The simpler question is to study the 2-groupoid $\mathrm{Aut}(\mathsf{Cat})$ of all autoequivalences of the category of (small) categories (i.e. the groupoid of equivalences of categories $\mathsf{Cat} \to \mathsf{Cat}$). This is equivalent to $\mathbb{Z}/2$ considered as a locally discrete 2-groupoid: the terminal object $\mathbb{1}$ is fixed up to unique isomorphism, and the arrow category $\mathbb{2}$ is the unique minimal generator (i.e. it is a generator and no proper subobject is a generator) so also fixed up to isomorphism. Since every category is functorially a colimit of copies of $\mathbb{2}$, once we decide whether our autoequivalence fixes or swaps the two maps $\mathbb{1} \to \mathbb{2}$ we've determined the autoequivalence up to a canonical isomorphism. And of course the nontrivial autoequivalence is the "opposite category" functor $\mathcal{C} \mapsto \mathcal{C}^\mathrm{op}$.

This is a somewhat artificial question, since an autoequivalence of the category $\mathsf{Cat}$ must respect the notion of isomorphism of categories, whereas all we should really want to respect is equivalence of categories. One way to fix this would be to consider $\mathsf{Cat}$ as a 2-category, and ask about the 3-groupoid of autobiequivalences of $\mathsf{Cat}$. But this starts to involve heavy machinery, and the problem will only get worse if we want to go up the categorical ladder. Hence a more "homotopical" approach:

**Question:** What is the biequivalence type of the 2-groupoid $\mathrm{Aut}(\mathrm{ho}\mathsf{Cat})$ of autoequivalences of $\mathrm{ho}\mathsf{Cat}$, the category of small categories localized at the equivalences of categories? Equivalently, $\mathrm{ho}\mathsf{Cat}$ is the category of small categories modulo the congruence identifying naturally isomorphic functors. In particular, is $\mathcal{C} \mapsto \mathcal{C}^\mathrm{op}$ isomorphic to the identity (presumably the answer is no)? Are there any other autoequivalences, essentially (probably not, but you never know)?

A variant would be to ask about the 2-groupoid of functors $\mathsf{Cat} \to \mathsf{Cat}$ which preserve equivalences of categories and induce equivalences on $\mathrm{ho}\mathsf{Cat}$.

I'm aware that Toën showed in some sense that the $(\infty,1)$-category of $(\infty,1)$ categories has an automorphism group of $\mathbb{Z}/2$, and Barwick and Schommer-Pries generalized this to $(\infty,n)$-categories for larger $n$. This question is meant to be an easier / more elementary warmup to those questions.

The approach that worked for $\mathsf{Cat}$ certainly isn't going to work for $\mathrm{Ho}\mathsf{Cat}$: Freyd showed ("On the concreteness of certain categories." Symposia Mathematica. Vol. 4. 1968-9, pp. 431-56) that $\mathrm{Ho}\mathsf{Cat}$ is not concrete, so it certainly can't have a generating object, never mind a colimit-dense object.

By playing around, I've convinced myself at least that the indiscrete categories (the equivalence class of the terminal category) can be characterized in $\mathrm{Ho}\mathsf{Cat}$ by the fact that all functors into it are isomorphic (consider functors from the walking equalizer diagram to see that such a category is a preorder and functors from the terminal category to see that all objects are isomorphic). So $\mathrm{Ho}\mathsf{Cat}(\mathbb{1},-)$ tells us how many isomorphism classes of objects a category has, and what isomorphism classes of functors are doing to them. But that's about all I've got.

havea generator. Plus, colimits in ho(Cat) are hopeless, so that part of the argument isn't going to carry over either. $\endgroup$1more comment