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

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 $\mathbf{Cat}$.