Corollary 7.7.0.6 (05SK)

Corollary 7.7.0.6. Let $\mathscr {F}: K \rightarrow \operatorname{\mathcal{S}}$ be a small diagram. Suppose we are given a Kan complex $X \in \operatorname{\mathcal{S}}$ and a natural transformation $\beta : \mathscr {F} \rightarrow \underline{X}$. The following conditions are equivalent:

$(1)$

The natural transformation $\beta $ exhibits $C$ as a colimit of the diagram $\mathscr {F}$.

$(2)$

For every map of Kan complexes $f: X' \rightarrow X$ and every levelwise pullback square

\[ \xymatrix { \mathscr {F}' \ar [r]^{ \beta ' } \ar [d] & \underline{X}' \ar [d]^{ \underline{f} } \\ \mathscr {F} \ar [r]^{\beta } & \underline{X} } \]

the natural transformation $\beta '$ exhibits $X'$ as a colimit of $\mathscr {F}'$.

$(3)$

For every vertex $x \in X$ and every levelwise pullback square

\[ \xymatrix { \mathscr {F}' \ar [r]^{ \beta ' } \ar [d] & \underline{ \{ x\} } \ar [d] \\ \mathscr {F} \ar [r]^{\beta } & \underline{X} } \]

the natural transformation $\beta '$ exhibits $\{ x\} $ as a colimit of $\mathscr {F}'$ (here the right vertical map is induced by the inclusion $\{ x\} \hookrightarrow X$).

$(4)$

For every vertex $x \in X$, the diagram $\mathscr {F}' = \underline{ \{ x\} } \times _{ \underline{X} } \mathscr {F}$ has a contractible colimit in $\operatorname{\mathcal{S}}$.

Proof. The implication $(1) \Rightarrow (2)$ follows from the universality of colimits in $\operatorname{\mathcal{S}}$ (Theorem 7.7.0.4) and the implications $(2) \Rightarrow (3) \Rightarrow (4)$ are immediate. We will complete the proof by showing that $(4)$ implies $(1)$. Choose a Kan complex $Y = \varinjlim (\mathscr {F} )$ and a natural transformation $\alpha : \mathscr {F} \rightarrow \underline{Y}$ which exhibits $Y$ as a colimit of $\mathscr {F}$. The natural transformation $\beta $ induces a morphism of Kan complexes $f: Y \rightarrow X$, which is characterized (up to homotopy) by the existence of a commutative diagram

\[ \xymatrix { & \underline{ Y } \ar [dr]^{ \underline{f} } & \\ \mathscr {F} \ar [ur]^{\alpha } \ar [rr]^{ \beta } & & \underline{X} } \]

in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{S}})$. To prove $(1)$, we must show that $f$ is a homotopy equivalence. By virtue of Remark 3.4.0.6, this is equivalent to the requirement that for each vertex $x \in X$, the homotopy fiber $Y_{x} = \{ x\} \times _{X}^{\mathrm{h}} Y$ is a contractible Kan complex. Using Example 7.6.3.12, we see that there is a pullback diagram $\tau :$

\[ \xymatrix { Y_{x} \ar [d] \ar [r] & \{ x\} \ar [d] \\ Y \ar [r]^{f} & X } \]

in the $\infty $-category $\operatorname{\mathcal{S}}$. As in Remark 7.7.0.2, we can choose a commutative diagram

\[ \xymatrix { \mathscr {F}' \ar [r]^{\alpha '} \ar [d] & \underline{ Y_ x } \ar [r] \ar [d] & \underline{ \{ x\} } \ar [d] \\ \mathscr {F} \ar [r]^{ \alpha } & \underline{Y} \ar [r]^{ \underline{f}} & \underline{X}, } \]

in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{S}})$, where the bottom row is given by the $2$-simplex $\sigma $, the right side of the diagram is the image of $\tau $ under the diagonal map $\operatorname{\mathcal{S}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{S}})$, and the left side of the diagram is a levelwise pullback square. It follows that the outer rectangle is also a levelwise pullback square (Proposition 7.6.2.28), so condition $(4)$ guarantees that the colimit of $\mathscr {F}'$ is a contractible Kan complex. Since $K$-indexed colimits in $\operatorname{\mathcal{S}}$ are universal (Theorem 7.7.0.4), the natural transformation $\alpha '$ exhibits $Y_{x}$ as a colimit of $\mathscr {F}'$, so that $Y_{x}$ is contractible as desired. $\square$