Definition 1.3.5 (Contractible category). An \(\infty \)-category \(C\) is called contractible if the functor \(p_{C}\colon C \to *\) is an equivalence.
Observe that an \(\infty \)-category \(C\) is contractible if and only if there exists an object \(x\colon * \to C\) such that the composite \(\const _x\colon C \to C\) is naturally isomorphic to the identity \(\id _C\colon C \to C\). A natural isomorphism \(H\colon \const _x \cong \id _C\) is called a contraction of \(C\) onto \(x\).
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