About (co)limits of accessible categories

Question

I am reading the paper colimits of accessible categories. In the introduction, the authors summarize what is known about limits and colimits of accessible categories. I believed that there was something known about homotopy limits or colimits of accessible categories as well. Browsing zbMATH or MR give nothing for "accessible category AND ((ho)motopy) limit". Could I have a reference please (or better keywords) ?

Answer 1

I can only offer a partial answer:

First, have a look at an explicit description of limits of categories.

Namely, strong (or pseudo-) limits can be modelled as the category of cartesian sections into the Grothendieck construction. Denote this category $\Gamma_{\mathrm{Gr}(F)}$.

[Part I] Existence of $\kappa$-directed limits:

Now; Let $F:X\to\mathrm{Cat}$ be a strong functor such that all the functors $F(f):F(x)\to F(y)$ involved are $\kappa$-accessible. Then $\kappa$-filtered colimits can be computed pointwise.

More generally, consider a functor $\gamma:I\to \Gamma_{\mathrm{Gr}(F)}$ and assume all the functors $F(f)$ to preserve $I$-indexed colimits. Then for every $x\in X$ we get a functor $\gamma_{(-)}(x):I\to F(x)$. Then take $\gamma^*:X\to\Gamma_{\mathrm{Gr}(F)}$ to be the functor given on objects by $$x\mapsto \mathrm{colim}_i\,\gamma_i(x).$$ The morphism part of this functor is defined using the universal properties of the colimits involved. If i'm not mistaken, $\gamma^*$ is a colimit of $\gamma$ and again a cartesian section.

As a special case take $I$ to be any $\kappa$-directed poset.

[Part II] Generated by $\kappa$-compact objects?

I'm not so sure about the second part of the definition of $\kappa$-accessible category.

My idea would be to consider the case where all the $F(f)$ preserve $\kappa$-compact objects. This is certainly the case in the article you linked to as Paré and Rosický consider fully faithful functors.

The problem now is that i don't see how one could extend a $\kappa$-compact object in one fiber $F(x)$ to a global cartesian section: Even though the stalks/fibers might be generated by $\kappa$-compact objects i don't know how to even of the existence of such objects in the category of (cartesian) sections.

But perhaps one of the other posters can say something in this direction.