Equivalent models of derived stacks (2303.12699v2)

Abstract: In the present paper, we establish an equivalence between several models of derived geometry. That is, we show that the categories of higher derived stacks they produce are Quillen equivalent. As a result, we tie together a model of derived manifolds constructed by Spivak--Borisov--Noel, a model of Carchedi--Roytenberg, and a model of Behrend--Liao--Xu. By results of Behrend--Liao--Xu the latter model is also equivalent to the classical Alexandrov--Kontsevich--Schwarz--Zaboronsky model. This equivalence allows us to show that weak equivalences of derived manifolds in the sense of Behrend--Liao--Xu correspond to weak equivalences of their algebraic models, thus proving the conjecture of Behrend--Liao--Xu. Our results are formulated in the framework of Fermat theories, allowing for a simultaneous treatment of differential, holomorphic, and algebraic settings.