  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.
