Theorem 24.1.10 ([Joyal and Tierney (2007), Section 4], [Lurie (2009), Corollary 4.3.16], [Hebestreit and Steinebrunner (2023)]). The nerve functor \[ N\colon \Cat _{\infty } \hookrightarrow \sAn \] is fully faithful. Its image consists of the complete Segal animae. In particular, the nerve functor induces an equivalence \[ N\colon \Cat _{\infty } \iso \CSeg (\An ) \] between \(\Cat _{\infty }\) and the \(\infty \)-category of complete Segal animae.
Proof. Since the nerve functor lands in \(\CSeg (\An )\), the adjunction \(\ac \dashv N\) induces an adjunction
We have to show that both the unit and the counit are equivalences. For the unit, let \(X\) be a complete Segal anima. The unit map \(\eta \colon X \to N(\ac (X))\) is a Dwyer-Kan equivalence by Proposition 24.1.9 and thus an equivalence by Lemma 24.1.8, since both sides are complete Segal animae. For the counit, let \(C\) be an \(\infty \)-category. To show that \(\varepsilon \colon \ac (N(C)) \to C\) is an equivalence, it suffices by Remark 24.1.5 to show that \(N(\varepsilon ) \colon N(\ac (N(C))) \to N(C)\) is an equivalence. But this is a consequence of the triangle identity and the fact that \(\eta _{N(C)}\colon N(C) \to N(\ac (N(C)))\) is an equivalence. □
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