Kerodon

Remark 7.5.6.16 (The Homotopy Colimit as a Left Derived Functor). The preceding results can be interpreted in the language of model categories. For every small category $\operatorname{\mathcal{C}}$, the category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ can be equipped with a model structure in which the fibrations are levelwise Kan fibrations and weak equivalences are levelwise weak homotopy equivalences (see Example ). Combining Propositions 7.5.6.9 and 7.5.6.12, we deduce that the homotopy colimit functor $ \underset { \longrightarrow }{\mathrm{holim}}: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow \operatorname{Set_{\Delta }}$ can be viewed as a left derived functor of the usual colimit $\varinjlim : \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow \operatorname{Set_{\Delta }}$ (see Definition ).