Corollary 7.1.8.4 (Fubini's Theorem for Colimits). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ and $K$ be simplicial sets, and let $F: B \times K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Assume that, for each object $b \in B$, the diagram $F(b, \bullet ): K \rightarrow \operatorname{\mathcal{C}}$ admits a colimit. Then:
- $(1)$
The diagram
\[ f: K \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \quad \quad k \mapsto F(\bullet , k) \]
admits a colimit in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$.
- $(2)$
Let $G$ be an object of $\operatorname{Fun}(B, \operatorname{\mathcal{C}})$ and let $\alpha : f \rightarrow \underline{G}_{K}$ be a natural transformation from $f$ to the constant diagram $\underline{G}_{K}: K \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$. Then $\alpha $ exhibits $G$ as a colimit of $f$ in the $\infty $-category $\operatorname{Fun}(B, \operatorname{\mathcal{C}})$ if and only if, for every vertex $b \in B$, the induced map $\alpha _{b}: F(b, \bullet ) \rightarrow \underline{G(b)}_{K}$ exhibits $G(b)$ as a colimit of the diagram $F(b, \bullet ): K \rightarrow \operatorname{\mathcal{C}}$.
- $(3)$
Let $G \in \operatorname{Fun}(B, \operatorname{\mathcal{C}})$ be a colimit of $f$. Then an object $C \in \operatorname{\mathcal{C}}$ is a colimit of $G$ if and only if it is a colimit of $F$.
Proof.
Assertions $(1)$ and $(2)$ are a reformulation of Proposition 7.1.8.2 (see Remarks 7.1.3.6 and 7.1.3.7). To prove $(3)$, fix a diagram $G: B \rightarrow \operatorname{\mathcal{C}}$ and a natural transformation $\beta : F' \rightarrow \underline{G}_{K}$ satisfying the condition described in $(2)$ (so that $G$ is given on objects by the formula $G(b) = \varinjlim _{k \in K}( F(b,k) )$). Fix an object $C \in \operatorname{\mathcal{C}}$, and let $\beta : G \rightarrow \underline{C}_{B}$ be a natural transformation from $G$ to the constant functor $\underline{C}_{B}: B \rightarrow \operatorname{\mathcal{C}}$. Applying the diagonal embedding $\operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(B \times K, \operatorname{\mathcal{C}})$, we obtain a natural transformation $\underline{\beta }_{K}: \underline{G}_{K} \rightarrow \underline{C}_{B \times K}$. Let $\gamma : F \rightarrow \underline{C}_{B \times K}$ be a composition of $\beta $ with $\alpha $. Then, for every object $D \in \operatorname{\mathcal{C}}$, we have a homotopy commutative diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{Hom}_{ \operatorname{Fun}(B,\operatorname{\mathcal{C}}) }( G, \underline{D}_{B} ) \ar [dr]^{ \circ [\alpha ] } & \\ \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( C, D) \ar [ur]^{ \circ [\beta ] } \ar [rr]^{ \circ [\gamma ] } & & \operatorname{Hom}_{ \operatorname{Fun}(B \times K, \operatorname{\mathcal{C}}) }( F, \underline{D}_{B \times K} ), } \]
where the diagonal map on the right is a homotopy equivalence. Allowing the object $D$ to vary, we conclude that $\beta $ exhibits $C$ as a colimit of $G$ if and only if $\gamma $ exhibits $C$ as a colimit of $F$. In particular, if the object $C$ is a colimit of $G$, then it is also a colimit of $F$. The converse follows from the observation that every natural transformation $\gamma : F \rightarrow \underline{C}_{B \times K}$ can be obtained in this way (since precomposition with $\alpha $ induces a homotopy equivalence $\operatorname{Hom}_{ \operatorname{Fun}(B,\operatorname{\mathcal{C}})}( G, \underline{C}_{B} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(B \times K, \operatorname{\mathcal{C}}) }( F, \underline{C}_{B \times K} )$).
$\square$