On the loop space of a 2-category

Abstract

Every small category $C$ has a classifying space $BC$ associated in a natural way. This construction can be extended to other contexts and set up a fruitful interaction between categorical structures and homotopy types. In this paper we study the classifying space $B_{2}C$ of a 2-category $C$ and prove that, under certain conditions, the loop space ${\Omega }_c{B}_{2}C$ can be recovered up to homotopy from the endomorphisms of a given object. We also present several subsidiary results that we develop to prove our main theorem.

21 pages, final version. Section 8 concerning the main theorem was rewritten. In particular, a partial converse for the main theorem was added

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