Warning 7.5.6.15 (03CK)

Warning 7.5.6.15. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of simplicial sets, let $\alpha : \mathscr {F}_{+} \twoheadrightarrow \mathscr {F}$ be the natural transformation of Construction 7.5.6.8, and let $\lambda : \varinjlim ( \mathscr {F}_{+} ) \xrightarrow {\sim } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ be the isomorphism of Proposition 7.5.6.12. Then we have a diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}_{+} ) \ar [r]^-{ \underset { \longrightarrow }{\mathrm{holim}}( \alpha ) } \ar [d] & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [d] \\ \varinjlim ( \mathscr {F}_{+} ) \ar [r]_{\varinjlim (\alpha ) } \ar [ur]^{\lambda }_{\sim } & \varinjlim ( \mathscr {F} ), } \]

where the outer square and the lower right triangle are commutative (Remark 7.5.6.13). Beware that the upper left triangle is usually not commutative. That is, $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ and $\varinjlim ( \mathscr {F}_{+} )$ are isomorphic when viewed as abstract simplicial sets, but not when viewed as quotients of the simplicial set $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}_{+} )$ (compare with Warning 7.5.3.14).