Lemma 4.1.9. Let \(C\) be an \(\infty \)-category with terminal object \(*\). Then:
- (1)
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The \(\infty \)-category \(C_*\) is pointed, with zero object given by \(0 := (*,\id _*\colon * \to *)\).
- (2)
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The \(\infty \)-category \(C\) is pointed if and only if the forgetful functor \[ \fgt \colon C_* = C_{*/} \longrightarrow C \] is an equivalence.
Proof. (1) For initiality of \(0\), note that the map \(\Hom _C(*,Y) \to \Hom _C(*,Y)\) given by precomposing with the identity is an equivalence, hence its fiber \(\Hom _{C_*}(0,Y)\) is contractible for every \(Y \in C_*\). For terminality, we observe that \[ \Hom _{C_*}(X,0) \simeq \fib \big (\Hom _C(X,*) \to \Hom _C(*,*)\big ) \] is the fiber of a map between two contractible animae, which thus is itself contractible.
(2) The ‘only if’ part is clear from (1). Conversely, assume that \(C\) is pointed. Then the object \(*\) is also an initial object in \(C\), and hence the target functor \(C_* = C_{*/} \to C\) is an equivalence by Lemma 21.3.4. □
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