Construction 7.5.3.3 (03AS): Explicit Isofibrant Replacement

Construction 7.5.3.3 (Explicit Isofibrant Replacement). Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a (strictly commutative) diagram of $\infty $-categories. Let $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ denote the $\mathscr {F}$-weighted nerve of $\operatorname{\mathcal{C}}$ (Definition 5.3.3.1), and let $\mathscr {F}^{+} = \operatorname{sTr}_{ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}}$ denote the strict transport representation of the projection map $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. It follows from Remark 5.3.4.10 that there is a unique natural transformation $\alpha : \mathscr {F} \rightarrow \mathscr {F}^{+}$ for which the diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}^{+} ) \ar [dr]^{ \lambda _{u} } & \\ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [ur]^{ \underset { \longrightarrow }{\mathrm{holim}}(\alpha ) } \ar [rr]^{ \lambda _{t} } & & \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}), } \]

is commutative, where $\lambda _{u}$ denotes the universal scaffold of Construction 5.3.4.7 and $\lambda _{t}$ denotes the taut scaffold of Construction 5.3.4.11. We will refer to $\mathscr {F}^{+}$ as the isofibrant replacement of $\mathscr {F}$.