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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?

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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.)

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