Supersymmetric Derived Stacks

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${}_{\ast }$) following the exposition of Toen and Vezzosi on ungraded modules in HAG I & II. We then define $$\text{Top}${* \centerdot}$-valuedmapsonthosesupermodules($$\text{Top}${* \centerdot}$ ${\mathbb{Z}}_2$-bi-graded), and show how they behave under supersymmetry transformations in the base. For $\mathrm{\Psi }:M\to X$ one such map, $M\in$ sk-sMod${}_{\ast }$, $X\in {\text{Top}}_{\ast \cdot }$, we argue that defining a prestack $F$ of simplicial sets over simplicial graded k-superalgebras object-wise by $F(M)=\{\mathrm{\Psi }(\sigma ,\theta )\mathrm{\mid }\sigma ,\theta \in M\}$ with the induced topology, one can call $F$ a supersymmetric stack if it is a derived stack.