Interchanging homotopy limits and colimits in CAT

TL;DR

This paper proves that under certain conditions, homotopy limits and colimits can be interchanged in the category of small categories, establishing a strong homotopy equivalence.

Contribution

It establishes a condition under which homotopy limits and colimits can be interchanged in CAT, specifically when the index category has a final object.

Findings

Homotopy colimits and limits are interchangeable under the given conditions.

The canonical map between these constructions is a strong homotopy equivalence.

The result applies to functors from product categories to CAT.

Abstract

Let $I$,$J$ be small categories and $C:I\times J@>>>\CAT$ a functor to the category of small categories. We show that if $I$ has a final object then the canonical map $\operatorname{hocolim}_J\operatorname{holim}_IC@>>>\operatorname{holim}_I \operatorname{hocolim}_JC$ is a strong homotopy equivalence.