Construction 7.5.6.8 (03CC): Explicit Cofibrant Replacement

Construction 7.5.6.8 (Explicit Cofibrant Replacement). Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, and let $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ denote the homotopy colimit of $\mathscr {F}$ (Construction 5.3.2.1). For each object $C \in \operatorname{\mathcal{C}}$, we let $\mathscr {F}_{+}(C)$ denote the simplicial set given by the fiber product

\[ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{ /C} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) = \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}|_{ \operatorname{\mathcal{C}}_{/C} } ). \]

The construction $C \mapsto \mathscr {F}_{+}(C)$ determines a diagram of simplicial sets $\mathscr {F}_{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. This diagram is equipped with a natural transformation $\alpha : \mathscr {F}_{+} \twoheadrightarrow \mathscr {F}$, which carries each object $C \in \operatorname{\mathcal{C}}$ to the comparison map

\[ \mathscr {F}_{+}(C) = \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}|_{ \operatorname{\mathcal{C}}_{/C} } ) \twoheadrightarrow \varinjlim ( \mathscr {F}|_{ \operatorname{\mathcal{C}}_{/C} } ) \simeq \mathscr {F}(C) \]

of Remark 5.3.2.9.