-Stacks and their Function Algebras – with applications to -Lie theory
MSc thesis
Utrecht 2010
Some time after the author wrote this thesis, he realized he had made a mistake in theorem 30. There is a sequence of isomorphisms in that proof for which the isomorphism from the second to the third line was unjustified. A natural bijection is guaranteed by the Dold-Puppe correspondence, but that does not imply an isomorphism of chain complexes. The work of Tashi Walde ''Homotopy coherent theorems of Dold-Kan type'' repairs this by proving that the Dold-Puppe correspondence is a natural equivalence of quasi-categories.
For any abelian Lawvere theory, we establish a Quillen adjunction between model category structures on cosimplicial T-algebras and on simplicial presheaves over duals of -algebras, whose left adjoint forms algebras of functions with values in the canonical -line object. We find mild general conditions under which this descends to the local model structure that models ∞-stacks over duals of -algebras.
For the theory of associative algebras this reproduces the situation in Toën‘s Champs affine. We consider the case where is the theory of smooth algebras: the case of synthetic differential geometry. In particular, we work towards a definition of smooth -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.
Last revised on August 14, 2026 at 14:26:20. See the history of this page for a list of all contributions to it.