Fibrations and homotopy colimits of simplicial sheaves

TL;DR

This paper proves that homotopy pullbacks of simplicial sheaves distribute over colimits and that inverse image functors preserve these pullbacks, introducing the concept of sharp maps analogous to quasi-fibrations.

Contribution

It generalizes Puppe's result to sheaves of simplicial sets and introduces the notion of sharp maps, enhancing understanding of homotopy limits and colimits in this context.

Findings

Homotopy pullbacks distribute over homotopy colimits for simplicial sheaves.

Inverse image functors preserve homotopy pullback squares.

Introduction of the sharp map concept related to quasi-fibrations.

Abstract

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 notion of a quasi-fibration of spaces, and seems to be of independent interest.