Shifted symplectic structures on derived Quot-stacks I: Differential graded manifolds

TL;DR

This paper develops a theory of dg schemes as a homotopy site, establishing an equivalence of infinity categories of stacks, and shows that stacks represented by dg schemes are derived schemes, advancing derived algebraic geometry.

Contribution

It introduces a new framework for dg schemes as a homotopy site and proves their equivalence with existing stack categories, connecting dg schemes to derived schemes.

Findings

Established a homotopy site for dg schemes

Proved the equivalence of stack categories

Demonstrated dg scheme stacks are derived schemes

Abstract

A theory of dg schemes is developed so that it becomes a homotopy site, and the corresponding infinity category of stacks is equivalent to the infinity category of stacks, as constructed by Toen and Vezzosi, on the site of dg algebras whose cohomologies have finitely many generators in each degree. Stacks represented by dg schemes are shown to be derived schemes under this correspondence.