lemma 5.4.9 does not hold for \(j > n \geq k\): For instance, for \(j=1\) and \(n=k=0\) and an \((\infty,1)\)-category \(\mathcal C\), the space \(\tau_0 \iota_0 \mathcal C\) is the set of isomorphism classes of objects of \(\mathcal C\). On the other hand, \(\iota_0 \tau_0 \mathcal C\) is the set of connected components of \(\mathcal C\), i.e the quotient of the set of isomorphism classes of object by the equivalence relation that \(c\sim d\) if there exists a zigzag of morphisms between \(c\) and \(d\).
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2