Schreiber
master thesis Stel

• Herman Stel,

  $\mathrm{\infty }$-Stacks and their Function Algebras – with applications to $\mathrm{\infty }$-Lie theory

  master thesis (2010)

  (pdf)

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 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 $\mathrm{\infty }$-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 ∞-categorification of Grothendieck’s de Rham space functor.

Further discussion

More on the topics discussed in this thesis can be found at function algebras on ∞-stacks .