Stacks and their function algebras.

Abstract

For T any abelian Lawvere theory, we establish a Quillen adjunction between model category structures on cosimplicial T-algebras and on simplicial presheaves over duals of T-algebras, whose left adjoint forms algebras of functions with values in the canonical T-line object. We find mild general conditions under which this descends to the local model structure that models ∞-stacks over duals of T-algebras.

For T the theory of commutative algebras this reproduces the situation in Toën's Champs Affines. We consider the case where T is the theory of C∞-rings: the case of synthetic differential geometry. In particular, we work towards a deffnition of smooth ∞-vector bundles with flat connection. To that end we analyse the tangent category of the category of C∞-rings and Kock's simplicial model for synthetic combinatorial differential forms which may be understood as an ∞-categorification of Grothendieck's de Rham space functor.