# Cocomplete and finitely complete category with nice pullbacks that is not locally presentable

I have a result that holds for cocomplete and finitely complete categories such that pullbacks preserve directed colimits, by which I mean $$A \times_B (\operatorname{colim}_{i \in I} C_i) = colim_{i \in I} (A \times_B C_i)$$, where $$I$$ is a directed category.

Thanks to a result by Adamek and Rosicky [locally presentable and accessible categories, Proposition 1.59] , I know that this holds in any locally (finitely) presentable category.

Is there any other category that satisfies those conditions?

If you look at the article on quasitoposes, which are locally cartesian closed and therefore satisfy your exactness condition, you'll find a number of examples that are not locally presentable. For example, the category of pseudotopological spaces, the category of bornological sets, the category of equilogical spaces, and (I think) the category of quasitopological spaces in the sense of Spanier. I might think of more (preferably a whole class of more) later.

• All responses are helpful, but I'll mark this one as accepted because the examples feel more natural to me. Thank you! – Maxime Lucas Apr 1 at 9:33

Filtered colimits commute with finite limits in any Grothendieck topos. A Grothendieck topos does not need to be locally finitely presentable; the presentability rank of a topos is tightly related to the structure of its site presentation, as shown in Prop. 5.5 of the preprint

Gabriel-Ulmer duality for topoi and its relation with site presentations, Ivan Di Liberti and Julia Ramos González, arXiv:1902.09391.

Indeed I must confess a conflict of interests, as I am one of the authors of that preprint.

• This looks interesting, thank you! Do you have an example of a "weakly k-ary site" for k > aleph_0 ? – Maxime Lucas Mar 31 at 8:26
• Actually Shulman's paper contains lots of examples. Thanks! – Maxime Lucas Mar 31 at 8:57

Every localization (= full reflective subcategory such that the reflector preserves finite limits) of a locally finitely presentable category satisfies this property. More can be found in Localisation of locally presentable categories (Brian Day and Ross Street, 1989).