A Dual Representation in Spectral Algebraic Geometry

Abstract

Given a spectral Deligne-Mumford stack $X$, we define a perception of $X$ to be a collection of a certain class of morphisms $Y \rightarrow X$. For the class of affine morphisms in SpDM, we show that from QCoh($X$) on can extract the affine perception $\text{Aff}_X$ of $X$ on the one hand, and a subcategory of an $\infty$-category of representations $\text{Rep}_{\mathfrak{g}_*}$ of a dg Lie algebra $\mathfrak{g}_*$ associated with $X$ on the other. For the class of local morphisms $\text{Sp\'et } R \rightarrow X$, the local perception of $X$ is given by the functor $\mathbf{X} = \text{Hom}(\text{Sp\'et}(-), X)$ it represents. If $\mathbf{X}$ is a geometric stack, Tannaka duality allows us to recover $\mathbf{X}$ from $\text{QCoh}(\mathbf{X})$, from which we can also get, after base change, a subcategory of $\text{Rep}_{\mathfrak{g}_*}$. We generalize those results by considering functors $\mathbf{X}: \text{CAlg}^{\text{cn}} \rightarrow \mathcal{S}$ that are representable in accordance with the spectral Artin representability theorem of Lurie.