The n-Category Café

Function T-Algebras on ∞-Stacks

Posted by Urs Schreiber

Over the course of this year, Herman Stel has been working on a master thesis and I have been advising him. Yesterday was the defense. See

• Herman Stel, $\infty$-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 $\infty$-stacks over duals of $T$-algebras.

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