Let $j: K\to K′$ be a categorical equivalence of simplicial sets. By [HTT, Remark 2.1.4.11], we have a Quillen equivalence (with the covariant model structures)
$$
j_!:\mathsf{sSet}_{/K}\rightleftarrows\mathsf{sSet}_{/K'}:j^*.
$$
On the other hand, we have the functor $\mathbf{\Delta}\to\mathsf{sSet}, [n]\mapsto\Delta^n$, thus a functor $\mathbf{\Delta}_{/K}\to\mathbf{\Delta}_{/K'}$ on categories of simplices.
I want to know: is this functor also a categorical equivalence (i.e., equivalence of ordinary categories)?
The initial motivation for asking this is: we have $\operatorname{colim}((\mathbf{\Delta}_{/K})^{\rm op}\xrightarrow{*}\mathcal{S})\simeq\operatorname{colim}_{[n]\in\mathbf{\Delta}^{\rm op}}K_n$, viewing each $K_n$ as a discrete/$0$-truncated object in $\mathcal{S}$ (I think, by Bousfield-Kan). I want to see if this can also be identified with $\operatorname{colim}(K\xrightarrow{*}\mathcal{S})$ in a canonical way (in the $\infty$-category of spaces $\mathcal{S}$) so that $\operatorname{colim}_{[n]\in\mathbf{\Delta}^{\rm op}}K_n$ in $\mathcal{S}$ depends on $K$ only up to categorical equivalence, which I can confirm when $K$ is an $\infty$-category.
(* means constant functor with value *.)
Best Answer
No it is not. But there is a replacement for this. There is another model of $\infty$-categories: marked simplicial sets. If $X$ is a simplicial set with a set of marked $1$-simplices $S$, the fibrant replacement of $(X,S)$ is a model of the localization by $S$ of the $\infty$-category corresponding to $X$ (as a mere simplicial set). There is always a canonical map $\tau_K:\Delta_{/K}\to K$ sending an $n$-simplex over $K$ to its value at $n$. We can mark those $1$-simplices of $\Delta_{/K}$ that are sent to the identites in $K$. Then the map $\tau_K:\Delta_{/K}\to K$ is an equivalence in the model structure of marked simplicial sets (where the marked simplices in $K$ are the identities). By a 2 out of 3 property, a morphism of simplicial sets $K\to K'$ is a categorical equivalence if and only if the induced functor $\Delta_{/K}\to\Delta_{/K'}$ is a weak equivalence of marked simplicial sets. The main properties of the comparison map $\tau_K:\Delta_{/K}\to K$ needed to prove what I claim above are proved in Kerodon or in my book Higher categories and homological algebra (Prop. 7.3.15) for instance. The model structure on marked simplicial sets is discussed in HTT, of course.