bbbk^{p|q},\Bbbk^{m|n})$ as a subset of
$\mathrm{Hom}(X,A^{m|n})$ via the embedding of Lemma 2.1.(i). It will then
suffice to show the following map

(13)
$\Gamma^{d}\mathrm{Hom}_{A}(A^{m|n},Y)\otimes\Gamma^{d}\mathrm{Hom}_{\Bbbk}(\Bbbk^{p|q},\Bbbk^{m|n})\rightarrow\Gamma^{d}\mathrm{Hom}_{A}(X,Y),$

obtained by restricting composition, is surjective.

Suppose that $a_{1},\dots,a_{\mu}$ (resp. $a_{\mu+1},\dots,a_{\mu+\nu}$) is a
$\Bbbk$-basis of $A_{\bar{0}}$ (resp. $A_{\bar{1}}$). Then, using the notation
in the proof of Lemma 2.1, it follows that there exist (homogeneous) bases
$(e(j,i))$, $(e^{(r)}(k,j))$ and $(e^{(r)}(k,i))$ of
$\mathrm{Hom}(\Bbbk^{p|q},\Bbbk^{m|n})$, $\mathrm{Hom}_{A}(A^{m|n},Y)$ and
$\mathrm{Hom}_{A}(X,Y)$, respectively, such that:

$e^{(r)}(k,j_{1})\circ e(j_{2},i)\,=\ e^{(r)}(k,i)\cdot\delta_{j_{1},j_{2}},$

where $\delta_{j_{1},j_{2}}$ is the Kronecker delta.

Suppose that
$r_{1},\dots,r_{a},r_{1}^{\prime},\dots,r_{b}^{\prime}\in\\{1,\dots,\mu+\nu\\}$,
for some $1\leq a,b\leq d$, and assume the basis elements

$e^{(r_{1})}(k_{1},i_{1}),\dots,e^{(r_{a})}(k_{a},i_{a})\quad(\text{resp.~{}}e^{(r_{1}^{\prime})}(k_{1}^{\prime},i_{1}^{\prime}),\dots,e^{(r_{b}^{\prime})}(k_{b}^{\prime},i_{b}^{\prime}))$

are even (odd). Then to prove surjectivity, it suffices to show that each
elemen of the form

(14)
$(e^{(r_{1})}(k_{1},i_{1}),\dots,e^{(r_{a})}(k_{a},i_{a});I)_{0}\cdot(e^{(r_{1}^{\prime})}(k_{1}^{\prime},i_{1}^{\prime}),\dots,e^{(r_{b}^{\prime})}(k_{b}^{\prime},i_{b}^{\prime}))_{1}$

belongs to the image of (13), since $\Gamma^{d}\mathrm{Hom}_{A}(X,Y)$ is
spanned by such elements according to Lemma 3.2.

Now suppose $I=(d_{1},\dots,d_{a})\in(\mathbb{Z}_{>0})^{a}$. Then, since
$m,n\geq d=|I|+b\geq a+b$, we may choose distinct indices
$j_{1},\dots,j_{a},j_{1}^{\prime},\dots,j_{b}^{\prime}$ such that
$e(j_{1},i_{1}),\dots,e(j_{a},i_{a}),\
e(j_{1}^{\prime},i_{1}^{\prime}),\dots,e(j_{b}^{\prime},i_{b}^{\prime})$ are
all even elements. We are thus able to form the element

$\displaystyle(e^{(r_{1})}(k_{1},j_{1}),\dots,e^{(r_{a})}(k_{a},j_{a});I)_{0}\cdot(e^{(r_{1}^{\prime})}(k_{1}^{\prime},j_{1}^{\prime}),\dots,e^{(r_{b}^{\prime})}(k_{b}^{\prime},j_{b}^{\prime}))_{1}$
$\displaystyle\quad\otimes(e(j_{1},i_{1}),\dots,e(j_{s},i_{s}),\
e(j_{1}^{\prime},i_{1}^{\prime}),\dots,e(j_{t}^{\prime},i_{t}^{\prime});J)_{0},$

where $J=(d_{1},\dots,d_{a},1,\dots,1)\in(\mathbb{Z}_{>0})^{a+b}$, which is
sent to the element (14) under the map induced by composition in
$\Gamma^{d}\boldsymbol{\mathcal{V}}_{A}$.

The proof of equivalence (ii) is similar. ∎

We also have the following non-graded analogue of Theorem 4.3.

######  Theorem 4.4.

Let $A\in\mathcal{V}_{\Bbbk}$ be an ordinary $\Bbbk$-algebra which is free as
a $\Bbbk$-module, and suppose $n,d\in\mathbb{N}$. Then there is an equivalence
of categories:

$\mathcal{P}_{d,\mathcal{V}_{A}}\
\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\ S^{A}(n,d)\kern
1.0pt\mathchar 45\relax\mathrm{mod},$

provided $n\geq d$.

###### Proof.

The proof is analagous to Theorem 4.3, using [14, Prop. A.1] in place of Lemma
4.1. ∎

It follows by Section 3.4 and Example 4.2 that Theorem 4.3 generalizes Theorem
4.2 of [1], while Theorem 4.4 generalizes Theorem 3.2 of [8].

##  5\. Generalized Schur-Weyl Duality

Throughout this section, we fix a superalgebra
$A\in\boldsymbol{\mathcal{V}}_{\Bbbk}$.

###  5.1. Wreath products

Consider the group algebra of the symmetric group, $\Bbbk\mathfrak{S}_{d}$, as
a superalgebra concentrated in degree zero. Then the wreath product
superalgebra, $A\wr\mathfrak{S}_{d}$, is the $\Bbbk$-supermodule $A^{\otimes
d}\otimes\Bbbk\mathfrak{S}_{d}$, with multiplication defined by

(15) $(x\otimes\rho)\cdot(y\otimes\sigma):=x(y\rho^{-1})\otimes\rho\sigma$

for all $x,y\in A^{\otimes d}$ and $\rho,\sigma\in\mathfrak{S}_{d}$. If $G$ is
a finite group, then note for example that $(\Bbbk G)\wr\mathfrak{S}_{d}$ is
isomorphic to the group algebra of the classical wreath product,
$G\wr\mathfrak{S}_{d}:=G^{d}\rtimes\mathfrak{S}_{d}$.

Assume in the remainder that $A$ is free as a $\Bbbk$-module. We then identify
the tensor power $A^{\otimes d}$ and group algebra $\Bbbk\mathfrak{S}_{d}$ as
subalgebras of $A\wr\mathfrak{S}_{d}$ by setting

$A^{\otimes d}=A^{\otimes d}\otimes
1_{\mathfrak{S}_{d}},\quad\Bbbk\mathfrak{S}_{d}=1_{A^{\otimes
d}}\otimes\Bbbk\mathfrak{S}_{d}$

respectively.

###  5.2. Tensor products

Given nonnegative integers $d$ and $e$, we have an embedding
$\mathfrak{S}_{d}\times\mathfrak{S}_{e}\hookrightarrow\mathfrak{S}_{d+e}$.
This induces an embedding

(16) $\Gamma^{d+e}M\hookrightarrow\Gamma^{d}M\otimes\Gamma^{e}M$

for any $M\in\boldsymbol{\mathcal{V}}_{\Bbbk}$, given by the composition of
the following maps:

$\displaystyle\Gamma^{d+e}M=(M^{\otimes d+e})^{\mathfrak{S}_{d+e}}\ $
$\displaystyle\hookrightarrow\,(M^{\otimes
d+e})^{\mathfrak{S}_{d}\times\mathfrak{S}_{e}}$ $\displaystyle\,\simeq\
\,(M^{\otimes d})^{\mathfrak{S}_{d}}\otimes(M^{\otimes
e})^{\mathfrak{S}_{e}}=\Gamma^{d}M\otimes\Gamma^{e}M.$

We then have a corresponding diagonal embedding

(17)
$\delta:\Gamma^{d+e}\,\mathcal{C}\hookrightarrow\Gamma^{d}\mathcal{C}\otimes\Gamma^{e}\mathcal{C},\qquad$

which sends $X\mapsto(X,X)$ and which acts on morphisms via the embedding
induced by (16).

Given $F\in\boldsymbol{\mathcal{P}}_{d,\mathcal{C}}$,
$G\in\boldsymbol{\mathcal{P}}_{e,\mathcal{C}}$, let $F\boxtimes
G\in\mathbf{rep}(\Gamma^{d}\mathcal{C}\otimes\Gamma^{e}\mathcal{C})$ denote
the functor which sends

$(X_{i},Y_{i})\mapsto F(X_{i})\otimes G(Y_{i})\ \mbox{ and }\
\varphi_{1}\otimes\varphi_{2}\mapsto F(\varphi_{1})\boxtimes G(\varphi_{2}),$

for all $X_{i},Y_{i}\in\mathcal{C}$ and
$\varphi_{i}\in\mathrm{Hom}_{\mathcal{C}}(X_{i},Y_{i})$, for $i=1,2$,
respectively. We then define the tensor product

(18)
$-\otimes-:\boldsymbol{\mathcal{P}}_{d,\mathcal{C}}\times\boldsymbol{\mathcal{P}}_{e,\mathcal{C}}\rightarrow\boldsymbol{\mathcal{P}}_{d+e,\,\mathcal{C}}$

as the composition of functors

$\boldsymbol{\mathcal{P}}_{d,\mathcal{C}}\times\boldsymbol{\mathcal{P}}_{e,\mathcal{C}}\xrightarrow{\
-\boxtimes-\
}\mathbf{rep}(\Gamma^{d}\mathcal{C}\otimes\Gamma^{e}\mathcal{C})\xrightarrow{\
\delta_{*}\ }\boldsymbol{\mathcal{P}}_{d+e,\mathcal{C}}.$

where $\delta_{*}$ is the functor induced by (17).

###  5.3. Generalized Schur-Weyl duality

Let us write
$\boldsymbol{\mathcal{P}}^{A}_{d}:=\boldsymbol{\mathcal{P}}_{d,\boldsymbol{\mathcal{E}}_{A}}$.
Then the (external) tensor product defined above induces a bifunctor

$-\otimes-:\boldsymbol{\mathcal{P}}^{A}_{d}\times\boldsymbol{\mathcal{P}}^{A}_{e}\rightarrow\boldsymbol{\mathcal{P}}^{A}_{d+e}.$

This gives $\boldsymbol{\mathcal{P}}^{A}:=\bigoplus_{d\geq
0}\boldsymbol{\mathcal{P}}^{A}_{d}$ the structure of a monoidal category.

Given any object $X\in\boldsymbol{\mathcal{E}}_{A}$, we associate the
corresponding object

$\Gamma^{d,X}:=\mathrm{Hom}_{\Gamma^{d}\boldsymbol{\mathcal{E}}_{A}}(X,-)$

in the category $\boldsymbol{\mathcal{P}}^{A}_{d}$. It follows by Yoneda’s
lemma (11) that $\Gamma^{d,X}$ is a projective object.

Let us write $\Gamma_{A}^{d,n}:=\Gamma^{d,A^{n}}$. Then it follows from (7)
that we have a decomposition

(19)
$\Gamma_{A}^{d,m+n}\simeq\bigoplus_{i+j=d}\Gamma_{A}^{i,m}\otimes\Gamma_{A}^{j,n}$

of strict polynomial functors.

Now let $\Lambda(n,d)$ denote the set of all tuples
$\lambda=(\lambda_{1},\dots,\lambda_{n})\in(\mathbb{Z}_{\geq 0})^{n}$ such
that $\sum\lambda_{i}=d$. Given $\lambda\in\Lambda(n,d)$, we will write
$\Gamma^{\lambda}_{A}:=\Gamma^{\lambda_{1},1}_{A}\otimes\cdots\otimes\Gamma^{\lambda_{n},1}_{A}$.
By (19) and induction, we have a canonical isomorphism

(20)
$\Gamma_{A}^{d,n}\simeq\bigoplus_{\lambda\in\Lambda(n,d)}\Gamma_{A}^{\lambda}.$

It follows that the objects $\Gamma^{\lambda}$ are projective in
$\boldsymbol{\mathcal{P}}^{A}_{d}$.

Let $\omega=(1,\dots,1)\in\Lambda(d,d)$. Notice that $\Gamma^{\omega}(X)\simeq
X^{\otimes d}$ for any $X\in\boldsymbol{\mathcal{E}}_{A}$, since
$\mathrm{Hom}_{A}(A,X)\simeq X$. It follows that
$\otimes^{d}:=\Gamma^{\omega}$ is a projective object of
$\boldsymbol{\mathcal{P}}^{A}_{d}$. We then have the following analogue of the
generalized Schur Weyl duality described in [6].

######  Theorem 5.1.

Assume $n\geq d$, and write $V=A^{n}$.

  * (i)

The left $S^{A}(n,d)$-supermodule $V^{\otimes
d}\simeq\mathrm{Hom}_{\boldsymbol{\mathcal{P}}^{A}_{d}}(\Gamma^{d,n},\otimes^{d})$
is a projective object of $S^{A}(n,d)\kern 1.0pt\mathchar
45\relax\mathbf{mod}$.

  * (ii)

There is a canonical isomorphism of superalgebras:

$\mathrm{End}_{\boldsymbol{\mathcal{P}}^{A}_{d}}(\otimes^{d})\cong
A\wr\mathfrak{S}_{d}.$

  * (iii)

We have an exact functor

$\mathrm{Hom}_{\boldsymbol{\mathcal{P}}^{A}_{d}}(\otimes^{d},-):\boldsymbol{\mathcal{P}}^{A}_{d}\rightarrow
A\wr\mathfrak{S}_{d}\,\kern 1.0pt\mathchar 45\relax\mathbf{mod}.$

###### Proof.

Part (i) follows from Theorem 4.3.(i) and the fact that $\bigotimes^{d}$ is a
projective object of $\boldsymbol{\mathcal{P}}^{A}_{d}$.

Recall from [6, §5] that there is an even idempotent $\xi_{\omega}\in
S^{A}(n,d)$ such that $V^{\otimes d}\simeq S^{A}(n,d)\xi_{\omega}$, as left
supermodules. Since $\xi_{\omega}$ is an idempotent, it follows that the
superalgebras $\xi_{\omega}S^{A}(n,d)\xi_{\omega}$ and
$\mathrm{End}_{S^{A}(n,d)}(S^{A}(n,d)\xi_{\omega})$ are isomorphic. Part (ii)
thus follows from Theorem 4.3.(i), together with [6, Lemma 5.15] and [6,
Proposition 5.17].

Finally, (iii) is a direct consequence of (i) and (ii). ∎

######  Remark 5.2.

One may refer to the functor in Theorem 5.1.(iii) as the generalized Schur-
Weyl duality functor. A similar functor related to classical Schur-Weyl
duality was studied in [15] in the context of $\mathfrak{g}$-categorification.

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