Revisiting colimits in $\mathbf{Cat}$ and homotopy category

TL;DR

This paper provides a systematic, explicit approach to establishing (co)limits in the category of small categories, connecting homotopy categories, weighted colimits, and the nerve functor.

Contribution

It clarifies and formalizes an elementary approach to (co)limits in Cat, linking homotopy categories and weighted colimits via explicit constructions.

Findings

Established the equivalence between the homotopy category functor and weighted colimits in Cat.

Constructed weighted colimits explicitly using properties of simplicial sets and the nerve functor.

Showed that the nerve embedding is reflective, implying (co)completeness of Cat.

Abstract

In this paper, we justify and make precise an elementary approach that establishes the existence of (co)limits in $\mathbf{Cat}$. This approach, while conceptually evident, has not been made fully explicit or systematically described in the literature. We first demonstrate an equivalence between the existence of the homotopy category functor $h : \mathbf{sSet} \rightarrow \mathbf{Cat}$ and the existence of a specific class of weighted colimits in $\mathbf{Cat}$. We then construct these weighted colimits explicitly by using certain properties of simplicial sets and the nerve functor. Consequentially, the embedding $N : \mathbf{Cat} \hookrightarrow \mathbf{sSet}$ is reflective, and can be used to infer the (co)completeness of $\mathbf{Cat}$. Finally, we use this approach to reformulate the construction of coequalizers and localizations in $\mathbf{Cat}$.