Iterated colimits

Theorem 21.2.11. Let \(J\) be a small \(\infty \)-category and consider a \(J\)-indexed family of small \(\infty \)-categories \(I_{\bullet }\colon J \to \Cat _{\infty }\). Define \(I := \colim _j I_j\), and let \(\varepsilon \colon I_{\bullet } \to \const _{I}\) denote the colimit cone, giving functors \(\varepsilon _j\colon I_j \to I\). Let \(C\) be an \(\infty \)-category and let \(F\colon I \to C\) be a functor.

(1)

Assume that the restricted diagram \[ F\vert _{I_j} \colon I_j \xrightarrow {\varepsilon _j} I \xrightarrow {F} C \] admits a colimit for every \(j \in J\). Then these colimits assemble into a functor \[ F'\colon J \to C, \qquad j \mapsto \colim (F\vert _{I_j}\colon I_j \to C). \]

(2)

In this case, the functor \(F\) admits a colimit in \(C\) if and only if the functor \(F'\) admits a colimit in \(C\), and then we have \[ \colim _{i \in I} F(i) \cong \colim _{j \in J} \colim (F\vert _{I_j}). \]

Dually, assume that \(F\vert _{I_j}\) admits a limit for every \(j \in J\). Then these limits assemble into a functor \[ F''\colon J\catop \to C, \qquad j \mapsto \lim (F\vert _{I_j}\colon I_j \to C). \] The functor \(F\) admits a limit if and only if \(F''\) admits a limit, and then \[ \lim _{i \in I} F(i) \cong \lim _{j \in J\catop } \lim _{i \in I_j} F(i). \]