Colimits of categories, zig-zags and necklaces

TL;DR

This paper provides a combinatorial framework for understanding colimits of small categories using zig-zags and necklaces, offering a new proof of the necklace theorem through double categories.

Contribution

It introduces double categories of zig-zags to describe colimits of categories and applies this to give a shorter proof of Dugger and Spivak's necklace theorem.

Findings

A combinatorial description of colimits of categories.

Double categories effectively track necessary identifications.

Shorter proof of the necklace theorem in simplicial settings.

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.