Equivalent models of derived stacks

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.