Lemma 4.3.8. Let \(I\) be an \(\infty \)-category.
- (1)
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If \(C\) admits \(I\)-indexed limits, then \(\PSp (C)\) admits \(I\)-indexed limits and the projection \(\PSp (C) \to C_*^{\N }\) creates them.
- (2)
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If \(C_*\) admits \(I\)-indexed colimits, then \(\PSp (C)\) admits \(I\)-indexed colimits and the projection \(\PSp (C) \to C_*^{\N }\) creates them.
Proof. This is just the definition of \(\PSp (C)\) as the lax equalizer of \(\id \) and \(\Omega \sh \). Indeed, a morphism \(X\to Y\) in \(\PSp (C)\) is a map of the underlying sequences together with a homotopy filling the square
Equivalently, it is a point of the stated equalizer. □
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