Internal and local homotopy theory

Abstract

There is a well-established homotopy theory of simplicial objects in a Grothendieck topos, and folklore says that the weak equivalences are axiomatisable in the geometric fragment of $L_{{\omega }_1,\omega }$. We show that it is in fact a theory of presheaf type, i.e. classified by a presheaf topos. As a corollary, we obtain a new proof of the fact that the local Kan fibrations of simplicial presheaves that are local weak homotopy equivalences are precisely the morphisms with the expected local lifting property.

27 pages, LaTeX

Cited by in corpus (2)

• Homotopical Algebra in Categories with Enough Projectives
• Eilenberg--Mac Lane Spaces for Topological Groups