A colimit decomposition for homotopy algebras in Cat

TL;DR

This paper proves that in the 2-category of categories, homotopy colimits can be decomposed into homotopy sifted colimits of finite coproducts, extending a known result from simplicial sets to Cat.

Contribution

The paper demonstrates a colimit decomposition for homotopy colimits in the 2-category of categories, generalizing a key simplicial set result to Cat.

Findings

Homotopy colimits in Cat decompose into homotopy sifted colimits of finite coproducts.

Extension of Badzioch's result from simplicial sets to the 2-category of categories.

Provides a structural understanding of homotopy colimits in categorical contexts.

Abstract

Badzioch showed that in the category of simplicial sets each homotopy algebra of a Lawvere theory is weakly equivalent to a strict algebra. In seeking to extend this result to other contexts Rosicky observed a key point to be that each homotopy colimit in simplicial sets admits a decomposition into a homotopy sifted colimit of finite coproducts, and asked the author whether a similar decomposition holds in the 2-category of categories Cat. Our purpose in the present paper is to show that this is the case.