Internal and local homotopy theory

2014/04/30 by Low, Zhen Lin
#03G30 #18B25 #18F10 #18G30 (Primary) 18C10 #55U10 (Secondary) #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1404.7788

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_{ω1, ω}. 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.

Related

• Weak equivalences of simplicial presheaves
• Model topoi and motivic homotopy theory
• The topological realization of a simplicial presheaf
• Some Properties of Internal Locale Morphisms Externalised
• Homotopy theory of simplicial presheaves in completely decomposable topologies