Stacks and their function algebras.
Summary
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.