Colimits of categories, zig-zags and necklaces

Abstract

Given a diagram of small categories $F : J \rightarrow \textbf{Cat}$, we provide a combinatorial description of its colimit in terms of the indexing category $J$ and the categories and functors in the diagram $F$. We introduce certain double categories of zig-zags in order to keep track of the necessary identifications. We found these double categories necessary, but also explanatory. When applied pointwise in the simplicially enriched setting, our constructions offer a shorter proof of the necklace theorem of Dugger and Spivak by direct computation.