Supersymmetric Derived Stacks

TL;DR

This paper develops a theory of supersymmetric derived stacks on $ ext{Z}_2$-bi-graded modules, extending existing stack frameworks to incorporate supersymmetry and supergeometry relevant to String Theory.

Contribution

It introduces the concept of supersymmetric derived stacks on $ ext{Z}_2$-bi-graded modules, defining maps and prestacks that respect supersymmetry transformations, expanding the mathematical toolkit for supergeometry.

Findings

Defined derived stacks on $ ext{Z}_2$-bi-graded modules.

Established behavior of supersymmetry transformations on these stacks.

Proposed a criterion for supersymmetric stacks as derived stacks.

Abstract

Stacks have become a prevalent tool in studying problems with connections to String Theory, hence we see a need to develop a theory of supersymmetric stacks proper. We first define derived stacks on $\mathbb{Z}_2$-bi-graded k-modules (objects of sk-sMod$_*$) following the exposition of Toen and Vezzosi on ungraded modules in HAG I & II. We then define $\text{Top}_{* \centerdot}$-valued maps on those supermodules ($\text{Top}_{* \centerdot}$ $\mathbb{Z}_2$-bi-graded), and show how they behave under supersymmetry transformations in the base. For $\Psi: M \rightarrow X$ one such map, $M \in $ sk-sMod$_*$, $X \in \text{Top}_{* \centerdot}$, we argue that defining a prestack $F$ of simplicial sets over simplicial graded k-superalgebras object-wise by $F(M) = \{\Psi(\sigma, \theta) | \sigma, \theta \in M \}$ with the induced topology, one can call $F$ a supersymmetric stack if it is a derived stack.

Equations

$$|x|=\left\{\begin{array}[]{ll}0,&\hbox{$x\in M_{0}$}\\ 1,&\hbox{$x\in M_{1}$}\end{array}\right.$$

$${\sigma }_{M,N}(x\otimes y)=(-1{)}^{|x||y|}y\otimes x$$

$$\lambda :$$

$$\rho :$$

$${\alpha }_{M,N,P}^{-1}:M\otimes (N\otimes P)\stackrel{\cong }{\to }(M\otimes N)\otimes P$$

$$(M\otimes N)\otimes (P\otimes Q)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(M\otimes k)\otimes N\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$M\otimes (N\otimes P)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(x\otimes y)\otimes z$$

$$=[x_0{y}_0+x_0{y}_1+x_1{y}_0+x_1{y}_1]\otimes z$$

$$[y_0{x}_0+y_1{x}_0+y_0{x}_1-y_1{x}_1]\otimes z$$

$$y_0(x_0\otimes z)+y_1(x_0\otimes z)+y_0(x_1\otimes z)-y_1(x_1\otimes z)$$

$$y_0(x_0{z}_0+x_0{z}_1)+y_1(x_0{z}_0+x_0{z}_1)+y_0(x_1{z}_0+x_1{z}_1)-y_1(x_1{z}_0+x_1{z}_1)$$

$$y_0(z_0{x}_0+z_1{x}_0)+y_1(z_0{x}_0+z_1{x}_0)+y_0(z_0{x}_1-z_1{x}_1)-y_1(z_0{x}_1-z_1{x}_1)$$

$$(x_0+x_1)\otimes [y_0{z}_0+y_0{z}_1+y_1{z}_0+y_1{z}_1]$$

$$x_0\otimes y_0{z}_0+x_0\otimes y_0{z}_1+x_0\otimes y_1{z}_0+x_0\otimes y_1{z}_1+x_1\otimes y_0{z}_0+x_1\otimes y_0{z}_1+x_1\otimes y_1{z}_0+x_1\otimes y_1{z}_1$$

$$|x\otimes y|=|x|+|y|$$

$$y_0{z}_0\otimes x_0+y_0{z}_1\otimes x_0+y_1{z}_0\otimes x_0+y_1{z}_1\otimes x_0+y_0{z}_0\otimes x_1-y_0{z}_1\otimes x_1-y_1{z}_0\otimes x_1+y_1{z}_1\otimes x_1$$

$$y_0(z_0{x}_0)+y_0(z_1{x}_0)+y_1(z_0{x}_0)+y_1(z_1{x}_0)+y_0(z_0{x}_1)-y_0(z_1{x}_1)-y_1(z_0{x}_1)+y_1(z_1{x}_1)$$

$$x\otimes (y\otimes z)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(M\otimes N{)}_0$$

$$(M\otimes N{)}_1$$

$$M\otimes N\ni x\otimes y$$

$$+x_0{y}_1+x_1{y}_0\stackrel{{\sigma }_1}{\to }+y_1{x}_0+y_0{x}_1\stackrel{{\sigma }_1}{\to }{x}_0{y}_1+x_1{y}_0\in (N\otimes M{)}_1$$

$${\rho }_M$$

$$={\oplus }_{i=0,1}{\lambda }_{M,i}\circ ({\sigma }_{M,k}{)}_i$$

$$={\lambda }_M\circ {\sigma }_{M,k}$$

$$M\otimes k\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$M⊠N=(M_0\otimes N_0)\oplus (M_1\otimes N_1)$$

$${\sigma }_{M,N}:M⊠N\to N⊠M$$

$${\sigma }_{M,N}:(M_0\otimes N_0)\oplus (M_1\otimes N_1)\stackrel{{\sigma }_0\oplus {\sigma }_1}{\to }(N_0\otimes M_0)\oplus (N_1\otimes M_1)=N⊠M$$

$${\sigma }_{i0}:$$

$${\sigma }_{i1}:$$

$$\lambda ={\lambda }_0\oplus {\lambda }_1:\Delta k⊠x=(k\otimes x_0)\oplus (k\otimes x_1)\stackrel{\cong }{\to }x$$

Full text

1 Introduction

Moduli spaces have for a long time been a pervasive object in String Theory. Moduli stacks in particular have been very useful, and with the advent of Derived Algebraic Geometry ([Lu], [TV], [TV4]), derived moduli stacks. Geometric Langlands is another avenue of research that has deep connections with Mathematical Physics ([AT], [EY]), and makes heavy use of the stacks formalism. Naturally then it seems appropriate to develop a theory of supersymmetric stacks in its own right since that notion ought to surface at some point.

One typically starts with a superspace, a graded vector space. To be more general, one can consider graded modules instead, over a commutative ring $k$, giving rise to a category of super-$k$-modules $k\text{-sMod}$. We work in Derived Algebraic Geometry, so we enlarge that to the category of simplicial graded $k$-supermodules $\text{s}k\text{-sMod}_{*}$. From that point onward we follow [TV] and [TV4] by showing $\text{s}k\text{-sMod}_{*}$ can be made into a symmetric monoidal model category, with a notion of supercommutativity using the parity function on homogeneous elements that leads to defining $\text{s}k\text{-sAlg}_{*}=\text{Comm}(\text{s}k\text{-sMod}_{*})$ the category of simplicial graded $k$-superalgebras, the opposite category of which is the category of affine graded $k$-superschemes $k\text{-D}^{-}\text{sAff}_{*}$, on which we put a bi-graded version of the étale topology of [TV4]. We consider bi-graded stacks on this site, functors valued in $S_{*\centerdot}=(\text{Set}_{\Delta})_{2\times 2}$, a bi-graded generalization of the notion of stacks on the ungraded site $(\text{k-D}^{-}\text{Aff},\acute{e}t.)$. The passage from the classical theory as expounded in [TV] and [TV4] to the bi-graded setting is made possible precisely because of our choice of working with bi-graded objects, along with our choice of tensor product on $\text{s}k\text{-sMod}_{*}$.

For each $M\in\text{s}k\text{-sMod}_{*}$, $M=(M_{ip})$, $i$ referred to as the level index, $p$ the parity index, $\sigma=(\sigma^{1},\sigma^{2})$, $\sigma^{i}\in M_{i0}$, $\theta=(\theta^{1},\theta^{2})$, $\theta^{A}\in M_{A1}$, we consider maps $\Psi:M\rightarrow\mathcal{X}$, $\mathcal{X}=(\mathcal{X}_{ip})\in\text{Top}_{*\centerdot}=(\text{Top})_{2\times 2}$, $\mathcal{X}$ determined by additional data such as constraint equations, $\Psi=X\oplus\psi$, $X:M_{0}\rightarrow\mathcal{X}_{0}$, $\psi:M_{1}\rightarrow\mathcal{X}_{1}$, so that $X$ is a function of $\sigma=(\sigma^{1},\sigma^{2})$, and $\psi$ is a function of $\theta=(\theta^{1},\theta^{2})$. Instead of using the traditional supersymetry transformations, say those in [GSW] for which $M_{0}=<\sigma^{1},\sigma^{2}>$, $M_{1}=<\theta^{1},\theta^{2}>$, $\delta\sigma^{\alpha}=\epsilon^{T}\rho^{\alpha}\theta$, with $\epsilon$ anticommuting, $\delta\theta^{A}=\epsilon^{A}$, we introduce more general transformations that are fully symmetric, both algebraically, but also parity-wise, i.e. transformations $\delta=\delta_{0}\oplus\delta_{1}$, $\delta_{0}:M_{0}\rightarrow M_{1}$ and $\delta_{1}:M_{1}\rightarrow M_{0}$ such that $\delta_{0}\sigma^{\alpha}=\epsilon^{T}\rho^{\alpha}\theta$, and $\delta_{1}\theta^{A}=\lambda^{T}\gamma^{A}\sigma$, for infinitesimal commuting parameters $\epsilon$ and $\lambda$, and matrices $\rho$ and $\gamma$. We then show that $\Psi$ transforms via a pullback under $\delta$.

For $A\in\text{s}k\text{-sAlg}_{*}$, define $F(A)=\{\Psi(\sigma,\theta)\,|\,\sigma\in A_{0},\theta\in A_{1}\}$ endowed with the subspace topology. Those functors $F$ that satisfy hyperdescent on $(k\text{-D}^{-}\text{sAff}_{*},\acute{e}t.)$ are called supersymmetric (derived) stacks.

2 Simplicial Super algebra

In a first time we remind the reader of various results pertaining to super-algebra that we will later generalize to the simplicial setting. Most of what is presented in the subsection below can be found in [V], and especially in [CCF], which we modify for our purposes.

2.1 Super Algebra

This subsection covers some basic conventions about superalgebra. We fix $k$ a commutative ring. We define a super $k$-module (also referred to as $k$-supermodule) to a be a $\mathbb{Z}_{2}$-graded $k$-module $M=M_{0}\oplus M_{1}$, endowed with a parity function $|\;|$ defined by:

[TABLE]

There is a (distributive) tensor product $\otimes$ on the set of $k$-supermodules $k\text{-sMod}$, for which the commutativity map $\sigma$ (also referred to as braiding map), defined on objects by $\sigma_{M,N}:M\otimes N\rightarrow N\otimes M$, $M=M_{0}\oplus M_{1}$, $N=N_{0}\oplus N_{1}$, satisfies:

[TABLE]

on homogeneous elements, and is extended by linearity. Morphisms of super modules are defined to be graded morphisms, or equivalently morphisms of graded objects of degree zero ([McL0]). In other terms if $f:M\rightarrow N$ is a morphism of super $k$-modules, then it decomposes into two morphisms $f_{0}:M_{0}\rightarrow N_{0}$ and $f_{1}:M_{1}\rightarrow N_{1}$. We denote by $\text{Hom}_{k}(M,N)$, or simply by $\text{Hom}(M,N)$ the set of graded morphisms from $M$ to $N$. It is a $k$-module. Each super module $M$ has an identity $id_{M}=id_{M_{0}}\oplus id_{M_{1}}\in\text{Hom}(M,M)$ that acts as a left and a right identity for composition. With obvious notations, composition is defined by $f\circ g=f_{0}\circ g_{0}\oplus f_{1}\circ g_{1}$, and is associative. This makes $k\text{-sMod}$ into a category.

The identity for the tensor product is $k$, and we have left and right identity maps:

[TABLE]

with $x\in M$, $M$ an object of $k\text{-sMod}$. Here for instance $k\otimes(x_{0}\oplus x_{1})=(k\otimes x_{0})\oplus(k\otimes x_{1})$, so that we can write $\lambda=\lambda_{0}\oplus\lambda_{1}$ and $\rho=\rho_{0}\oplus\rho_{1}$, $\lambda_{i}$ and $\rho_{i}$ the classical left and right identity maps in $k\text{-Mod}$ respectively. We regard elements of $k$ to be of degree zero so that they preserve parity with the tensor product. The associator is defined by:

[TABLE]

and satisfies the pentagon coherence diagram ([McL]):

[TABLE]

as well as the triangle coherence diagrams:

[TABLE]

and the bigon relations $\rho_{k}=\lambda_{k}$.

The braiding map $\sigma$ is required to satisfy the hexagon coherence condition:

[TABLE]

We investigate what does the parity function have to satisfy for this to hold: we start with an element $(x\otimes y)\otimes z$ in $(M\otimes N)\otimes P$, with $x=x_{0}\oplus x_{1}$, $y=y_{0}\oplus y_{1}$ and $z=z_{0}\oplus z_{1}$. Thus, abbreviating tensor products $x\otimes y$ by $xy$ and direct sums $x\oplus y$ by $x+y$ for simplicity of notation:

[TABLE]

maps to:

[TABLE]

under $\sigma\otimes z$, which then maps to:

[TABLE]

under $\alpha$. This expands as:

[TABLE]

and this maps under $1\otimes\sigma$ to:

[TABLE]

Now starting from the same object $(x\otimes y)\otimes z$, under $\alpha$ this maps to:

[TABLE]

which expands as:

[TABLE]

Now if we have:

[TABLE]

then this expansion maps to:

[TABLE]

under $\sigma$, which itself maps to:

[TABLE]

under $\alpha$, and this is exactly (1), showing that the hexagonal diagram does commute with the definition of the parity on tensor products given by (2). To see that (2) is indeed the correct relation, it suffices to circle the hexagon diagram both ways by starting from a same element $(x\otimes y)\otimes z$, $x$, $y$, $z$ homogeneous elements, as in the diagram below:

[TABLE]

It follows from (2) that if the parity function is calculated mod 2, the tensor product on $k\text{-sMod}$ is defined by:

[TABLE]

where $M_{i}\otimes N_{j}$, $i,j=0,1$ is the usual tensor product in $k\text{-Mod}$ on the underlying $k$-modules. Finally, the braiding $\sigma=\sigma_{0}\oplus\sigma_{1}$ on $k\text{-sMod}$ satisfies $\sigma^{2}=\sigma_{0}^{2}\oplus\sigma_{1}^{2}=1$. Indeed:

[TABLE]

where we have omitted tensor products for ease of reading. Further:

[TABLE]

meaning, taking $k$ to have even parity, the following diagram commutes:

[TABLE]

At this point we have shown that $(k\text{-sMod},\otimes_{k},\alpha,\lambda,\rho,k)$ is a symmetric monoidal category.

2.2 Diagonal super algebra

2.2.1 Box tensor product $\boxtimes$

In order to develop a theory within an Algebraic Geometry context over graded $k$-modules, we need commutative monoids whose definition really makes sense only if we take a diagonal version of $(k\text{-sMod},\otimes_{k},\alpha,\lambda,\rho,k)$. One way to achieve this is by considering $k\text{-sMod}_{*}$, the category of graded $k$-supermodules. We take the grading to be $\mathbb{Z}_{2}$ as well. An object of $k\text{-sMod}_{*}$ is of the form $M=M_{0}\oplus M_{1}$, $M_{i}=M_{i0}\oplus M_{i1}$ for $i=0,1$, where objects of $M_{ip}$ have parity $p$. Thus one can write $M=\oplus_{i,p=0,1}M_{ip}$, $i$ referring to the graded index, or level, and $p$ referring to parity. We will adopt the shorthand matricial notation $M=(M_{ip})$ to denote such objects. It follows that we will refer to the components $M_{ip}$ as entries of the object $M\in k\text{-sMod}_{*}$. In the same manner, morphisms $f:M\rightarrow N$ in $k\text{-sMod}_{*}$ are defined entrywise: $f_{ip}:M_{ip}\rightarrow N_{ip}$. The tensor product on $k\text{-sMod}_{*}$ is a diagonal tensor product defined by:

[TABLE]

where the tensor products used on the right hand side are those of $k\text{-sMod}$. Observe that this is performed levelwise and each level is closed under $\otimes$: $M_{ip}\otimes N_{iq}\in k\text{-sMod}_{i}$, $i=0,1$. It follows $M_{i}\otimes N_{i}=(M\boxtimes N)_{i}$. The advantage of having such a definition of the tensor product on $k\text{-sMod}_{*}$ is that one can use classical module theory with elements that have a parity. We have a braiding:

[TABLE]

defined levelwise on supermodules:

[TABLE]

with $\sigma_{i}=\sigma_{i0}\oplus\sigma_{i1}:M_{i}\otimes N_{i}\rightarrow N_{i}\otimes M_{i}$. To be more precise:

[TABLE]

2.2.2 $k\text{-sMod}_{*}$ symmetric monoidal category

Recall that classically morphisms of supermodules $M\rightarrow N$ are graded morphisms $f=f_{0}\oplus f_{1}$, $f_{0}:M_{0}\rightarrow N_{0}$, $f_{1}:M_{1}\rightarrow N_{1}$. In the bi-graded case with the box tensor product defined above, morphisms in $k\text{-sMod}_{*}$ are defined levelwise, and on each level, morphisms are morphisms in $k\text{-sMod}$. Thus $f:M\rightarrow N$ in $k\text{-sMod}_{*}$ decomposes as $f=f_{0}\oplus f_{1}$, with $f_{i}:M_{i}\rightarrow N_{i}$, $i=0,1$, and for $i$ fixed, $f_{i}=f_{i0}\oplus f_{i1}$, with $f_{ip}:M_{ip}\rightarrow N_{ip}$, with $p=0,1$. Thus $f=\oplus_{i,p=0,1}f_{ip}$, which we can represent matricially by $f=(f_{ip})$.

We still denote by $\text{Hom}(M,N)$ the set $\text{Hom}_{k\text{-sMod}_{*}}(M,N)$. We have $\text{Hom}_{k\text{-sMod}_{*}}=\text{Hom}_{(k\text{-sMod}_{0},\otimes)}\oplus\text{Hom}_{(k\text{-sMod}_{1},\otimes)}$. It is a $k$-module. For any $M\in k\text{-sMod}_{*}$, $id_{M}=id_{M_{0}}\oplus id_{M_{1}}\in\text{Hom}(M,M)$. Composition is associative. This makes $k\text{-sMod}_{*}$ into a category. We have:

[TABLE]

To be more precise:

[TABLE]

with $\alpha_{0}$ and $\alpha_{1}$ associators in $k\text{-sMod}$, so we do have:

[TABLE]

We have a pentagon diagram:

[TABLE]

and triangle coherence diagrams:

[TABLE]

with bigons $\rho_{\Delta k}=\lambda_{\Delta k}$. For the hexagonal coherence condition:

[TABLE]

with $\sigma^{2}=1$ and $\lambda_{M}=\rho_{M}\circ\sigma_{M,\Delta k}$ for symmetry. We have proved:

Theorem 2.2.2.1**.**

$(k\text{-sMod}_{*},\boxtimes,\alpha,\lambda,\rho,\Delta k)$ is a symmetric monoidal category.

2.2.3 Commutative monoids in $k\text{-sMod}_{*}$

A monoid in $\mathcal{C}=k\text{-sMod}_{*}$ is an object $M$ of $\mathcal{C}$ with an associative binary operation $\mu:M\boxtimes M\rightarrow M$, $\mu=\mu_{0}\oplus\mu_{1}$, $\mu_{i}$ the ordinary binary operation in $k\text{-sMod}_{i}$ for $i=0,1$. To be more precise:

[TABLE]

where $\mu_{i}$ and $m_{i}\otimes m_{i}^{\prime}$ decompose as:

[TABLE]

and:

[TABLE]

since each $\mu_{ip}$ is defined on homogeneous elements. The terms on the right hand side recombine as $(m_{i}m_{i}^{\prime})_{0}\oplus(m_{i}m_{i}^{\prime})_{1}=m_{i}m_{i}^{\prime}$.

We also have a unit map $\eta:\Delta k\rightarrow M$, $\eta=\eta_{0}\oplus\eta_{1}$, $\eta_{i}$ the unit in $k\text{-sMod}_{i}$ for $i=0,1$. $\mu$ and $\eta$ satisfy, as in [McL], the coherence diagrams:

[TABLE]

and:

[TABLE]

where $f\boxtimes g=\oplus_{i=0,1}f_{i}\otimes g_{i}$.

A monoid $A$ in $k\text{-sMod}_{*}$ is said to be super-commutative, which we will just refer to as being commutative, if $\mu\circ\sigma=\mu$, i.e. if levelwise, for all $x,y$ homogeneous in $A_{i}$ for $i=0,1$, we have:

[TABLE]

Another way of saying this is $A\in k\text{-sMod}_{*}$ is supercommutative if it is so level wise.

We define the category of graded $k$-super algebras to be the category of commutative monoids (with commutativity defined by $\sigma$) in $k\text{-sMod}_{*}$:

[TABLE]

For $A\in k\text{-sAlg}_{*}$, we denote by $A\text{-sMod}_{*}$ the category of elements of $k\text{-sMod}_{*}$ that are $A$-modules in the classical sense, where the action of $A$ on $M\in k\text{-sMod}_{*}$ is defined levelwise. A morphism of $A$-modules is a morphism $f:M\rightarrow N$ of graded supermodules such that $f(am)=af(m)$ for $a\in A$, $m\in M$. More specifically, $am$ is defined via:

[TABLE]

with $A\boxtimes M=(A_{0}\otimes M_{0})\oplus(A_{1}\otimes M_{1})$, $\mu=\mu_{0}\oplus\mu_{1}$ as we saw above, so that $am=a_{0}m_{0}\oplus a_{1}m_{1}$, with $a_{i}m_{i}\in M_{i}$, $i=0,1$. Then $f(am)=af(m)$ reads:

[TABLE]

where $f_{i}\in\text{Mor}\,(A_{i}\text{-sMod}_{i})$ for $i=0,1$. We denote by $\text{Hom}_{A}(M,N)$ the set of morphism of $A$-modules. There is a tensor product on $A\text{-sMod}_{*}$ defined by:

[TABLE]

where the equivalence relation $\sim$ is defined by $ma\boxtimes n=m\boxtimes an$ for $a\in A$, $m\in M$, $n\in N$. However we will use a simplified product that will allow us to use the work of Toen and Vezzosi; we define:

[TABLE]

that is:

[TABLE]

which is fullly diagonalized.

2.3 Simplicial Generalization

We now define the simplicial counterparts to all the above definitions. We will not worry about universe considerations, and if needed they can be transcribed to our case from [TV] and [TV4]. We denote by $\text{s}k\text{-sMod}_{*}$ the category of simplicial objects in $k\text{-sMod}_{*}$:

[TABLE]

whose objects will be referred to as simplicial graded $k$-supermodules. We endow this category with the (simplicial) levelwise tensor product $\boxtimes$, that is for $M,N\in\text{s}k\text{-sMod}_{*}$, we have $M\boxtimes N=\oplus_{n\geq 0}M_{n}\boxtimes N_{n}$, where:

[TABLE]

We will put a model structure on this category. Following [GoJa] it becomes evident that we will need a notion of model category structure on the category of bi-graded simplicial sets. We do this first.

2.3.1 Model category of $\mathbb{Z}_{2}$-bi-graded simplicial sets

In this subsection, we define a model category structure on the category of bi-graded simplicial sets, following [GoJa]. For completeness’ sake, we briefly remind the reader of a few results regarding simplicial sets, all of which can be found in [GoJa]. Recall that a map $p:X\rightarrow Y$ of simplicial sets is said to be a fibration if for any commutative diagram:

[TABLE]

where as usual $\Delta^{n}=\text{Hom}_{\Delta}(-,[n])$, there is a dotted map $\Delta^{n}\rightarrow X$ making the above diagram commute. A fibrant simplicial set, or Kan complex, is a simplicial set $X$ for which $X\rightarrow*$ is a fibration, it being the unique map to the final object $*$. For $X$ a fibrant simplicial set, $v\in X_{0}$, we define the simplicial homotopy group $\pi_{n}(X,v)$ for $n\geq 1$ to be the set of homotopy classes of maps $\alpha:\Delta^{n}\rightarrow X$ rel $\partial\Delta^{n}$ that fit in diagrams of the form:

[TABLE]

and define $\pi_{0}(X)$ to be the set of homotopy classes of vertices of $X$. A cofibration is an inclusion of simplicial sets. An equivalence is defined as follows: $f:X\rightarrow Y$ between fibrant simplicial sets is said to be a weak equivalence if for any vertex $x$ of $X$, $f_{*}:\pi_{n}(X,x)\rightarrow\pi_{n}(Y,f(x))$ is an isomorphism for all $n\geq 1$ and $f_{*}:\pi_{0}(X)\rightarrow\pi_{0}(Y)$ is a bijection.

We now define all those concepts in the bi-graded case. Let $\text{Top}_{*\centerdot}$ denote the category of $\mathbb{Z}_{2}$-bi-graded topological spaces. Denote by $S_{*\centerdot}$ the category of $\mathbb{Z}_{2}$-bi-graded simplicial sets. We define the realization functor $|\;|:S_{*\centerdot}\rightarrow\text{Top}_{*\centerdot}$ entrywise. Introduce the bi-graded simplex category $\Delta_{*\centerdot}\downarrow X_{*\centerdot}$ of a bi-graded simplicial set $X_{*\centerdot}$, whose objects are bi-graded maps which entry-wise read $\sigma_{ip}:\Delta^{n}\rightarrow X_{ip}$, $i,p=0,1$.

An arrow of $\Delta_{*\centerdot}\downarrow X_{*\centerdot}$ consists of commutative diagrams of simplicial maps for $i,p=0,1$:

[TABLE]

We have the following result from [GoJa]:

Lemma 2.3.1.1**.**

$X_{ip}\cong\operatorname*{\text{colim}}_{\begin{subarray}{c}\Delta^{n}\rightarrow X_{ip}\\ \text{ in }\Delta\downarrow X_{ip}\end{subarray}}\Delta^{n}\in\text{Set}_{\Delta}$.

Using the following definition on usual simplicial sets:

[TABLE]

we define the realization $|X_{*\centerdot}|$ of a bi-graded simplicial set $X_{*\centerdot}$ to be $|X_{*\centerdot}|=(|X_{ip}|)\in\text{Top}_{*\centerdot}$.

We define the singular functor $S:\text{Top}_{*\centerdot}\rightarrow S_{*\centerdot}$ entrywise. For $T_{*\centerdot}\in\text{Top}_{*\centerdot}$, $S(T_{*\centerdot})=(S(T_{ip}))$, where $S(T_{ip})$, $i,p=0,1$ is the simplicial set given by:

[TABLE]

and $|\Delta^{n}|$ is the standard $n$-simplex:

[TABLE]

Proposition 2.3.1.2**.**

For all $X_{*\centerdot}\in S_{*\centerdot}$, $Y_{*\centerdot}\in\text{Top}_{*\centerdot}$, we have $\text{Hom}_{\text{Top}_{*\centerdot}}(|X_{*\centerdot}|,Y_{*\centerdot})\cong\text{Hom}_{S_{*\centerdot}}(X_{*\centerdot},SY_{*\centerdot})$, that is $|\;|\dashv S$.

Proof.

It suffices to write:

[TABLE]

∎

We define a map $p_{*\centerdot}:X_{*\centerdot}\rightarrow Y_{*\centerdot}$ of bi-graded simplicial sets to be a fibration if for $i,q=0,1$ the maps $p_{iq}:X_{iq}\rightarrow Y_{iq}$ are fibrations, i.e. if it is so entrywise. Thus a fibrant bi-graded simplicial set, or bi-graded Kan complex, is a bi-graded simplicial set $Y_{*\centerdot}$ such that $Y_{ip}\rightarrow*$ is a fibration for $i,p=0,1$. In the same manner, a bi-graded continuous map $f_{*\centerdot}:T_{*\centerdot}\rightarrow U_{*\centerdot}$ is said to be a bi-graded Serre fibration if it is so entry-wise. For $f_{*\centerdot},g_{*\centerdot}:K_{*\centerdot}\rightarrow X_{*\centerdot}$ bi-graded simplicial maps, we say there is a bi-graded simplicial homotopy $f_{*\centerdot}\xrightarrow{\simeq}g_{*\centerdot}$ if we have a simplicial homotopy $f_{ip}\xrightarrow{\simeq}g_{ip}$ for $i,p=0,1$, which we recall means there is a commutative diagram:

[TABLE]

in which case we say $h_{*\centerdot}=(h_{ip})$ is a bi-graded simplicial homotopy $f_{*\centerdot}\rightarrow g_{*\centerdot}$. If $j_{*\centerdot}:L_{*\centerdot}\subset K_{*\centerdot}$ denotes a bi-graded inclusion such that $f_{*\centerdot}|_{L_{*\centerdot}}=g_{*\centerdot}|_{L_{*\centerdot}}$, then we say we have a bi-graded simplicial homotopy $f_{*\centerdot}\rightarrow g_{*\centerdot}$ rel $L_{*\centerdot}$ if we have a simplicial homotopy $f_{ip}\rightarrow g_{ip}$ rel $L_{ip}$ for $i,p=0,1$, that is such that the following diagrams commute:

[TABLE]

For $X_{*\centerdot}$ a fibrant bi-graded simplicial set, $v_{*\centerdot}=(v_{ip})\in(X_{*\centerdot})_{0}$, we define $\pi_{n}(X_{*\centerdot},v_{*\centerdot})=(\pi_{n}(X_{ip},v_{ip}))$ where $\pi_{n}(X_{ip},v_{ip})$ is the set of homotopy classes of maps $\alpha_{ip}:\Delta^{n}\rightarrow X_{ip}$ rel $\partial\Delta^{n}$ for $i,p=0,1$, such that the following diagram is commutative:

[TABLE]

and $\pi_{0}(X_{*\centerdot})=(\pi_{0}(X_{ip}))$, $\pi_{0}(X_{ip})$ the set of path components of $X_{ip}$, $i,p=0,1$. Now a map $f_{*\centerdot}:X_{*\centerdot}\rightarrow Y_{*\centerdot}$ between fibrant bi-graded simplicial sets is said to be a weak equivalence if for all $x_{*\centerdot}\in(X_{*\centerdot})_{0}$, the induced map $(f_{*\centerdot})_{*}:\pi_{k}(X_{*\centerdot},x_{*\centerdot})\rightarrow\pi_{k}(Y_{*\centerdot},f_{*\centerdot}(x_{*\centerdot}))$ is an isomorphism for all $k\geq 1$, that is $(f_{ip})_{*}:\pi_{k}(X_{ip},x_{ip})\xrightarrow{\cong}\pi_{k}(Y_{ip},f_{ip}(x_{ip}))$ for $i,p=0,1$, and $(f_{*\centerdot})_{*}:\pi_{0}(X_{*\centerdot})\rightarrow\pi_{0}(Y_{*\centerdot})$ is a bijection, that is $(f_{ip})_{*}:\pi_{0}(X_{ip})\rightarrow\pi_{0}(Y_{ip})$ is a bijection for $i,p=0,1$, or in other terms a weak equivalence is so if it is a weak equivalence entry-wise.

Proposition 2.3.1.3**.**

A map $f_{*\centerdot}:X_{*\centerdot}\rightarrow Y_{*\centerdot}$ between fibrant bi-graded simplicial sets is a trivial fibration if and only if $f_{*\centerdot}$ has the right lifting property with respect to all maps $\partial\Delta^{n}\subset\Delta^{n}$ for $n\geq 0$, entry-wise.

Proof.

The result holds entry-wise ([GoJa]), hence is true for the corresponding bi-graded objects by definition. ∎

Proposition 2.3.1.4**.**

Suppose $X_{*\centerdot}$ is a bi-graded Kan complex. Then the canonical map $\eta_{X_{*\centerdot}}:X_{*\centerdot}\rightarrow S|X_{*\centerdot}|$ is a weak equivalence.

Proof.

The proof follows immediately from a similar proposition of [GoJa] in the ungraded case because of the definitions of $S$ and $|\,\,|$, and the fact that $\eta_{X_{*\centerdot}}$ is a weak equivalence if it is so entry-wise, which is the case ([GoJa]). ∎

Now if $X_{*\centerdot}$ is a bi-graded Kan complex, $x_{*\centerdot}$ any vertex of $X_{*\centerdot}$, then by virtue of the above proposition we have:

[TABLE]

so a map $f_{*\centerdot}:X_{*\centerdot}\rightarrow Y_{*\centerdot}$ of bi-graded Kan complexes is a weak equivalence if and only if the induced map $|f_{*\centerdot}|:|X_{*\centerdot}|\rightarrow|Y_{*\centerdot}|$ is a bi-graded topological weak equivalence, which leads us to defining, as in [GoJa], that a map $f_{*\centerdot}:X_{*\centerdot}\rightarrow Y_{*\centerdot}$ of bi-graded simplicial sets be a weak equivalence if the induced map $|f_{*\centerdot}|:|X_{*\centerdot}|\rightarrow|Y_{*\centerdot}|$ is a weak equivalence of bi-graded spaces. We define a cofibration of bi-graded simplicial sets to be an entry-wise inclusion.

Theorem 2.3.1.5**.**

$S_{*\centerdot}$ together with the classes of Kan fibrations, cofibrations and weak equivalences defined above is a model category.

Proof.

$\text{Set}_{\Delta}$ is complete and cocomplete, hence so is $S_{*\centerdot}$. For the 2 out of 3 property, suppose we have $f_{*\centerdot}=g_{*\centerdot}\circ h_{*\centerdot}$. If two of $f_{*\centerdot}$, $g_{*\centerdot}$ or $h_{*\centerdot}$ is a weak equivalence, so are their entries, so writing $f_{ip}=g_{ip}\circ h_{ip}$ since the 2 out of 3 property holds at that level for $i,p=0,1$, then the third function would be an equivalence for $i,p=0,1$, hence so would be the resulting bi-graded function. For the retract property, say $f_{*\centerdot}$ is a retract of $g_{*\centerdot}$, and $g_{*\centerdot}$ is a weak equivalence, fibration or cofibration. The retract breaks up into diagrams for $i,p=0,1$ as:

[TABLE]

$g_{*\centerdot}$ having one of the three properties mentioned above, it is so entry-wise, at which level $f_{ip}$ being a retract of $g_{ip}$ the former map shares the same property, for $i,p=0,1$, hence $f_{*\centerdot}$ shares that same property $g_{*\centerdot}$ had. For the lifting property suppose we have:

[TABLE]

with $i_{*\centerdot}$ a cofibration, $p_{*\centerdot}$ a fibration. We show there is a dotted arrow as shown making the diagram commutative if either $i_{*\centerdot}$ or $p_{*\centerdot}$ is a weak equivalence. That diagram breaks up into diagrams for $j,q=0,1$:

[TABLE]

Suppose for argument’s sake, $p_{*\centerdot}$ is a weak equivalence. Then it is so entry-wise, so in the diagrams above for $j,q=0,1$ there is a dotted arrow, hence the original bi-graded square has a diagonal arrow as claimed. Finally for the functorial factorization property, let $f_{*\centerdot}:X_{*\centerdot}\rightarrow Y_{*\centerdot}$ be a map in $S_{*\centerdot}$. We will show half of the property, namely that $f_{*\centerdot}$ can factor as $p_{*\centerdot}\circ i_{*\centerdot}$ with $p_{*\centerdot}$ a trivial fibration, $i_{*\centerdot}$ a cofibration, the other factorization as a trivial cofibration followed by a fibration being proved in like manner. $f_{*\centerdot}=(f_{ip})$, and entry-wise the factorization property holds, so we can write: $f_{jq}=p_{jq}\circ i_{jq}$, with $p_{jq}$ a trivial fibration for $j,q=0,1$, $i_{jq}$ a cofibration for $j,q=0,1$. Writing $i=(i_{jq})$ and $p=(p_{jq})$ we have $f_{*\centerdot}=p_{*\centerdot}\circ i_{*\centerdot}$ with $p_{*\centerdot}$ a trivial fibration, and $i_{*\centerdot}$ a cofibration. This completes the proof. ∎

2.3.2 Model category structure on $\text{s}k\text{-sMod}_{*}$

We adapt Thm II.4.1 of [GoJa] to our setting. Let $\mathcal{C}_{*\centerdot}$ be a $\mathbb{Z}_{2}$-bi-graded category, $s\mathcal{C}_{*\centerdot}$ the category of simplicial objects in $\mathcal{C}_{*\centerdot}$, $s\mathcal{C}_{*\centerdot}=\mathcal{C}_{*\centerdot}^{\Delta^{\text{op}}}$. We assume there is a functor $G_{*\centerdot}:s\mathcal{C}_{*\centerdot}\rightarrow S_{*\centerdot}$ with a left adjoint $F_{*\centerdot}:S_{*\centerdot}\rightarrow s\mathcal{C}_{*\centerdot}$. We define a morphism $f_{*\centerdot}:M_{*\centerdot}\rightarrow N_{*\centerdot}$ in $\text{s}k\text{-sMod}_{*}$ to be a weak equivalence (resp. a fibration) if $G_{*\centerdot}f_{*\centerdot}$ is a weak equivalence (resp. a fibration) of bi-graded simplicial sets, and $f_{*\centerdot}$ is a cofibration if it has the left lifting property with respect to all trivial fibrations in $s\mathcal{C}_{*\centerdot}$. Entry-wise, that gives us $G_{ip}:s\mathcal{C}\rightarrow\text{Set}_{\Delta}$ with a left adjoint $F_{ip}:\text{Set}_{\Delta}\rightarrow s\mathcal{C}$. Given the model category structure we put on $S_{*\centerdot}$, fibrations, cofibrations and weak equivalences in $s\mathcal{C}_{*\centerdot}$ are defined entry-wise in $s\mathcal{C}$. We will apply this formalism to the case $\mathcal{C}=k\text{-Mod}$, so that $s\mathcal{C}=\text{s}k\text{-Mod}$ and $s\mathcal{C}_{*\centerdot}=\text{s}k\text{-sMod}_{*}$.

Note also that we have a natural map:

[TABLE]

which decomposes into entry maps:

[TABLE]

for $i,p=0,1$.

Theorem 2.3.2.1**.**

Suppose $\mathcal{C}$ is bicomplete and $G_{*\centerdot}:s\mathcal{C}_{*\centerdot}\rightarrow S_{*\centerdot}$ commutes with filtered colimits. Then with the classes of fibrations, cofibrations and weak equivalences defined above, along with the assumption that a cofibration with the left lifting property with respect to all fibrations be a weak equivalence, $s\mathcal{C}_{*\centerdot}$ is a model category.

Proof.

Since $G_{*\centerdot}$ commutes with filtered colimits, it does so entry-wise, so the hypotheses of Thm II.4.1 of [GoJa] are met, hence $s\mathcal{C}$ is a model category, something we will use to prove $s\mathcal{C}_{*\centerdot}$ itself is a model category. $s\mathcal{C}_{*\centerdot}$ is clearly bicomplete. For the 2 out of 3 property, suppose we have a factorization in $s\mathcal{C}_{*\centerdot}$:

[TABLE]

This breaks up into diagrams for $i,p=0,1$:

[TABLE]

in $s\mathcal{C}$, where the 2 out of 3 property holds since it is a model category. Suppose for illustrative purposes $g_{*\centerdot}$ and $f_{*\centerdot}$ are weak equivalences, then $g_{ip}$ and $f_{ip}$ are so for $i,p=0,1$, hence $h_{ip}$ is a weak equivalence for $i,p=0,1$, thus so is $h_{*\centerdot}=(h_{ip})$. For the retract property, $f_{*\centerdot}$ a retract of $g_{*\centerdot}$ is a property defined by a diagram in $s\mathcal{C}_{*\centerdot}$, which breaks up into retract diagrams in $s\mathcal{C}$, where the retract property holds, so starting from $g_{*\centerdot}$ with a property $P$, it being a weak equivalence, a fibration, or a cofibration, it is so entry-wise, so by the retract property in $s\mathcal{C}$ $f_{ip}$ shares the same property for $i,p=0,1$, and those entries recombine into $f_{*\centerdot}$ with that same property $P$ $g_{*\centerdot}$ had. For the lifting property and the functorial factorization property, the argument is the same, we work entry-wise and use the fact that $s\mathcal{C}$ is a model category from [GoJa]. ∎

Corollary 2.3.2.2**.**

With the above notions of fibrations, cofibrations and weak equivalences, $\text{s}k\text{-sMod}_{*}$ is a model category.

Proof.

Let $\mathcal{C}=k\text{-Mod}$, so that $s\mathcal{C}=\text{s}k\text{-Mod}$ and $s\mathcal{C}_{*\centerdot}=\text{s}k\text{-sMod}_{*}$. We know $\mathcal{C}$ is bicomplete. Start from the forgetful functor $G:k\text{-Mod}\rightarrow\text{Set}$, with a left adjoint $F$, which we both prolong to the simplicial case to get maps which we will again denote by $G$ and $F$: $G:\text{s}k\text{-Mod}\rightarrow\text{Set}_{\Delta}$ and its left adjoint $F$, by defining $G(X)_{n}=G(X_{n})$. $G$ preserving filtered colimits, so will its bi-graded generalization $G_{*\centerdot}:\text{s}k\text{-sMod}_{*}\rightarrow S_{*\centerdot}$, which has a bi-graded generalization of $F$ for left adjoint, denoted $F_{*\centerdot}$. At this point we just use the previous theorem. ∎

2.3.3 bi-graded simplicial categories

About notations, if $\mathcal{C}$ is a bi-graded category, write $\underline{\mathcal{C}}$ for its ungraded counterpart. For instance if $\mathcal{C}=\text{s}k\text{-sMod}_{*}$, then $\underline{\mathcal{C}}=\text{s}k\text{-Mod}$, which generically refers to either $\text{s}k\text{-Mod}_{0}$ or $\text{s}k\text{-Mod}_{1}$. Also, for ease of reading, we will just write $M$ for $M_{*\centerdot}\in k\text{-sMod}_{*}$. We also define the enhanced Hom set $\text{Hom}^{+}$ as follows: $(\text{Hom}^{+}(M,N))_{0}=\text{Hom}(M,N)$, while $(\text{Hom}^{+}(M,N))_{1}$ consist of the set of parity reversing morphisms from $M$ to $N$.

Definition 2.3.3.1**.**

A bi-graded category $\mathcal{C}$ is a bi-graded simplicial category, following the ungraded definition in [GoJa], if there is a mapping space functor:

[TABLE]

such that $\forall M,N\in\text{Ob\,}(\mathcal{C})$:

$\underline{\text{Hom}}_{\mathcal{C}}(M,N)_{0\times 0}=\text{Hom}^{+}_{\mathcal{C}}(M,N)$ 2. 2.

$\underline{\text{Hom}}_{\mathcal{C}}(M,-):\mathcal{C}\rightarrow S_{*\centerdot}$ has a left-adjoint $M\boxtimes-:S_{*\centerdot}\rightarrow\mathcal{C}$. Adjointness means:

[TABLE]

with associativity $M\boxtimes(K\times L)\cong(M\boxtimes K)\boxtimes L$. 3. 3.

$-\boxtimes K:\mathcal{C}\rightarrow\mathcal{C}$ has a right adjoint $\exp_{-}(K)=\underline{\hom}_{\mathcal{C}}(K,-):\mathcal{C}\rightarrow\mathcal{C}$ i.e.

[TABLE]

Theorem 2.3.3.2**.**

$s\underline{\mathcal{C}}=\text{s}k\text{-Mod}$ being a simplicial category ([TV4]), it follows that $s\mathcal{C}=\text{s}k\text{-sMod}_{*}$ becomes a bi-graded simplicial category with:

[TABLE]

Proof.

Here, as in [TV]:

[TABLE]

is defined by:

[TABLE]

For $\psi:[m]\rightarrow[n]$ in $\Delta$, we have an induced map $\psi^{*}:(M\otimes K)_{m}\rightarrow(M\otimes K)_{n}$ that comes from:

[TABLE]

We determine $\text{Hom}_{s\mathcal{C}}(M\boxtimes K,N)$. In a first time, $\text{s}k\text{-Mod}$ being a simplicial category, if $M,N\in\text{s}k\text{-Mod}$, $K\in\text{Set}_{\Delta}$, we have $\text{Hom}(M\otimes K,N)\cong\text{Hom}(K,\underline{\text{Hom}}_{\text{s}k\text{-Mod}}(M,N))$. If now $M,N\in\text{s}k\text{-sMod}$, $K\in S_{\centerdot}=(\text{Set}_{\Delta})_{0}\oplus(\text{Set}_{\Delta})_{1}$:

[TABLE]

if we define:

[TABLE]

Now for $M,N\in\text{s}k\text{-sMod}_{*}=s\mathcal{C}$, $K\in S_{*\centerdot}$, we have:

[TABLE]

if we define:

[TABLE]

Thus (3) shows $M\boxtimes-\dashv\underline{\text{Hom}}_{s\mathcal{C}}(M,-)$. We also have:

[TABLE]

From (3) again, we have:

[TABLE]

as claimed. For the associativity, we use the associativity of $\otimes$ in the ungraded case:

[TABLE]

For the exponent map:

[TABLE]

if we define:

[TABLE]

From there

[TABLE]

if we define $N^{K}=\oplus_{i}N_{i}^{K_{i}}$. This shows $-\boxtimes K\dashv\exp_{-}(K)$. This completes the proof. ∎

2.3.4 Bi-graded simplicial model categories

We axiomatize the definition of bi-graded simplicial model category as done in [GoJa] in the ungraded case. We first need to define pullbacks in $S_{*\centerdot}$. It being a bi-graded category, pullbacks are defined entrywise. In what follows $\mathcal{C}=\text{s}k\text{-sMod}_{*}$. We also use the abbreviation $\underline{\text{Hom}}_{\mathcal{C}}(X,Y)_{ip}=XY_{ip}$ for $i,p=0,1$. Consider the cartesian square:

[TABLE]

it breaks up into individual cartesian squares:

[TABLE]

for $i,p=0,1$, giving $\Gamma=\oplus_{i,p}AX_{ip}\times_{AY_{ip}}BY_{ip}=(AX_{ip}\times_{AY_{ip}}BY_{ip})$.

Graded simplicial model category Axiom - grsModCat 2.3.4.1**.**

Let $\mathcal{C}$ be a bi-graded model category and a bi-graded simplicial category. Suppose $j:A\rightarrow B$ is a cofibration, $q:X\rightarrow Y$ a fibration. Then:

[TABLE]

is a fibration in $S_{*\centerdot}$, which is trivial if either of $j$ or $q$ is.

Definition 2.3.4.2**.**

A category satisfying the axiom grsModCat above will be called a bi-graded simplicial model category

This definition follows exactly the definition of such categories in the ungraded case as laid out in [GoJa]. We prove a preliminary result found in the same reference, that will be instrumental in proving that $\text{s}k\text{-sMod}_{*}$ is a bi-graded simplicial model category.

Proposition 2.3.4.3**.**

Let $\mathcal{C}$ be a bi-graded model category and a bi-graded simplicial category, $i:K\rightarrow L$ a cofibration in $S_{*\centerdot}$, $q:X\rightarrow Y$ a fibration in $\mathcal{C}$. Then the grsModCat axiom is equivalent to:

[TABLE]

being a fibration, trivial if either of $i$ or $q$ is. Here we have denoted $N^{K}=\underline{\hom}_{\mathcal{C}}(K,N)$ for ease of reading.

Proof.

It suffices to work entry-wise. We have:

[TABLE]

a fibration. For $p=0$, we have a fibration:

[TABLE]

which decomposes into:

[TABLE]

for $p=0,1$, fibration in $\text{Set}_{\Delta}$, trivial if either of $j$ or $q$ is, in particular if $j_{ip}$ or $q_{ip}$ is. Using the ungraded counterpart of the proposition from [GoJa], this is equivalent to:

[TABLE]

being a fibration, trivial if either $i_{ip}:K_{ip}\rightarrow L_{ip}$ or $q_{ip}:X_{ip}\rightarrow Y_{ip}$ is, for $p=0,1$.

For $p=1$, we have:

[TABLE]

which decomposes as:

[TABLE]

and:

[TABLE]

both fibrations, and trivial if $j_{ik}$, or $q_{il}$ trivial with $k\neq l$. Again, by the ungraded counterpart of this result from [GoJa], this is equivalent to:

[TABLE]

fibration, trivial if $i_{ik}:K_{ik}\rightarrow L_{ik}$ or $q_{il}:X_{il}\rightarrow Y_{il}$ is, for $k\neq l$. Recombining (4) and (5) yields:

[TABLE]

fibration, trivial if $i:K\rightarrow L$ or $q:X\rightarrow Y$ is. ∎

Theorem 2.3.4.4**.**

$\text{s}k\text{-sMod}_{*}$ is a bi-graded simplicial model category.

Proof.

We use the functor $G:\mathcal{C}=\text{s}k\text{-sMod}_{*}\rightarrow S_{*\centerdot}$ used to put a model structure on $\text{s}k\text{-sMod}_{*}$. We wish to show $\text{s}k\text{-sMod}_{*}$ satisfies the grsModCat axiom, which we just showed is equivalent to saying in particular the map

[TABLE]

is a fibration for $i:K\rightarrow L$ a cofibration in $S_{*\centerdot}$, $q:X\rightarrow Y$ a fibration in $\mathcal{C}$ (trivial if either of $i$ or $q$ is), which means $G(X^{L})\rightarrow G(X^{K}\times_{Y^{K}}Y^{L})$ is a fibration in $S_{*\centerdot}$ (trivial if either of $i$ or $q$ is). $G$ being a right adjoint it commutes with finite limits so this is equivalent to showing that:

[TABLE]

is a fibration, trivial if either of $i$ or $q$ is. Now $F\dashv G:S_{*\centerdot}\rightarrow\mathcal{C}$ satisfies, for $L,K\in S_{*\centerdot}$:

[TABLE]

where we have used the fact that $F_{ip}\dashv G_{ip}$, $i,p=0,1$, so it commutes with colimits as a left adjoint. Note that we used (for $p=q+r$):

[TABLE]

throughout. We need the following lemma:

Lemma 2.3.4.5**.**

If for any $K,L\in S_{*\centerdot}$ there is a natural isomorphism $F(L\boxtimes K)\cong F(L)\boxtimes K$, then for any $X\in\mathcal{C}=\text{s}k\text{-sMod}_{*}$:

[TABLE]

Proof.

It suffices to write:

[TABLE]

with $S=\text{Set}_{\Delta}$. ∎

From there,

[TABLE]

that is:

[TABLE]

is equivalent to:

[TABLE]

Now $q:X\rightarrow Y$ fibration in $\mathcal{C}$ means $G(q):GX\rightarrow GY$ fibration in $S_{*\centerdot}$, and $S_{*\centerdot}$ being a bi-graded simplicial model category it satisfies the grsModCat axiom, so the above map is a fibration, trivial if either of $q$ or $i$ is, that is $\text{s}k\text{-sMod}_{*}$ is a bi-graded simplicial model category. ∎

2.3.5 Internal hom

From [CCF] and [V] we know $k\text{-sMod}$ has an internal hom $\underline{\mathcal{H}\text{om}}_{k\text{-sMod}}$. This comes from the fact that the tensor product $\otimes_{k}$ on $k\text{-sMod}$ involves terms of mixed parity. In those references the internal hom is defined as follows:

[TABLE]

while $\underline{\mathcal{H}\text{om}}_{k\text{-sMod}}(M,N)_{1}$ is the set of morphisms $\phi:M\rightarrow N$ that reverse parity, i.e. those morphisms $M_{0}\rightarrow N_{1}$ and $M_{1}\rightarrow N_{0}$. We can see this is indeed the correct definition by starting from the formal definition for the internal hom:

[TABLE]

Expanding the left hand side in full:

[TABLE]

where we used the fact that $k\text{-Mod}$ is a closed monoidal category in the sense of [Ho], with internal hom $\text{Hom}_{k}$. It follows as claimed above that:

[TABLE]

Note that this shows what we called the enhanced hom $\text{Hom}^{+}$ earlier is actually the internal hom: $\text{Hom}^{+}=\underline{\mathcal{H}\text{om}}$. Henceforward we will use the notation $\underline{\mathcal{H}\text{om}}$.

For the tensor product in $k\text{-sMod}_{*}$:

[TABLE]

if $\underline{\mathcal{H}\text{om}}_{\text{s}k\text{-sMod}_{*}}(N,P)=\oplus_{i=0,1}\underline{\mathcal{H}\text{om}}_{\text{s}k\text{-sMod}}(N_{i},P_{i})$. We prolong this to the simplicial case levelwise in the simplicial index:

[TABLE]

At this point $\mathcal{C}=\text{s}k\text{-sMod}_{*}$ the category of simplicial graded $k$-supermodules is endowed with the level-wise tensor product $\boxtimes$ for which we have an internal hom $\underline{\mathcal{H}\text{om}}_{\text{s}k\text{-sMod}_{*}}$, and is endowed with the ”usual” model structure whereby weak equivalences and fibrations are defined on the underlying bi-graded simplicial sets. We have also proved $\mathcal{C}$ is a bi-graded simplicial model category with:

[TABLE]

Let:

[TABLE]

where again commutativity is defined on superalgebras by $ab=(-1)^{|a||b|}ba$ on homogeneous elements.

Now for $A\in\text{s}k\text{-sAlg}_{*}$, denote by $A\text{-sMod}_{*}$ the category of objects of $\text{s}k\text{-sMod}_{*}$ that are $A$-modules, with $(A\text{-sMod}_{*})_{n}=A_{n}\text{-sMod}$. A morphism $f$ of $A$-supermodules is a simplicial morphism of supermodules $f:M\rightarrow N$ such that $f(am)=af(m)$ for $a\in A$, $m\in M$, $M\in A\text{-sMod}_{*}$. To be more specific $f$ has components $f_{n}:M_{n}\rightarrow N_{n}$ in $\text{Hom}_{A_{n}}(M_{n},N_{n})$ and:

[TABLE]

There is a tensor product $M\tilde{\boxtimes}_{A}N$ defined level-wise:

[TABLE]

3 Pre-homotopical Algebraic context

The notion of Homotopical Algebraic context is introduced in [TV4]. The reader is referred to that reference for a full definition. We call it pre-homotopical for the simple reason that we do not use $\mathcal{C}_{0}$, nor do we need $\mathcal{A}$, a sub-category of good objects, or equivalently we just work with a symmetric monoidal model category $\mathcal{C}$ satisfying only the first four assumptions of [TV4] which we will adapt to our bi-graded setting.

3.1 $\text{s}k\text{-sMod}_{*}$ symmetric monoidal model category

We use the definition of symmetric monoidal model category as presented in [Ho]. We already have a monoidal structure $(\boxtimes_{k},\alpha,\lambda,\rho,\Delta k)$ on $k\text{-sMod}_{*}$ that we prolong to a monoidal structure on $\text{s}k\text{-sMod}_{*}$ level-wise. For that tensor product, we have an internal hom $\underline{\mathcal{H}\text{om}}_{\text{s}k\text{-sMod}_{*}}$. We have half of an adjunction of two variables:

[TABLE]

with:

[TABLE]

We have the braiding $\sigma:M\boxtimes N\xrightarrow{\cong}N\boxtimes M$ in $k\text{-sMod}_{*}$ that we prolonged to $\text{s}k\text{-sMod}_{*}$. Now:

[TABLE]

so $(\boxtimes,\underline{\mathcal{H}\text{om}}_{\text{s}k\text{-sMod}_{*}},\underline{\mathcal{H}\text{om}}_{\text{s}k\text{-sMod}_{*}},\phi_{r},\phi_{r}\circ\sigma^{*})$ is an adjunction of two variables, hence we have a closed monoidal structure on $\text{s}k\text{-sMod}_{*}$ making it into a closed monoidal category, as defined in [Ho].

We now show $\boxtimes$ is a Quillen bifunctor in $\mathcal{C}=\text{s}k\text{-sMod}_{*}$. Let $f:U\rightarrow V$ be a cofibration in $\mathcal{C}$, $g:W\rightarrow X$ a cofibration in $\mathcal{C}$ as well. This means they are cofibrations on the underlying bi-graded simplicial sets, so entry-wise cofibrations of simplicial sets, i.e. inclusions. More generally, maps in $\text{s}k\text{-sMod}_{*}$ are fibrations, cofibrations or weak equivalences if and only if they are respectively fibrations, cofibrations or weak equivalences in $\text{s}k\text{-Mod}$, or equivalenty said, if they are so entry-wise.

We need:

[TABLE]

cofibration in $\mathcal{C}$, trivial if either of $f$ or $g$ is.

$f:U\rightarrow V$ and $g:W\rightarrow X$ being cofibrations in $\text{s}k\text{-sMod}_{*}$ means we have entry-wise cofibrations in $\text{s}k\text{-Mod}$ $f_{ip}:U_{ip}\rightarrow V_{ip}$ and $g_{ip}:W_{ip}\rightarrow X_{ip}$ for $i,p=0,1$, with $U=(U_{ip})$, $V=(V_{ip})$, $W=(W_{ip})$ and $X=(X_{ip})$. Consider the coproduct $(V\boxtimes W)\coprod_{U\boxtimes W}(U\boxtimes X)=\oplus_{i=0,1}(V_{i}\otimes W_{i})\coprod_{U_{i}\otimes W_{i}}(U_{i}\otimes X_{i})$, which we simply denote by $\coprod$, computed in $\text{s}k\text{-sMod}_{*}$. Levelwise, it reads:

[TABLE]

further decomposing into parity wise coproducts:

[TABLE]

with a similar coproduct defining $\coprod_{i1}$, in such a manner that $\coprod=(\coprod_{ip})$. Note that the maps in this diagram are induced by $f_{ip}$ and $g_{ip}$. Thus the diagram defining $\coprod_{i0}$ further decomposes into two coproduct diagrams:

[TABLE]

so that $\coprod_{i0}=C_{i00}\oplus C_{i11}$. One would obtain a similar decomposition for $\coprod_{i1}$. Working levelwise, we want:

[TABLE]

to be a cofibration. But we know $f_{ip}:U_{ip}\rightarrow V_{ip}$ and $g_{iq}:W_{iq}\rightarrow X_{iq}$ are cofibrations, so $C_{ipq}=V_{ip}\otimes W_{iq}\coprod_{U_{ip}\otimes W_{iq}}U_{ip}\otimes X_{iq}\rightarrow V_{ip}\otimes X_{iq}$ is a cofibration for $p,q=0,1$, $\otimes$ being a Quillen bifunctor on $\text{s}k\text{-Mod}$, and all such maps recombine into (7), which is therefore a cofibration. Now if either of $f$ or $g$ is trivial, so are its entries. For instance if this is true of $f$, the entries $f_{ip}$ are trivial, making $U_{ip}\otimes W_{iq}\coprod_{U_{ip}\otimes W_{iq}}U_{ip}\otimes X_{iq}\rightarrow V_{ip}\otimes X_{iq}$ trivial for $i,p,q=0,1$, and those maps recombine into $f\Box g$ trivial. This shows that $\boxtimes$ is a Quillen bifunctor.

Since $k$ is the unit for $k\text{-Mod}$, $\Delta k$ the unit in $k\text{-sMod}_{*}$, if $c_{*}$ is the constant simplicial functor then, $c_{*}(k)=k_{*}$ is the unit for $\text{s}k\text{-Mod}$ and $c_{*}(\Delta k)=\Delta k_{*}$ is the unit for $\text{s}k\text{-sMod}_{*}$, a constant bi-graded simplicial object. If we call $Q$ the cofibrant replacement functor in $\text{s}k\text{-sMod}_{*}$, entrywise that can be obtained from the cofibrant replacement functor on $\text{s}k\text{-Mod}$. We want:

[TABLE]

a weak equivalence for any cofibrant object $X$ in $\text{s}k\text{-sMod}_{*}$. That means the entries of $X$ are cofibrant in $\text{s}k\text{-Mod}$ as well, and this latter being a symmetric monoidal category ([TV], [TV4]) we have $Qk_{*}\otimes X_{ip}\rightarrow k_{*}\otimes X_{ip}$ cofibrations for $i,p=0,1$, all of which recombine into (8), a cofibration, and this for all cofibrant $X$ in $\text{s}k\text{-sMod}_{*}$. Finally, $\boxtimes_{k}$ being symmetric in $k\text{-sMod}_{*}$, so is its prolongation to $\text{s}k\text{-sMod}_{*}$. Thus we have proved:

Theorem 3.1.0.1**.**

$\text{s}k\text{-sMod}_{*}$ is a symmetric monoidal model category.

3.2 Assumption 1

This is referred to as Assumption 1.1.0.1. in [TV4]. A homotopical algebraic context has 6 assumptions, we use the first four, that is why we call our theory as being based on a pre-homotopical context. In what follows, we use the fact that $\text{s}k\text{-Mod}$ satisfies our four assumptions ([SS]).

Assumption 1 3.2.1**.**

$\text{s}k\text{-sMod}_{*}$ is proper, pointed, and for any $X,Y\in\text{s}k\text{-sMod}_{*}$, the natural morphism:

[TABLE]

is an equivalence. Further $\text{Ho}\,(\text{s}k\text{-sMod}_{*})$ is an additive category.

Proof.

$\text{s}k\text{-Mod}$ is proper and pointed, hence so is $\text{s}k\text{-sMod}_{*}$. Let $X,Y\in\text{s}k\text{-sMod}_{*}$ we show the map $QX\coprod QY\rightarrow X\coprod Y\rightarrow RX\times RY$ is a natural equivalence. This composition decomposes levelwise into:

[TABLE]

which itself decomposes as follows:

[TABLE]

thus entrywise, for $p,q$ fixed:

[TABLE]

This however is an equivalence since $\text{s}k\text{-Mod}$ is assumed to satisfy this assumption. All such maps recombine into (9), which is therefore an equivalence. Hence the assumption holds in $\text{s}k\text{-sMod}_{*}$. Finally $\text{Ho}\,(\text{s}k\text{-sMod}_{*})$ is an additive category since this is true of $\text{Ho}\,(\text{s}k\text{-Mod})$. ∎

3.3 Assumption 2

Assumption 2 3.3.1**.**

Let $\mathcal{C}=\text{s}k\text{-sMod}_{*}$ and $A\in\text{Comm}(\mathcal{C})=\text{s}k\text{-sAlg}_{*}$. Define a morphism in $A\text{-sMod}_{*}$ to be a fibration or an equivalence if it is so on the underlying objects of $\mathcal{C}$. With this notion, $A\text{-sMod}_{*}$ is a combinatorial, proper model category, on which a tensor product $X\tilde{\boxtimes}_{A}Y$ is defined as $\oplus_{i,p}X_{ip}\otimes_{A_{ip}}Y_{ip}$. With the monoidal structure defined by $-\tilde{\boxtimes}_{A}-$, $A\text{-sMod}_{*}$ is a symmetric monoidal model category.

Proof.

We first show properness, that is, left-properness and right-properness. Starting with left-properness, we must show any pushout of a weak equivalence along a cofibration is a weak equivalence. Consider any such pushout in $A\text{-sMod}_{*}$:

[TABLE]

those maps being defined on the underlying objects in $\text{s}k\text{-sMod}_{*}$ we have that same diagram in $\text{s}k\text{-sMod}_{*}$, which is proper, so $U\rightarrow V$ is a weak equivalence in $\text{s}k\text{-sMod}_{*}$, hence so it is in $A\text{-sMod}_{*}$. We show right properness in like manner.

Regarding being combinatorial, we first show $A\text{-sMod}_{*}$ is cofibrantly generated. Since we will use the notations of [Ho] where that concept is discussed, we remind the reader about those notations: recall that if $M$ is a model category, to say that $M$ is cofibrantly generated means there are small sets $I$ and $J$ (again not worrying about universe considerations) of morphisms in $M$, and a small cardinal $\alpha$ such that:

domains and codomains of maps in $I$ and $J$ are $\alpha$-small. 2. 2.

the class of fibrations is $J$-inj. 3. 3.

the class of trivial fibrations is $I$-inj.

See [Ho] for the relation between $I$ and $J$. Since $A\text{-Mod}$ is cofibrantly generated for $A\in\text{s}k\text{-CAlg}$, it follows $A_{ip}\text{-Mod}_{iq}$, $i,p,q=0,1$ is cofibrantly generated for $A=(A_{ip})\in\text{s}k\text{-sAlg}_{*}$, with small sets $I_{ipq}$ and $J_{ipq}$ such that domains and codomains of maps in those sets are $\alpha_{ipq}$-small for some $\alpha_{ipq}$. Then:

[TABLE]

where each summand is of the form $A\text{-Mod}$ for some $A\in\text{s}k\text{-CAlg}$, cofibrantly generated. Then define $I=(I_{ipq})$, $J=(J_{ipq})$, with $\alpha=\text{max}\{\alpha_{ipq}\}$, with $I$-inj the fibrations in $A\text{-sMod}_{*}$, $J$-inj the trivial fibrations in $A\text{-sMod}_{*}$, thereby cofibrantly generated. Finally recall from [TV] for example, that a category $\mathcal{C}$ is locally presentable if there exists a small set of $\alpha$-small objects $\mathcal{C}_{0}\subset\mathcal{C}$ for some cardinal $\alpha$, such that any object in $\mathcal{C}$ is an $\alpha$-filtered colimit of objects in $\mathcal{C}_{0}$. It is clear a bi-graded category is locally presentable if it is so entry-wise. Then a combinatorial model category is a cofibrantly generated model category whose underlying category is locally presentable. Since for $A\in\text{s}k\text{-CAlg}$, $A\text{-Mod}$ is combinatorial, in particular it is locally presentable, so is $A\text{-sMod}_{*}$ for $A\in\text{s}k\text{-sAlg}_{*}$, and being cofibrantly generated as well, it is therefore combinatorial.

We now show $A\text{-sMod}_{*}$ is a symmetric monoidal model category. We first prove $-\tilde{\boxtimes}_{A}-$ gives a monoidal structure on $A\text{-sMod}_{*}$. $A\text{-Mod}$ has an internal hom $\underline{\text{Hom}}_{A}$ for $A\in\text{s}k\text{-CAlg}$:

[TABLE]

Now for $A\in\text{s}k\text{-sAlg}_{*}$, $M,N,P\in A\text{-sMod}_{*}$, we have:

[TABLE]

so there is an internal hom $\underline{\text{Hom}}_{A}$ in $A\text{-sMod}_{*}$, with

[TABLE]

So far we have half of an adjunction of two variables:

[TABLE]

Recall that we have the braiding $\sigma:M\boxtimes N\rightarrow N\boxtimes M$ in $\text{s}k\text{-sMod}_{*}$ defined entry-wise, so accordingly we also have a braiding operator $\sigma_{A}:M\tilde{\boxtimes}_{A}N\rightarrow N\tilde{\boxtimes}_{A}M$. We have:

[TABLE]

so $(\tilde{\boxtimes}_{A},\underline{\text{Hom}}_{A},\underline{\text{Hom}}_{A},\phi_{r},\phi_{r}\circ\sigma_{A}^{*})$ is an adjunction of two variables, hence we have a closed monoidal structure on $A\text{-sMod}_{*}$ making it into a closed monoidal category.

We now show $\tilde{\boxtimes}_{A}$ is a Quillen bifunctor. Write $\mathcal{C}=A\text{-sMod}_{*}$, let $f:U\rightarrow V$ and $\mathcal{C}$, $g:W\rightarrow X$ be cofibrations in $\mathcal{C}$. We need:

[TABLE]

to be a cofibration in $\mathcal{C}$, trivial if either of $f$ or $g$ is. This decomposes entry-wise into:

[TABLE]

for $i,p=0,1$, to be a cofibration in $\mathcal{C}$, trivial if either of $f$ or $g$ is. This follows from the same computation done when showing that $\boxtimes$ was a Quillen bifunctor on $\text{s}k\text{-sMod}_{*}$.

Now $A$ being the unit in $A\text{-sMod}_{*}$ for $\tilde{\boxtimes}_{A}$, $QA\rightarrow A$ a cofibrant replacement, for any cofibrant object $X$, we show $QA\tilde{\boxtimes}_{A}X\rightarrow A\tilde{\boxtimes}_{A}X$ is a weak equivalence. This follows from the definition of $\tilde{\boxtimes}_{A}$ and the fact that for $A\in\text{s}k\text{-CAlg}$, $X$ cofibrant in $\text{s}k\text{-Mod}$, $QA\otimes_{A}X\rightarrow A\otimes_{A}X$ is a weak equivalence. Finally, since $M\tilde{\boxtimes}_{A}N=\oplus_{i,p}M_{ip}\otimes_{A_{ip}}N_{ip}$ and $-\otimes_{A}-$ defines a symmetric monoidal structure on $A\text{-Mod}$, for $A\in\text{s}k\text{-CAlg}$, it follows that $-\tilde{\boxtimes}_{A}-$ defines a symmetric monoidal structure on $A\text{-sMod}_{*}$ as well for $A\in\text{s}k\text{-sAlg}_{*}$. This completes the proof. ∎

3.4 Assumption 3

Assumption 3 3.4.1**.**

For $A\in\text{Comm}(\mathcal{C})=\text{s}k\text{-sAlg}_{*}$, for any $M$ fibrant in $A\text{-sMod}_{*}$, the functor $-\tilde{\boxtimes}_{A}M:A\text{-sMod}_{*}\rightarrow A\text{-sMod}_{*}$ preserves equivalences.

Proof.

For $M$ fibrant in $A\text{-sMod}_{*}$, writing $1$ for the terminal object of $A\text{-sMod}_{*}$, $M\rightarrow 1$ being a fibration, we have $M_{ip}\rightarrow 1_{ip}$ a fibration in $\text{s}k\text{-Mod}$ for $i,p=0,1$, i.e. $M_{ip}$ fibrant in $A_{ip}\text{-Mod}$ for $i,p=0,1$. Let $f$ be an equivalence in $A\text{-sMod}_{*}$, $f=(f_{ip})$, $f_{ip}$ a weak equivalence in $A_{ip}\text{-Mod}$ for $i,p=0,1$. We know for $A\in\text{s}k\text{-CAlg}$, $M\in A\text{-Mod}$, $-\otimes_{A}M$ preserves weak equivalences, so having $M_{ip}$ fibrant and $f_{ip}$ an equivalence, $f_{ip}\otimes_{A_{ip}}M_{ip}$ is a weak equivalence for $i,p=0,1$, hence so is $f\tilde{\boxtimes}_{A}M$ for $M\in A\text{-sMod}_{*}$. ∎

3.5 Assumption 4

Assumption 4 3.5.1**.**

For $A\in\text{Comm}(\mathcal{C})=\text{s}k\text{-sAlg}_{*}$, there exist categories $A\text{-Comm}(\mathcal{C})$ and $A-\text{Comm}_{nu}(\mathcal{C})$ (non-unital) whose morphisms are fibrations and equivalences if they are so on the underlying objects of $\mathcal{C}$. This makes those categories into combinatorial proper model categories. If $B$ is cofibrant in $A\text{-Comm}(\mathcal{C})$, the functor $B\tilde{\boxtimes}_{A}-:A\text{-sMod}_{*}\rightarrow B\text{-sMod}_{*}$ preserves equivalences.

Proof.

Given how fibrations and equivalences are defined in those categories, proving they are combinatorial, proper model categories is done exactly the same way we proved those statements for $A\text{-sMod}_{*}$. Proving that $B\tilde{\boxtimes}_{A}-$ preserves equivalences for $B$ cofibrant is done in the same manner that we proved $-\tilde{\boxtimes}_{A}C$ preserves equivalences for $C$ fibrant in $A\text{-sMod}_{*}$. ∎

4 Bi-graded derived algebraic stacks

4.1 Finitely presented morphisms

Let $\mathcal{C}=\text{s}k\text{-sMod}_{*}$, so that $\text{Comm}(\mathcal{C})=\text{s}k\text{-sAlg}_{*}=M$, our model category of interest. We will need $M$ to be proper, but weak equivalences are defined on the underlying objects of $\text{s}k\text{-sMod}_{*}$, and it being proper, so is $M$. We fix a bi-graded cofibrant resolution functor $(\Gamma:M\rightarrow M^{\Delta},\iota)$, $\Gamma=(\Gamma_{ip})$ ([TV]), with $\Gamma_{ip}:\underline{M}\rightarrow\underline{M}^{\Delta}$ cofibrant resolution functors in $\text{s}k\text{-Mod}$ for $i,p=0,1$ where as usual the underlined object corresponds to an ungraded counterpart. Here $M=\text{Comm}(\text{s}k\text{-sMod}_{*})$, so $\underline{M}=\text{Comm}(\underline{\text{s}k\text{-sMod}_{*}})=\text{Comm}(\text{s}k\text{-Mod})=\text{s}k\text{-CAlg}$. We have $c^{*}=(c^{*})_{2X2}$, with weak equivalences: $\Gamma_{ip}(A)\xrightarrow{\iota_{ip}(A)}c^{*}A$ for $i,p=0,1$, with $c^{*}$ the constant cosimplicial object functor. Since we consider bi-graded morphisms in $M$, we have:

[TABLE]

so we define $\text{Map}_{\underline{M}}=\text{Map}_{\underline{M},0}\oplus\text{Map}_{\underline{M},1}$ so that we have:

[TABLE]

Now let $A\rightarrow B$ be a morphism in $M=\text{s}k\text{-sAlg}_{*}$. Consider any filtered diagram of objects under $A$, $\{C_{i}\}_{i\in I}\in A/M$. We have a natural morphism:

[TABLE]

and those objects are defined in $\text{Ho}\,(S_{*\centerdot})$. This shows a morphism in $M$ is finitely presented if and only if it is so in $\underline{M}=\text{s}k\text{-CAlg}$.

4.2 Bi-graded derivations and cotangent complexes

For $\mathcal{C}=\text{s}k\text{-sMod}_{*}$, $A\in\text{Comm}(\mathcal{C})$, $M\in$ $A\text{-Comm}(\mathcal{C})$, we define a commutative monoid $A\oplus M$ with $A\coprod M$ as underlying object, just as in [TV4], with multiplication:

[TABLE]

defined by:

[TABLE]

where:

[TABLE]

Here $\mu$ and $\rho$ have entries as defined in [TV4] and where in the last line we use the fact that $M$ is a commutative monoid in $A\text{-sMod}_{*}$. To show $A\oplus M$ is commutative, it suffices to write:

[TABLE]

Then:

[TABLE]

Now $A\in k\text{-sMod}_{*}$ is super-commutative if $\mu\circ\sigma=\mu$, that is for any homogeneous elements $x,y$ in $A$, $xy=\mu(x\otimes y)=(-1)^{|x||y|}yx$. Presently we have $\pi$ on:

[TABLE]

and on each entry $\pi$ is super-commutative, so $A\oplus M$ is super-commutative.

Now for $A\rightarrow B$ a morphism in $\text{Comm}(\mathcal{C})$, $M\in$ $B\text{-Comm}(\mathcal{C})$, the morphism $id\coprod*:B\oplus M\rightarrow B$ is a morphism in $A\text{-Comm}(\mathcal{C})$, hence we can regard $B\oplus M\in A\text{-Comm}(\mathcal{C})/B$.

Definition 4.2.1**.**

For a morphism $A\rightarrow B$ in $\text{Comm}(\mathcal{C})$, $M\in B\text{-Comm}(\mathcal{C})$, $\mathcal{C}=\text{s}k\text{-sMod}_{*}$, we define, as in [TV4], the set of $A$-derivations $B\rightarrow M$ as:

[TABLE]

where $B\rightarrow B\oplus M$ is a section of $id\coprod*$.

We have the following result, much as in [TV4]:

Proposition 4.2.2**.**

For any morphism $A\rightarrow B$ in $\text{s}k\text{-sAlg}_{*}$, there is an object $\mathbb{L}_{B/A}\in B\text{-sMod}_{*}$, there is a $d\in\pi_{0}(\mathbb{D}\text{er}_{A}(B,\mathbb{L}_{B/A}))$ such that for any $M=(M_{ip})$, $M_{ip}\in B_{ip}\text{-Mod}$, i.e. $M\in B\text{-sMod}_{*}$, the natural induced morphism:

[TABLE]

is an isomorphism in $\text{Ho}\,(S_{*\centerdot})$.

Note that for $\psi\in\text{Map}_{B\text{-sMod}_{*}}(\mathbb{L}_{B/A},M)$, the element $d^{*}\psi$ is constructed as follows:

[TABLE]

Proof.

A morphism $A\rightarrow B\in\text{Comm}(\mathcal{C})$ gives rise to entry maps $A_{ip}\rightarrow B_{ip}\in\text{Comm}(\underline{\mathcal{C}})$, with $\underline{\mathcal{C}}=\text{s}k\text{-Mod}$, for $i,p=0,1$, so we are guaranteed by [TV4] that there is some $\mathbb{L}_{B_{ip}/A_{ip}}\in B_{ip}\text{-Mod}$ and a $d_{ip}\in\pi_{0}(\mathbb{D}\text{er}_{A_{ip}}(B_{ip},\mathbb{L}_{B_{ip}/A_{ip}}))$ such that for any $M_{ip}\in B_{ip}\text{-Mod}$:

[TABLE]

is an isomorphism in $\text{Ho}\,(\text{Set}_{\Delta})$. We view $\mathbb{L}_{B/A}=(\mathbb{L}_{B_{ip}/A_{ip}})$ as an object of $(B_{ip}\text{-Mod}_{ip})\subset B\text{-sMod}_{*}$, and the same is true of $M=(M_{ip})$. Then it suffices to write:

[TABLE]

with $d=(d_{ip})\in\pi_{0}(\mathbb{D}\text{er}_{A}(B,\mathbb{L}_{B/A}))$. ∎

Definition 4.2.3**.**

Let $A\rightarrow B$ a morphism in $\text{Comm}(\text{s}k\text{-sMod}_{*})$. The $B$-module $\mathbb{L}_{B/A}\in\text{Ho}\,(B\text{-sMod}_{*})$ is called as in [TV4] the cotangent complex of $B$ over $A$. If $A=1=\Delta k_{*}$, write $\mathbb{L}_{B}=\mathbb{L}_{B/1}$, which we will call the cotangent complex of $B$.

4.3 étale morphisms

We have shown that $A\text{-sMod}_{*}$ is a symmetric monoidal model category with the tensor product $\tilde{\boxtimes}_{A}$. This means its homotopy category $\text{Ho}\,(A\text{-sMod}_{*})$ has a natural symmetric monoidal structure $\tilde{\boxtimes}_{A}^{\mathbb{L}}$.

Definition 4.3.1**.**

Let $f:A\rightarrow B$ be a morphism in $\text{Comm}(\text{s}k\text{-sMod}_{*})=\text{s}k\text{-sAlg}_{*}$. The morphism $f$ is said to be formally étale if:

[TABLE]

or entry-wise:

[TABLE]

for $i,p=0,1$, i.e. $f$ is formally étale if it is so entry-wise.

Note that this is a well-defined morphism: if $p=0$, the parity of $\mathbb{L}_{A_{ip}}\otimes_{A_{ip}}B_{ip}$ is $p+p+p=0$, so we do have $\mathbb{L}_{A_{i0}}\otimes_{A_{i0}}B_{i0}\rightarrow\mathbb{L}_{B_{i0}}$, and if $p=1$, the parity of $\mathbb{L}_{A_{i1}}\otimes_{A_{i1}}B_{i1}$ is 3 mod 2, which is 1, so we also have a well-defined map $\mathbb{L}_{A_{i1}}\otimes_{A_{i1}}B_{i1}\rightarrow\mathbb{L}_{B_{i1}}$.

Definition 4.3.2**.**

We define a morphism of commutative monoids in $\text{s}k\text{-sMod}_{*}$ to be étale if it is finitely presented and formally étale.

This is the same definition as initially presented in [TV4]. Following the same reference, we will use the same terminology for morphisms in $\text{Ho}\,(Comm(\mathcal{C}))$ and for the corresponding morphisms of representable stacks. We typically write $\mathcal{A}\text{ff}_{\mathcal{C}}=(Comm(\mathcal{C}))^{\text{op}}$ but if $\mathcal{C}=\text{s}k\text{-sMod}_{*}$, $Comm(\mathcal{C})=\text{s}k\text{-sAlg}_{*}$, $(\text{s}k\text{-sAlg}_{*})^{\text{op}}$ is then denoted, much as in [TV4], $k\text{-D}^{-}\text{sAff}_{*}$.

If $\mathcal{C}=\text{s}k\text{-sAlg}_{*}\subset\text{s}k\text{-sMod}_{*}$, $A\in\mathcal{C}$, we define the underlying bi-graded space of $A$ as:

[TABLE]

so that following [TV4], we define:

[TABLE]

Write $\text{Spec\,}A\in k\text{-D}^{-}\text{sAff}_{*}$ if $A\in\text{s}k\text{-sAlg}_{*}$. Following [TV4], we define families $\{\text{Spec\,}A_{i}\rightarrow\text{Spec\,}A\}_{i\in I}$ to be étale covering families if and only if there is some $J\subset I$ such that for all $i\in I$:

[TABLE]

and

[TABLE]

is étale and surjective, where $\text{Spec\,}A\rightarrow\text{Spec\,}B$ is étale if $B\rightarrow A$ is an étale morphism in $\text{s}k\text{-sAlg}_{*}$.

Note that (11) reads as follows entry-wise:

[TABLE]

Now if $A\in k\text{-sAlg}_{*}$:

[TABLE]

along with $\pi_{0}A=(\pi_{0}A_{ip})$, this gives us, for $A\in\text{s}k\text{-sAlg}_{*}$:

[TABLE]

It follows:

[TABLE]

so that entrywise:

[TABLE]

However, finitely presented morphisms in $\text{s}k\text{-sAlg}_{*}$ are so entry-wise, and given the definition of formally étale morphisms and the definition of $\tilde{\boxtimes}_{A}$, formally étale maps are so entry-wise as we argued. It follows étale morphisms in $\text{s}k\text{-sMod}_{*}$ are so entry-wise, so the above morphism is entry-wise étale. We conclude that a family of coverings in $k\text{-D}^{-}\text{sAff}_{*}$ is an étale covering if and only if it is so entry-wise in $\text{k-D}^{-}\text{Aff}$.

4.4 Descent on super modules

We now define the descent condition on supermodules, which is just an adaption to our setting of the work done in [TV4]. We define a cosimplicial object $A_{*}$ in $Comm(\mathcal{C})$, $\mathcal{C}=k\text{-sMod}_{*}$, to be an object of $Comm(\mathcal{C})^{\Delta}$, which to $[n]$ associates $A_{n}=(A_{n,ip})$. A cosimplicial $A_{*}$-module $M_{*}$ is given by a graded $A_{n}$-supermodule $M_{n}$ for all $n\in\Delta$, $M_{n}=(M_{n,ip})$, and for any morphism $u:[n]\rightarrow[m]$ in $\Delta$ of a morphism of graded $A_{n}$-supermodules $\alpha_{u}:M_{n}\rightarrow M_{m}$ satisfying the usual covariance condition. In the same manner, a morphism of cosimplicial graded $A_{*}$-supermodules $f:M_{*}\rightarrow N_{*}$ is given by a collection of morphisms $f_{n}:M_{n}\rightarrow N_{n}$ for all $n\in\Delta$ commuting with the $\alpha$’s. This defines a category $\text{cs}A_{*}\text{-sMod}_{*}$ of co-simplicial graded $A_{*}$-supermodules. This is a combinatorial, proper model category where equivalences (resp. fibrations) are morphisms $f:M_{*}\rightarrow N_{*}$ such that each $f_{n}:M_{n}\rightarrow N_{n}$ is an equivalence (resp. a fibration) in $A_{n}$-sMod∗, a level-wise projective model structure. It will also be convenient sometimes, for $A\in k\text{-sAlg}_{*}$, to regard $B_{*}$ a co-simplicial commutative graded $A$-superalgebra as a co-simplicial commutative monoid with a co-augmentation morphism $A\rightarrow B_{*}$, where we regard $A$ as a constant co-simplicial object. This latter defines a category $\text{cs}A\text{-sMod}_{*}$ of co-simplicial graded cc($A$)-supermodules, which is really cs($A\text{-sMod}_{*}$), along with its levelwise projective model structure.

For a co-simplicial graded $A$-super module $M_{*}$, we define a co-simplicial graded $B_{*}$-supermodule $B_{*}\tilde{\boxtimes}_{A}M_{*}$ by $(B_{*}\tilde{\boxtimes}_{A}M_{*})_{n}=B_{n}\tilde{\boxtimes}_{A}M_{n}$, with morphisms between different degrees given by the ones on $B_{*}$ and $M_{*}$. This gives a functor:

[TABLE]

with a right adjoint:

[TABLE]

sending a graded $B_{*}$-super module $M_{*}$ to its underlying graded $A$-super module, clearly a Quillen adjunction. We have an additional adjunction, as in [TV4]:

[TABLE]

and by composition:

[TABLE]

Definition 4.4.1**.**

Let $B_{*}$ be a co-simplicial commutative graded super monoid, $M_{*}$ a co-simplicial graded $B_{*}$ super module. We say $M_{*}$ is homotopy cartesian if for all $u:[n]\rightarrow[m]$ in $\Delta$, the map induced by $\alpha_{u}:M_{n}\rightarrow M_{m}$, $M_{n}\boxtimes_{B_{n}}^{\mathbb{L}}B_{m}\rightarrow M_{m}$ is an isomorphism in $\text{Ho}\,(B_{m}$-sMod∗).

Now if $A\in Comm(\mathcal{C})$, $\mathcal{C}=k\text{-sMod}_{*}$, $B_{*}$ co-simplicial commutative algebra over $A$, seen as a co-simplicial commutative graded super monoid, we say the augmentation $A\rightarrow B_{*}$ satisfies the descent condition if in the adjunction:

[TABLE]

$B_{*}\tilde{\boxtimes}_{A}^{\mathbb{L}}-$ is fully faithful and gives an equivalence between $\text{Ho}\,(A\text{-sMod}_{*})$ and the full subcategory of $\text{Ho}\,(\text{cs}B_{*}\text{-sMod}_{*})$ spanned by homotopy cartesian objects.

4.5 étale model topology

In this subsection we prove that étale covering families define a model topology on $k\text{-D}^{-}\text{sAff}_{*}$, which further satisfies an assumption that we will refer to as the ”Cover” assumption, the bi-graded generalization of Assumption 1.3.2.2 of [TV4].

4.5.1 Model topology

Recall that a family of morphisms $\{\text{Spec\,}A_{i}\rightarrow\text{Spec\,}A\}_{i\in I}$ in $k\text{-D}^{-}\text{sAff}_{*}$ is an étale covering family if and only if there is a finite subset $J\subset I$ such that for all $i\in I$, $\pi_{*}A\tilde{\boxtimes}_{\pi_{0}A}\pi_{0}A_{i}\xrightarrow{\cong}\pi_{*}A_{i}$ and $\coprod_{j\in J}\text{Spec\,}\pi_{0}A_{j}\rightarrow\text{Spec\,}\pi_{0}A$ is étale and surjective. Recall that being étale means finitely presented and formally étale, where $A\rightarrow B$ is formally étale if $\mathbb{L}_{A}\tilde{\boxtimes}_{A}B\xrightarrow{\cong}\mathbb{L}_{B}$, and finitely presented means for any familiy $\{C_{i}\}$, $\text{hocolim}\,_{i\in I}\text{Map}_{A/M}(B,C_{i})\xrightarrow{\cong}\text{Map}_{A/M}(B,\text{hocolim}\,C_{i})$. Recall also that a family is a covering étale family if and only if it is so entry-wise. We will use the fact, proven in [TV4], that étale covering families define a model topology in $\text{k-D}^{-}\text{Aff}=(\text{s}k\text{-CAlg})^{\text{op}}$. We refer the reader to [TV] for the notion of model topology. There are three conditions to be met. Let $M=k\text{-D}^{-}\text{sAff}_{*}$, let $\text{Spec\,}A\in M$. Suppose we have an isomorphism $\text{Spec\,}B\rightarrow\text{Spec\,}A$. Entry-wise, this reads $\text{Spec\,}B_{ip}\rightarrow\text{Spec\,}A_{ip}$ isomorphism for $i,p=0,1$ in $\text{k-D}^{-}\text{Aff}$, which gives $\{\text{Spec\,}B_{ip}\rightarrow\text{Spec\,}A_{ip}\}\in\text{cov}_{\text{\'{e}t}}(\text{Spec\,}A_{ip})$ for $i,p=0,1$. But since $\text{cov}_{\text{\'{e}t}}(X)=(\text{cov}_{\text{\'{e}t}}(X_{ip}))$, it follows that condition 1 is met. For the second point, let $\{\text{Spec\,}A_{i}\rightarrow\text{Spec\,}A\}_{i\in I}\in\text{cov}_{\text{\'{e}t}}(\text{Spec\,}A)$. This means $\{\text{Spec\,}A_{i,kp}\rightarrow\text{Spec\,}A_{kp}\}_{i\in I}\in\text{cov}_{\text{\'{e}t}}(\text{Spec\,}A_{kp})$ for $k,p=0,1$. Let $\{\text{Spec\,}B_{ij}\rightarrow\text{Spec\,}A_{i}\}_{j\in J}\in\text{cov}_{\text{\'{e}t}}(\text{Spec\,}A_{i})$ for $i\in I$, and again that means $\{\text{Spec\,}B_{ij,kp}\rightarrow\text{Spec\,}A_{i,kp}\}_{j\in J}\in\text{cov}_{\text{\'{e}t}}(\text{Spec\,}A_{i,kp})$ for $k,p=0,1$. Then $\{\text{Spec\,}B_{ij,kp}\rightarrow\text{Spec\,}A_{kp}\}_{i\in I,j\in J}\in\text{cov}_{\text{\'{e}t}}(\text{Spec\,}A_{kp})$ for $k,p=0,1$, i.e. $\{\text{Spec\,}B_{ij}\rightarrow\text{Spec\,}A\}_{i\in I,j\in J}\in\text{cov}_{\text{\'{e}t}}(\text{Spec\,}A)$. This is the second condition. Third point, for any $\{\text{Spec\,}A_{i}\rightarrow\text{Spec\,}A\}_{i\in I}\in\text{cov}_{\text{\'{e}t}}(\text{Spec\,}A)$, that is $\{\text{Spec\,}A_{i,jp}\rightarrow\text{Spec\,}A_{jp}\}_{i\in I}\in\text{cov}_{\text{\'{e}t}}(\text{Spec\,}A_{jp})$ for $j,p=0,1$, for any morphism $\text{Spec\,}B\rightarrow\text{Spec\,}A$, which entry-wise reads $\text{Spec\,}B_{jp}\rightarrow\text{Spec\,}A_{jp}$, then $\{\text{Spec\,}A_{i,jp}\times_{\text{Spec\,}A_{jp}}^{h}\text{Spec\,}B_{jp}\rightarrow\text{Spec\,}B_{jp}\}\in\text{cov}_{\text{\'{e}t}}(\text{Spec\,}B_{jp})$ for $j,p=0,1$, which recombine into $\{\text{Spec\,}A_{i}\times_{\text{Spec\,}A}^{h}\text{Spec\,}B\rightarrow\text{Spec\,}B\}\in\text{cov}_{\text{\'{e}t}}(\text{Spec\,}B)$. Hence étale covering families define a model topology on $k\text{-D}^{-}\text{sAff}_{*}$.

Now part of showing the cover assumption is satisfied hinges on the fact that $(s\text{Aff}_{\mathcal{C}})_{*}$ is a simplicial model category on which we can put a Reedy model structure.

4.5.2 Reedy model structure on $s\text{Aff}_{\mathcal{C}}$

$s\text{Aff}_{\mathcal{C}}=\mathcal{A}\text{ff}_{\mathcal{C}}^{\Delta^{\text{op}}}$ with $\mathcal{A}\text{ff}_{\mathcal{C}}=k\text{-D}^{-}\text{sAff}_{*}=(\text{s}k\text{-sAlg}_{*})^{\text{op}}$, is a simplicial category with $\mathcal{C}=\text{s}k\text{-sMod}_{*}$, thus it is cotensored over $\text{Set}_{\Delta}$ so for $X_{*}\in s\text{Aff}_{\mathcal{C}}$, $K\in\text{Set}_{\Delta}$, $X_{*}^{K}$ is well defined, and its zeroth part will just be denoted $X^{K}$. We have in particular:

[TABLE]

Indeed, for $X_{*},Y_{*}\in s\text{Aff}_{\mathcal{C}}$ and $K\in\text{Set}_{\Delta}$:

[TABLE]

since $s\text{Aff}_{\underline{\mathcal{C}}}$ is a simplicial category. This further recombines into:

[TABLE]

We also have:

[TABLE]

Thus the simplicial structure on $s\text{Aff}_{\mathcal{C}}$ is an entry-wise generalization of that on $s\text{Aff}_{\underline{\mathcal{C}}}$. One could also show that it is a bi-graded simplicial category and this is proved formally in exactly the same manner that we showed $\text{s}k\text{-sMod}_{*}$ is a bi-graded simplicial category. Then for the bi-graded simplicial structure, we have $X_{*}^{K}=\oplus_{i,p,q}K_{*,ip}^{K_{iq}}$. If $K=\Delta^{n}\times\Delta^{n}$, with $\Delta^{n}$ regarded as being of even parity, we then have:

[TABLE]

for the simplicial structure. For $Y\in\mathcal{A}\text{ff}_{\mathcal{C}}\hookrightarrow s\text{Aff}_{\mathcal{C}}$ regarded as a constant simplicial object via the constant simplicial functor cs${}_{*}(Y)$, with cs${}_{*}(Y)_{n}=Y$ for any $n$, when we write $Y^{K}$ we really mean $(\text{cs}_{*}(Y))^{K}$.

We now put a Reedy structure on $s\text{Aff}_{\mathcal{C}}$. Equivalences $X_{*}\rightarrow Y_{*}$ are such that $X_{n}\rightarrow Y_{n}$ are equivalences in $\mathcal{A}\text{ff}_{\mathcal{C}}$ for any $n$, and fibrations $X_{*}\rightarrow Y_{*}$ are those maps such that for any $n$:

[TABLE]

is a fibration in $\mathcal{A}\text{ff}_{\mathcal{C}}$. Entry-wise this decomposes into $X_{ip}^{\Delta^{n}}\rightarrow X_{ip}^{\partial\Delta^{n}}\times_{Y_{ip}^{\partial\Delta^{n}}}Y_{ip}^{\Delta^{n}}$, fibration in $\text{Aff}_{\underline{\mathcal{C}}}$ for $i,p=0,1$, i.e. the Reedy model structure on $s\text{Aff}_{\mathcal{C}}$ is an entry-wise Reedy model structure on $s\text{Aff}_{\underline{\mathcal{C}}}$. Observe that for $K\in\text{Set}_{\Delta}$, the functor:

[TABLE]

is a right Quillen functor for the Reedy model structure on $s\text{Aff}_{\mathcal{C}}$.

Indeed it decomposes into $X_{*}=(X_{*,ip})\mapsto(X_{ip}^{K})$, right Quillen in the Reedy model structure on $s\text{Aff}_{\underline{\mathcal{C}}}$ entry-wise, recombining into a right Quillen functor in $s\text{Aff}_{\mathcal{C}}$. Its right derived functor is given by:

[TABLE]

4.5.3 The Cover Assumption

This assumption, which mirrors Assumption 1.3.2.2 of [TV4], is just the bi-graded version of the latter, so the reader is referred to that reference for the full statement. Essentially we assume the étale topology on $\text{Ho}\,(k\text{-D}^{-}\text{sAff}_{*})$ is quasi-compact, projections onto coproducts form étale-covering families, both of which are fairly immediate to verify, and thirdly if $X_{*}\rightarrow Y$ is an augmented simplicial object in $\mathcal{A}\text{ff}_{\mathcal{C}}$ corresponding to a co-augmented co-simplicial object $A\rightarrow B_{*}$ in $Comm(\mathcal{C})$, if we assume that for all $n$ the morphism:

[TABLE]

by itself forms an étale covering family in $\mathcal{A}\text{ff}_{\mathcal{C}}$, then $A\rightarrow B_{*}$ satisfies descent. We discuss this third point.

We have:

[TABLE]

with this the map (12) reads entry-wise:

[TABLE]

for $i,p=0,1$. Now since étale covering families are so entry-wise, those maps correspond to étale covering families in $\text{Aff}_{\underline{\mathcal{C}}}$, so the maps $A_{ip}\rightarrow B_{*,ip}$ satisfy descent by the original ungraded cover assumption of [TV4].

We show this implies they can be recombined into $A\rightarrow B_{*}$ that also satisfies descent. This will imply our bi-graded cover assumption is just a diagonalization of the original cover assumption of [TV4].

Recall that to satisfy descent means

[TABLE]

is fully faithful and establishes an equivalence of $\text{Ho}\,(A\text{-sMod}_{*})$ with the subcategory of homotopy cartesian objects in $\text{Ho}\,(\text{cs}B_{*}\text{-sMod}_{*})$. That the entries $A_{ip}\rightarrow B_{*,ip}$ satisfy descent then means that $B_{*,ip}\otimes_{A_{ip}}^{\mathbb{L}}-:\text{Ho}\,(A_{ip}\text{-Mod})\xrightarrow{ff}\text{Ho}\,(\text{cs}B_{*,ip}\text{-Mod})$ for $i,p=0,1$. Those maps recombine into $B_{*}\tilde{\boxtimes}_{A}^{\mathbb{L}}-:\text{Ho}\,(A\text{-sMod}_{*})\rightarrow\text{Ho}\,(\text{cs}B_{*}\text{-sMod}_{*})$ fully faithful. Regarding homotopy cartesian objects, those are objects $M_{*}$ in $\text{Ho}\,(\text{cs}B_{*}\text{-Mod}_{*})$ such that $M_{n}\tilde{\boxtimes}^{\mathbb{L}}_{B_{n}}B_{m}\rightarrow M_{m}$ is an isomorphism in $\text{Ho}\,(B_{m}\text{-sMod}_{*})$. We quickly check $B_{*}\tilde{\boxtimes}_{A}^{\mathbb{L}}-$ maps objects of $\text{Ho}\,(A\text{-sMod}_{*})$ into cartesian objects of $\text{Ho}\,(\text{cs}B_{*}\text{-sMod}_{*})$: let $C_{*}\in\text{Ho}\,(A\text{-sMod}_{*})$. We know from [TV4] in the ungraded case that entry-wise:

[TABLE]

for $i,p=0,1$, since $B_{*}\otimes_{A}^{\mathbb{L}}-$ maps objects of $\text{Ho}\,(A\text{-Mod})$ into homotopy cartesian objects of $\text{Ho}\,(\text{cs}B_{*}\text{-Mod})$. Now we have:

[TABLE]

hence $B_{*}\tilde{\boxtimes}_{A}^{\mathbb{L}}-$ maps objects of $\text{Ho}\,(A\text{-sMod}_{*})$ into homotopy cartesian objects. The essential surjectivity is dealt with entry-wise. Thus the third point is satisfied.

To conclude this subsection on the model topology, the étale topology does define a model topology on $k\text{-D}^{-}\text{sAff}_{*}$ and makes $(k\text{-D}^{-}\text{sAff}_{*},\acute{e}t.)$ into a model site. Further this topology satisfies our technical cover assumption, bi-graded generalization of the cover assumption as originally introduced in [TV4].

4.6 Bi-graded simplicial presheaves

Let $M$ be a small bi-graded model category (the choice of universes $\mathbb{U}\in\mathbb{V}$ being implied, $M$ $\mathbb{V}$-small, $\mathbb{U}$-cofibrantly generated…), $W(M)$ its class of equivalences. For us $M=k\text{-D}^{-}\text{sAff}_{*}$. Define:

[TABLE]

the category of bi-graded simplicial presheaves on $M=(M_{ip})$. Equivalently:

[TABLE]

We put the entry-wise projective model structure on $\text{sPr}_{\mathbb{Z}^{2}_{2}}(M)$ where equivalences and fibrations are defined object-wise.

Our aim at this point is to define a model category of prestacks $M^{\wedge}$, obtained as a bi-graded left Bousfield localization of $\text{sPr}_{\mathbb{Z}^{2}_{2}}(M)$ along $\{h_{u}\,|\,u\in W(M)\}$ where $h:M\rightarrow\text{Pr}_{\mathbb{Z}_{2}}(M)\hookrightarrow\text{sPr}_{\mathbb{Z}^{2}_{2}}(M)$ is the bi-graded Yoneda embedding. In other terms we localize each individual $\text{sPr}(\underline{M})$, i.e. we do an entry-wise Bousfield localization. Then $\text{Ho}\,(M^{\wedge})$ is identified with the subcategory of $\text{Ho}\,(\text{sPr}_{\mathbb{Z}^{2}_{2}}(M))$ that consists of presheaves that preserve weak equivalences entry-wise. $M^{\wedge}$ is a bi-graded simplicial model category since $\text{sPr}(\underline{M})^{\wedge}=\text{sPr}(\text{k-D}^{-}\text{Aff})^{\wedge}$ is a simplicial model category from [TV], and we denote by $\mathbb{R}\underline{\text{Hom}}\in\text{Ho}\,(S_{*\centerdot})$ its derived hom. From there we define a model category $M^{\sim\,,\,\acute{e}t.}$ of stacks on $(M,\acute{e}t.)$, left bi-graded Bousfield localization of $M^{\wedge}$ along homotopy étale-hypercovers. We obtain a pair of adjoint Quillen functors:

[TABLE]

giving rise to:

[TABLE]

we finally define a bi-graded stack on $(k\text{-D}^{-}\text{sAff}_{*},\acute{e}t.)$ to be an object of $\text{sPr}_{\mathbb{Z}^{2}_{2}}(M)$ whose image in $\text{Ho}\,(M^{\wedge})$ is in the essential image of the functor $j$, exactly in the same way that derived stacks were defined in [TV4].

4.7 Bi-graded left Bousfield localization

The reference we use for Bousfield localizations is [Hi], which we follow closely since it is just a matter of adapting Hirschhorn’s definition to the bi-graded case. Let $M=(M_{ip})$ be a $\mathbb{Z}_{2}$-graded model category, such as $k\text{-D}^{-}\text{sAff}_{*}$. Let $\mathcal{C}$ be a class of maps in $M$, $\mathcal{C}=(\mathcal{C}_{ip})$. An object $W$ of $M$ is $\mathcal{C}$-local if it is fibrant and for any $f:A\rightarrow B$ in $\mathcal{C}$, or entry-wise $f_{ip}:A_{ip}\rightarrow B_{ip}$ in $\underline{\mathcal{C}}$ for $i,p=0,1$, the induced map of mapping spaces is a weak equivalence:

[TABLE]

or entry-wise:

[TABLE]

In other terms, $\mathcal{C}$-local objects are entry-wise $\underline{\mathcal{C}}$-local objects. We define a map $g:X\rightarrow Y$ in $M$ to be a $\mathcal{C}$-local equivalence if for any $\mathcal{C}$-local object $W$, the induced map of mapping spaces:

[TABLE]

is a weak equivalence, or equivalently entry-wise:

[TABLE]

for $i,p=0,1$, so again $\mathcal{C}$-local equivalences are entry-wise $\underline{\mathcal{C}}$-local equivalences.

Finally we define the bi-graded left Bousfield localization of $M$ with respect to $\mathcal{C}$ to be a model category structure $\text{L}_{\mathcal{C}}M$ on the underlying bi-graded category of $M$ such that the class of weak equivalences is the class of $\mathcal{C}$-local equivalences, cofibrations are those of $M$, and fibrations have the right lifting property with respect to those cofibrations that are also $\mathcal{C}$-local equivalences. If $A\rightarrow B$ is such a cofibration, $X\rightarrow Y$ a fibration, a diagram such as:

[TABLE]

breaks up into diagrams for $i,p=0,1$:

[TABLE]

with $A_{ip}\rightarrow B_{ip}$ a cofibration that’s also a $\underline{\mathcal{C}}$-local equivalence. This shows fibrations are also entry-wise fibrations, hence:

[TABLE]

4.8 Model category of prestacks $M^{\wedge}$

We first define restricted diagrams, following [TV]. For $M$ a small category (relative to $\mathbb{V}$), such as $k\text{-D}^{-}\text{sAff}_{*}$, given $S_{*\centerdot}$, bi-graded simplicial model category, cofibrantly generated, we consider $S_{*\centerdot}^{M}$ the category of bi-graded simplicial functors $M\rightarrow S_{*\centerdot}$ with its entry-wise projective model structure. For any $x\in M$ we have an induced map:

[TABLE]

with a left adjoint $(\iota_{x})_{!}:S_{*\centerdot}\rightarrow S_{*\centerdot}^{M}$ that’s a left Quillen functor. Let $I$ be a set of generating cofibrations in $S_{*\centerdot}$, $f:A\rightarrow B$ any morphism in $I$, $u:x\rightarrow y$ any morphism in $M$. Consider the natural morphism in $S_{*\centerdot}^{M}$:

[TABLE]

with $(\iota_{x})_{!jp}:S_{jp}\rightarrow S_{jp}^{M_{jp}}$, left adjoint to

[TABLE]

that is, $(\iota_{x}^{*})_{jp}=\iota_{x_{jp}}^{*}$, so that $(\iota_{x})_{!jp}=(\iota_{x_{jp}})_{!}$. Thus:

[TABLE]

which means $f\Box u=(f_{jp}\Box u_{jp})$, so that entry-wise:

[TABLE]

Definition 4.8.1**.**

The model category of restricted diagrams from $M$ to $S_{*\centerdot}$, denoted $M^{\text{op},\wedge}$, is defined to be the left Bousfield localization of $S_{*\centerdot}^{M}$ along the set of morphisms of the form $f\Box u$, for $f\in I$, and $u$ a weak equivalence in $M$.

If we denote by $W$ the set of weak equivalences, this gives:

[TABLE]

or in other terms:

[TABLE]

With this notion, we define $M^{\wedge}=L_{I\Box W}S_{*\centerdot}^{M^{\text{op}}}$ to be the model category of prestacks on $M$.

4.9 Hyperdescent

There is a slew of pseudo-representable objects we have to define before we get to the definition of hyperdescent. Those objects were initially defined in [TV4] in the ungraded case, and we briefly reproduce their bi-graded definitions here for convenience’s sake.

We say $F\in M^{\wedge}$ is pseudo-representable if it is a small disjoint union of representable presheaves:

[TABLE]

that is it is entry-wise pseudo representable.

A pseudo-fibration is a morphism of pseudo-representable objects represented by a fibration in $M$. This breaks up into:

[TABLE]

with entries that are pseudo-fibrations, so pseudo-fibrations are so entry-wise in $\underline{M}$.

To avoid repetition, the reader is referred to [TV] for the definition of pseudo-covering. It is clearly entry-wise a pseudo-covering.

For pseudo-representable hypercovers, let $x$ be a fibrant object of $M$. Then a pseudo-representable hypercover is an object $F_{*}\rightarrow h_{x}$ in $sM^{\wedge}/h_{x}$ such that for any $n\geq 0$, the induced morphism:

[TABLE]

is a pseudo-fibration and pseudo-covering of pseudo-representable objects. We write $F_{*}=(F_{*,ip})$ with $F_{*,ip}\in s\underline{M}^{\wedge}/h_{x_{ip}}$ with $x=(x_{ip})$ in $M$, so that:

[TABLE]

showing that pseudo-representable hypercovers are defined entry-wise.

Note that $M^{\wedge}=L_{I\Box W}S_{*\centerdot}^{M^{\text{op}}}$ is naturally tensored and cotensored over $S_{*\centerdot}$ with external products and exponentials being defined objectwise. This makes $M^{\wedge}$ into a bi-graded simplicial model category with bi-graded simplicial hom $\underline{\text{Hom}}$ and derived bi-graded simplicial hom that we will denote as in [TV4] $\mathbb{R}_{\omega}\underline{\text{Hom}}$.

Regarding Yoneda, fix a cofibrant resolution functor $(\Gamma:M\rightarrow M^{\Delta},\iota)$ with $\Gamma=(\Gamma_{ip})$ and $\iota=(\iota_{ip})$, that is for all $x\in M$, $\Gamma(x)=(\Gamma_{ip}(x_{ip}))$ is a co-simplicial object in $M$, cofibrant for the Reedy model structure on $M^{\Delta}$, together with a natural equivalence $\iota(x):\Gamma(x)\rightarrow c^{*}(x)$, bi-graded constant co-simplicial object in $M$ at $x$. We have:

[TABLE]

since $\underline{h}_{ip}$ preserves fibrant objects and equivalences between them, so does $\underline{h}$, so $\mathbb{R}\underline{h}:\text{Ho}\,(M)\rightarrow\text{Ho}\,(M^{\wedge})$ is well-defined. Further:

[TABLE]

for $i,p=0,1$, leading to:

[TABLE]

This follows from the classical enriched Yoneda isomorphism $\mathbb{R}_{\omega}\underline{\text{Hom}}(h_{x},F)\cong F(x)$, and the fact that $\text{Ho}\,(h)\cong\mathbb{R}\underline{h}$ in $\text{Ho}\,(M^{\wedge})$ entry-wise ([TV]).

Finally, $F\in M^{\wedge}$ is said to have hyperdescent if for any fibrant object $x\in M$, for any pseudo-representable hypercover $H_{*}\rightarrow h_{x}$, with realization $|H_{*}|=(|H_{*,ip}|)$, the induced morphism:

[TABLE]

is an isomorphism in $\text{Ho}\,(S_{*\centerdot})$. A stack on $(M,\acute{e}t.)$ is a prestack $F\in M^{\wedge}$ that satisfies $\acute{e}t.$-hyperdescent. We denote by $M^{\sim\,,\,\acute{e}t.}$ the model category of stacks on $M$. Another way to say this is by considering $H_{\beta}(x)$ for $x$ fibrant in $M$ the set of representatives of the set of isomorphism classes of objects $F_{*}\rightarrow h_{x}$ in $sM^{\wedge}/h_{x}$ (see [TV4] for details), those morphisms that are pseudo-representable hypercovers with a bound on the cardinality of each $F_{n}$. We have:

[TABLE]

Then we can equivalently say $M^{\sim\,,\,\acute{e}t.}$ is the left Bousfield localization of $M^{\wedge}$ with respect to morphisms in $H_{\beta}$.

4.10 local equivalences

Another way to define stacks is as follows. First consider the presheaf of connected components of $RF$:

[TABLE]

leading to:

[TABLE]

factoring as:

[TABLE]

which further factors as $\pi_{0}^{\text{eq}}:\text{Ho}\,(M^{\wedge})\rightarrow\Pr_{\mathbb{Z}_{2}^{2}}(\text{Ho}\,(M))$. We also have an evaluation functor:

[TABLE]

with a left adjoint:

[TABLE]

so once right derived:

[TABLE]

Now for $F\in M^{\wedge}$, $x\in M$ fibrant, $s\in\pi_{0}^{\text{eq}}(F)(x)$ can be represented by a morphism $s:h_{x}\rightarrow F$ in $\text{Ho}\,(M^{\wedge})$, giving by pullback $\mathbb{R}j_{x}^{*}s:\mathbb{R}j_{x}^{*}(h_{x})\rightarrow\mathbb{R}j_{x}^{*}F$. Since we have a point $*\rightarrow\mathbb{R}j_{x}^{*}h_{x}$, by composition we get a global point $*\rightarrow\mathbb{R}j_{x}^{*}F$. All of this can be found in [TV]. We now define, much as in this reference, for any $n>0$, the sheaf $\pi_{n}(F,s)$, by:

[TABLE]

this decomposes as follows:

[TABLE]

where $s\in\pi_{0}^{\text{eq}}(F)(x)=(\pi_{0}^{\text{eq}}(F_{ip})(x_{ip}))$ so we write $s=(s_{ip})$.

We now define local equivalences exactly as in [TV]. Those are also referred to as $\pi_{*}$-equivalences. A morphism $f:F\rightarrow G$ in $M^{\wedge}$ is such an equivalence if the induced morphism $\pi_{0}(F)\rightarrow\pi_{0}(G)$ is an isomorphism on $\text{Ho}\,(M)$, i.e. if it is entry-wise so, and additionally if for any fibrant $x$ in $M$, any $s\in\pi_{0}^{\text{eq}}(F)(x)$, and $n>0$, we have a bijection of sheaves on $\text{Ho}\,(M/x)$: $\pi_{n}(F,s)\rightarrow\pi_{n}(G,f(s))$, and those are entry-wise isomorphisms as well. In other terms local equivalences are entry-wise local equivalences.

Finally given $M$, and for us this is really $k\text{-D}^{-}\text{sAff}_{*}$ we have in mind with the étale topology, there is a closed model structure on $\text{sPr}_{\mathbb{Z}^{2}_{2}}(M)$ called the entry-wise local projective model structure for which equivalences are local equivalences, cofibrations are cofibrations in $M^{\wedge}$ for its projective model structure. It turns out that $\text{sPr}_{\mathbb{Z}^{2}_{2}}(M)$ with this model structure is $M^{\sim\,,\,\acute{e}t.}$. The proof of the existence of such a model structure is done entry-wise, and at that level this is proved in [TV]. Now as pointed out above, we can also construct $M^{\sim\,,\,\acute{e}t.}$ as the left Bousfield localization of $M^{\wedge}$ with respect to $H=\{|F_{*}|\rightarrow h_{x}\,|\,x\in M^{f},F_{*}\in H_{\beta}(x)\}$. Maps in $H$ break down as $|F_{*,ip}|\rightarrow h_{x_{ip}}$ so that it’s clear we have $H=(H_{ip})$, hence:

[TABLE]

4.11 Truncations

Define $X\in S_{*\centerdot}$ to be $n$-truncated if it is entry-wise $n$-truncated. $x\in\text{Ho}\,(M_{*\centerdot})$ is said to be $n$-truncated if it is so entry-wise. Then define $\pi_{\leq n}$-equivalences as in [TV]. There is a model structure on $\text{sPr}_{\mathbb{Z}^{2}_{2}}(M)$ called the $n$-truncated bi-graded local projective model structure for which equivalences are bi-graded $\pi_{\leq n}$-equivalences, cofibrations are cofibrations for the bi-graded projective model structure on $M^{\wedge}$. That we have such a model structure follows directly for the same result in the ungraded case proved in [TV], since we can work entry-wise. This model category is denoted $M_{\leq n}^{\sim\,,\,\acute{e}t.}$, and can be seen as a substitute for working with compactified theories in Theoretical Physics. We also have as a Corollary that this model category can be obtained as the left Bousfield localization of $M^{\sim\,,\,\acute{e}t.}$ with respect to $\{\partial\Delta^{i}\otimes\underline{h}_{x}\rightarrow\Delta^{i}\otimes\underline{h}_{x},i>n,x\in M\}$.

5 Supersymmetric stacks

In what follows we will make use of the strictification theorem, which can be found in [HS], [TV] and [T]: for $C$ a category with a subcategory $S$ of morphisms, $M$ a cofibrantly generated model category, $M^{C,S}$ the localization of $M^{C}$ with respect to restricted diagrams, then:

[TABLE]

where $L$ stands for Segal localization. In particular if $T$ is an $S$-site ([TV]), then we just have $L(M^{T})\cong\mathbb{R}\underline{\text{Hom}}(T,LM)$. If $M=S_{*\centerdot}$, with $L(S_{*\centerdot})=\text{Top}_{*\centerdot}$, we have $L(S_{*\centerdot}^{T})\cong\mathbb{R}\underline{\text{Hom}}(T,\text{Top}_{*\centerdot})$. For us $T=L(\text{s}k\text{-sAlg}_{*})^{\text{op}}=L(k\text{-D}^{-}\text{sAff}_{*})$ will be denoted $\text{d}k\text{-sAff}_{*}$. We define by:

[TABLE]

the Segal localization of $\text{sPr}_{\mathbb{Z}^{2}_{2}}(k\text{-D}^{-}\text{sAff}_{*})$. Then we define the category of stacks $\text{d}k\text{-sAff}^{\;\sim\,,\,\acute{e}t.}_{*}$ as a left exact localization of $\widehat{\text{d}k\text{-sAff}}_{*}$, which can equivalently be obtained as the Segal localization of $k\text{-D}^{-}\text{sAff}^{\sim\,,\,\acute{e}t.}_{*}$, itself the left Bousfield localization of $\text{sPr}_{\mathbb{Z}^{2}_{2}}(k\text{-D}^{-}\text{sAff}_{*})$, regarded as a simplicial category for strictication, and then as a bi-graded simplicial category.. Diagrammatically:

[TABLE]

The stacks we have defined are objects of $k\text{-D}^{-}\text{sAff}^{\sim\,,\,\acute{e}t.}_{*}$, and after Segal localization, they become objects of the category $\text{d}k\text{-sAff}^{\;\sim\,,\,\acute{e}t.}_{*}$, where they become functors: $F:L(\text{s}k\text{-sAlg}_{*})\rightarrow\text{Top}_{*\centerdot}$. Now we argue for every $A\in\text{s}k\text{-sAlg}_{*}$, $\mathcal{X}\in\text{Top}_{*\centerdot}$ determined by constraint equations such as equations of motion, $\Psi:A\rightarrow\mathcal{X}$ a map satisfying those equations, define $F(A)=\{\Psi(\sigma_{i},\theta_{i})\,|\,\sigma_{i}\in A_{0},\theta_{i}\in A_{1},i=0,1\}$ with the induced topology. Suppose $F$ thus defined is a stack. If in addition we have a notion of supersymmetric transformation on each $A\in\text{s}k\text{-sAlg}_{*}$ under which $\Psi$ is well-behaved, then the resulting functor $F$ is called a supersymmetric stack.

In a first time we go over simple supersymmetry transformations using a standard reference such as [GSW]. This will motivate our modifications, which will come afterwards.

5.1 Ordinary Supersymmetry

Everything in this section is from [GSW], and the reader is referred to that reference for more details. Suppose we have, for $M\in\text{s}k\text{-sMod}$, two elements $\sigma^{\alpha}$ of $M_{0}\in\text{s}k\text{-Mod}$, $\alpha=1,2$, and two elements $\theta^{A}$ of $M_{1}$, $A=1,2$. Those are seen as vector components. We also introduce matrices:

[TABLE]

and

[TABLE]

We write $\bar{\theta}=\theta^{T}\rho^{0}$. Suppose elements of $M$ are variables. We can define the following derivatives:

[TABLE]

defining what is called a supersymmetry transformation:

[TABLE]

where $\epsilon$ is an odd parameter. If one defines a field on $A$ valued in Top, with $\mu$ being a dimensional index:

[TABLE]

then defining an ansatz for the supersymmetry transformation of $Y$ under $\delta$ as in:

[TABLE]

one finds:

[TABLE]

5.2 Supersymmetric maps

For our purposes we need to have $Y$ valued in a bi-graded topological space. Note that in the definition of $Y^{\mu}$ in the preceding section, there is no odd part. Additionally, the supersymmetry transformation of $\theta$ produces $\epsilon$, another anticommuting variable, though if we want symmetry in our formalism, we would like to obtain an even element. Finally the supersymmetry transformations on $\sigma$ and $\theta$ are not symmetric themselves, so this is something we might want to enforce. Finally $\delta$ on each of $X^{\mu}$, $\psi^{\mu}$ and $B^{\mu}$ in the previous section is defined via an ansatz, so those transformations are really specific to each field and do not correspond to a single algebraic operation $\delta$, something we would like to have for a more functorial treatment of supersymmetry transformations.

We introduce $\mathcal{X}=(\mathcal{X}_{ip})\in\text{Top}_{*\centerdot}$, $\Psi:M\rightarrow\mathcal{X}$, $M\in k\text{-sMod}_{*}$, with entries $\Psi_{ip}:M_{ip}\rightarrow\mathcal{X}_{ip}$ with $i,p=0,1$. We have seen that $k\text{-sMod}_{*}$ has an internal Hom, its odd parts being made of morphisms in $k\text{-sMod}_{*}$ that change parity. Let $\delta$ be one such morphism.

We consider those $\delta$’s that provide supersymmetric transformations in the sense that they are elements of $\underline{\mathcal{H}\text{om}}(M,M)_{1}$, and their definition displays some symmetry. For that purpose, we are led to introducing even parameters $\epsilon$ and $\lambda$, and matrices $\rho$ and $\gamma$, such that

[TABLE]

with $\alpha,A=0,1$. Thus $\delta_{0}:M_{0}\rightarrow M_{1}$ and $\delta_{1}:M_{1}\rightarrow M_{0}$. The induced transformation on $\Psi$ is given by:

[TABLE]

or $\delta=\delta_{0}\oplus\delta_{1}\in\underline{\mathcal{H}\text{om}}(M,M)_{1}$. Thus:

[TABLE]

that is $\delta^{*}=\delta_{0}^{*}\oplus\delta_{1}^{*}$. We now generalize this construction to the case $M\in\text{s}k\text{-sMod}_{*}$. This means we consider objects $\sigma_{n}$, $\theta_{n}$, $\gamma_{n}$, $\rho_{n}$, $\lambda_{n}$, $\epsilon_{n}$ and $\delta_{n}$. We have:

[TABLE]

where $d_{i}\sigma_{n}^{\alpha}=(d_{i}\sigma^{\alpha})[n-1]=d^{i*}\sigma^{\alpha}[n-1]=\sigma^{\alpha}\circ d^{i}[n-1]$. $\delta$ is a natural transformation of simplicial objects, breaking into $\delta_{0}:M_{*0}\rightarrow M_{*1}$ and $\delta_{1}:M_{*1}\rightarrow M_{*0}$ with:

[TABLE]

as well as:

[TABLE]

with $s_{j}$ the degeneracy map, defined by:

[TABLE]

where $d^{i}$ and $s^{j}$ are the usual connecting maps on $\Delta$ ([GoJa]). We have similar commutative diagrams for $\delta_{*1}$. In all those, we have:

[TABLE]

with $\epsilon$, $\lambda$, $\rho$, $\gamma$ fixed, independently of the coordinates $\sigma$ and $\theta$, and:

[TABLE]

the collection of which defines:

[TABLE]

thereby defining:

[TABLE]

objectwise presentation of:

[TABLE]

with $(\delta^{*}F)(A)=\delta^{*}F(A)$. If $\delta^{*}$ commutes with the left exact localization $\widehat{\text{d}k\text{-sAff}_{*}}\rightarrow\text{d}k\text{-sAff}^{\;\sim\,,\,\acute{e}t.}_{*}$, then $F$ is referred to as a supersymmetric stack. In so doing, we preserve the physical origin of such a functor: to say that it is supersymmetric is in reference to the supersymmetry transformation in the base, that is the context is supersymmetric insofar as we have supersymmetri transformations, not that $F$ itself transforms in a symmetric manner under such transformations.