Categorical Homotopy and Fibrations

Fibrations and Homotopy Colimits of Simplicial Sheaves

We show that homotopy pullbacks of sheaves of simplicial sets over a Grothendieck topology distribute over homotopy colimits; this generalizes a result of Puppe about topological spaces. In addition, we show that inverse image functors between categories of simplicial sheaves preserve homotopy pullback squares. The method we use introduces the notion of a sharp map, which is analogous to the no...

Categorical homotopy theory

Categorical Homotopy Theory

This paper is an exposition of the ideas and methods of Cisinksi, in the context of A-presheaves on a small Grothendieck site, where A is an arbitrary test category in the sense of Grothendieck. The homotopy theory for the category of simplicial presheaves and each of its localizations can be modelled by A-presheaves in the sense that there is a corresponding model structure for A-presheaves wi...

Homotopy Classification of Categorical Torsors

The long-known results of Schreier on group extensions are here raised to a categorical level by giving a factor set theory for torsors under a categorical group (G,⊗) over a small category B. We show a natural bijection between the set of equivalence classes of such torsors and [B(B),B(G,⊗)], the set of homotopy classes of continuous maps between the corresponding classifying spaces. These res...

Categorical aspects of equivariant homotopy