The homotopy \(n\)-category functor \(h_n\) takes a space \(X\) to its \(n\)-truncation \(\tau_{n} X\). Given a \((\infty, 1)\)-category \(\mathcal C\), \(h_0 \mathcal C= \tau_0 \iota_0 \mathcal C\) is the set of isomorphism classes of objects, and \(h_1 \mathcal C\) is its homotopy \(1\)-category [Lur09, Def. 1.1.3.2]. For \(n \geq 1\), \(h_n = \tau_n\) is equivalent to the ‘\(n\)-homotopy category’ of [SY20, Def. 2.9]. Of particular relevance to this paper will be the case of \((\infty,2)\)-categories \(\mathcal C\) where \(h_1\mathcal C= \tau_1 \iota_1 \mathcal C\).
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2