Lemma 23.7.7. Let \(T\) be an \(\infty \)-category with pullbacks and \(I\)-indexed colimits, and let \(X_{\bullet } \colon I \to T\) be a diagram with colimit \(X := \colim _i X_i\). Then \(T\) satisfies descent for \(I\)-indexed colimits if and only if the following two conditions hold:
- (1)
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For any map \(Y \to X\), the canonical map \(\colim _i (X_i \times _X Y) \to Y\) is an isomorphism.
- (2)
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Given a cartesian transformation \(Y_{\bullet } \to X_{\bullet }\), each map \(Y_i \to (\colim _j Y_j) \times _X X_i\) is an isomorphism.
Proof. By definition, \(T\) satisfies descent for \(I\)-indexed colimits if and only if the canonical functor \(T_{/X} \to \lim _i T_{/X_i}\) is an equivalence. By Lemma 23.7.6, this functor admits a left adjoint given by sending a cartesian transformation \(Y_{\bullet } \to X_{\bullet }\) to its colimit. The functor is an equivalence if and only if both the unit and counit of this adjunction are isomorphisms. The counit at \(Y \to X\) is precisely the map in (1), while the unit at a cartesian transformation \(Y_{\bullet } \to X_{\bullet }\) consists of the maps in (2). □
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