# Compact "non-discrete" objects in non-accessible categories

An object $$C$$ of a category $$\mathcal{C}$$ with filtered colimits is compact if its hom-functor $$\mathcal{C}(C, -) : \mathcal{C} \to \mathsf{Set}$$ preserves $$\alpha$$-filtered colimits for some regular cardinal $$\alpha$$. It is well known that every object of an accessible category is compact. There are also many non-accessible categories that have certain compact objects, but these objects tend to be "discrete", in the sense that they are small co-powers of the terminal object (e.g. I believe that any discrete topological space is compact). My question is, are there any (interesting) examples of non-accessible categories with compact objects that are $$not$$ discrete, i.e. that are not small co-powers of the terminal object?

Let $$\textbf{On}$$ be the category of ordinals and let $$\mathcal{E} = [\textbf{On}^\textrm{op}, \textbf{Set}]$$. It is an amusing exercise to check that $$\mathcal{E}$$ is an elementary topos. (Hint: every slice of $$\textbf{On}$$ is small!) However, $$\mathcal{E}$$ is not accessible: indeed, the terminal object is not $$\kappa$$-presentable for any $$\kappa$$. Nonetheless, $$\mathcal{E}$$ has a large supply of presentable objects:
Proposition. An object in $$\mathcal{E}$$ is $$\kappa$$-presentable if and only if it is the colimit of a $$\kappa$$-small diagram of representable presheaves.
(What's "really" going on is that $$\mathcal{E}$$ is "almost" an accessible category – there are "only" two problems. First, $$\mathcal{E}$$ has too many generating objects: the class of representable presheaves is a minimal generating class of objects, whereas an accessible category must be generated by a small set of objects. Second, $$\mathcal{E}$$ has objects that are too big: as we saw, even the terminal object fails to be presentable, whereas every object in an accessible category is presentable. If we permit ourselves to work in a larger Grothendieck universe we would see that $$\mathcal{E}$$ can be embedded in a Grothendieck topos in a very nice way and that is basically why it has many of the same good properties.)