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1. • CommentRowNumber1.
   • CommentAuthormaxsnew
   • CommentTimeMar 8th 2017
   • (edited Mar 8th 2017)
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   Author: maxsnew
   Format: MarkdownItexThere's a table on several nlab articles, for example at the bottom of [accessible categories](https://ncatlab.org/nlab/show/accessible+category) 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](https://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf), 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?

   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?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMar 8th 2017
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   Author: Mike Shulman
   Format: MarkdownItexIt 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthormaxsnew
   • CommentTimeMar 8th 2017
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   Author: maxsnew
   Format: MarkdownItexOh 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!

   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!