$\mathrm{\infty}$-Stacks and their Function Algebras – with applications to $\mathrm{\infty}$-Lie theory
master thesis (2010)
(pdf)
This text should not be blindly relied upon, it contains some errors in important places.
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.
More on the topics discussed in this thesis can be found at function algebras on ∞-stacks .
Herman Stel, Cosimplicial C∞ rings and the de Rham complex of Euclidean space (arXiv:1310.7407)
This note demonstrates that the normalization of the cosimplicial algebra of functions on the infinitesimal singular simplicial complex is the de Rham complex.