A classification of accessible categories
(2002) Journal of Pure and Applied Algebra — Vol. 175, n° 1-3, p. 7-30 (2002)
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- Abstract
- For a suitable collection D of small categories, we define the D-accessible categories, generalizing the A-accessible categories of Lair, Makkai, and Pare; here the A-accessible categories are seen as the D-accessible categories where D consists of the A-small categories. A small category W is called D-filtered when W-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.
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Adamek, J., Borceux, F., Lack, S., & Rosicky, J. (2002). A classification of accessible categories. Journal of Pure and Applied Algebra, 175(1-3), 7-30. https://doi.org/10.1016/S0022-4049(02)00126-3 (Original work published 2002)