For a suitable collection D of small categories, we define the D-accessible categories, generalizing the k-accessible categories (for a regular cardinal k) of Lair, Makkai, and Paré here the k-accessible categories are seen as the D-accessible ones where D consists of the k-small categories. A small category C is called D-filtered when C-colimits commute with D-limits in the category of sets. An object of a category is called D-presentable when the corresponding representable functor preserves D-filtered colimits. The D-accessible categories are then the categories with D-filtered colimits and a small set of D-presentable objects which is ``dense with respect to D-filtered colimits''.
We suppose always that D satisfies a technical condition called ``soundness'': this is the ``suitable'' case mentioned above. Every D-accessible category is accessible; thus the choice of different sound D provides a classification of accessible categories, as referred to in the title. A surprising number of the main results from the theory of accessible categories remain valid in the D-accessible context.
The locally D-presentable categories are defined as the cocomplete D-accessible ones. When D consists of the finite categories, these are precisely the locally finitely presentable categories are Gabriel and Ulmer. When D consists of the finite discrete categories, these are the finitary (many-sorted) varieties.
As a by-product of this theory, we prove that the free completion under D-filtered colimits distributes over the free completion under limits. This result is new, even in the case where D is empty and D-filtered colimits are just arbitrary (small) colimits.
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