Oh I see, I didn’t parse the “exists K” in the definition of accessible category correctly. I rephrased the definition slightly on the accessible category page to hopefully make that clearer.

Thanks!

]]>It has to do with cardinality. A category with directed colimits and a generating set of compact objects is a *finitely* accessible (or $\omega$-accessible) category. These are the ones that correspond to domains. A general accessible category is $\kappa$-accessible for some cardinal $\kappa$, and has $\kappa$-directed colimits and a generating set of $\kappa$-compact objects. Any (small) poset satisfies this property when $\kappa$ is the cardinality of the poset.

There’s a table on several nlab articles, for example at the bottom of accessible categories that relates “rich categories” with “rich preorders/posets”.

Most of it makes sense to me (Topos - Locale, Powerset - Presheaf), but I don’t understand the “accessible” column.
Specifically, the definition of accessible categories are categories as having directed colimits and a generating set of compact objects, is almost word for word the definition of *algebraic domain* used in for example Abramsky and Jung’s notes on domain theory, which is that it is a poset with directed suprema and a basis of compact objects.

It seems to me like these are the appropriate analogous concepts, for example the Ind-completion of a small category is the analogue of the Ideal completion of a poset.

So why is it that in the table posets are the analogue of accessible categories? I’m guessing there is a different analogy than I am thinking about?

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