Categorical equivalences vs. categories of simplices

Question

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 *.)

Answer 1

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.

Answer 2

Dénis-Charles Cisinski has given an excellent big-picture answer. But to flesh out his answer to the question itself, the functor $j/K : \Delta/K \to \Delta/K'$ will almost never be an equivalence of categories — in fact only when $j$ is an isomorphism. So any non-isomorphic equivalent categories give a counterexample to your question, e.g. $1$ and the “walking isomorphism”.

This is nothing specific to do with simplicial sets: generally for any map of presheaves $j : K \to K' : \newcommand{\C}{\textbf{C}} \C^{\mathrm{op}} \to \mathrm{Set}$, it’s straightforward to check that if $j/\C : K/\C \to K'/\C$ is full, then $j$ is injective, and if $j/\C$ is essentially surjective, then $j$ is surjective. So if $j/\C$ is an equivalence, then $j$ is an isomorphism.