Homotopy Colimits of Algebras Over Cat-Operads and Iterated Loop Spaces

TL;DR

This paper extends homotopy colimit constructions to categories of algebras over Cat-operads, establishing equivalences between n-fold monoidal categories, loop spaces, and infinite loop spaces, thus linking categorical and topological structures.

Contribution

It generalizes Thomason's homotopy colimit to algebras over arbitrary Cat-operads and proves new equivalences between monoidal categories and loop space categories.

Findings

Equivalence between n-fold monoidal categories and C_n-spaces.

Equivalence between n-fold monoidal categories and n-fold loop spaces.

Extension of Thomason's results to braided monoidal categories and 2-fold loop spaces.

Abstract

We extend Thomason's homotopy colimit construction in the category of permutative categories to categories of algebras over an arbitrary $\Cat$ operad and analyze its properties. We then use this homotopy colimit to prove that the classifying space functor induces an equivalence between the category of $n$-fold monoidal categories and the category of $\mathcal{C}_n$-spaces after formally inverting certain classes of weak equivalences, where $\mathcal{C}_n$ is the little $n$-cubes operad. As a consequence we obtain an equivalence of the categories of $n$-fold monoidal categories and the category of $n$-fold loop spaces and loop maps after localization with respect to some other class of weak equivalences. We recover Thomason's corresponding result about infinite loop spaces and obtain related results about braided monoidal categories and 2-fold loop spaces.