Revisiting colimits in $\mathbf{C}\mathbf{a}\mathbf{t}$ and homotopy category

In this paper, we justify and make precise an elementary approach that establishes the existence of (co)limits in $\mathbf{C}\mathbf{a}\mathbf{t}$. 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 $h:\mathbf{s}\mathbf{S}\mathbf{e}\mathbf{t}\to \mathbf{C}\mathbf{a}\mathbf{t}$ and the existence of a specific class of weighted colimits in $\mathbf{C}\mathbf{a}\mathbf{t}$. We then construct these weighted colimits explicitly by using certain properties of simplicial sets and the nerve functor. Consequentially, the embedding $N:\mathbf{C}\mathbf{a}\mathbf{t}\hookrightarrow \mathbf{s}\mathbf{S}\mathbf{e}\mathbf{t}$ is reflective, and can be used to infer the (co)completeness of $\mathbf{C}\mathbf{a}\mathbf{t}$. Finally, we use this approach to reformulate the construction of coequalizers and localizations in $\mathbf{C}\mathbf{a}\mathbf{t}$.