A classification of accessible categories

✍ Scribed by Jiřı́ Adámek; Francis Borceux; Stephen Lack; Jiřı́ Rosický

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 198 KB
• Volume: 175
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(02)00126-3

✦ Synopsis

For a suitable collection D of small categories, we deÿne the D-accessible categories, generalizing the -accessible categories of Lair, Makkai, and Parà e; here the -accessible categories are seen as the D-accessible categories where D consists of the -small categories. A small category C is called D-ÿltered 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-ÿltered colimits. The D-accessible categories are then the categories with D-ÿltered colimits and a small set of D-presentable objects which is "dense with respect to D-ÿltered colimits".

We suppose always that D satisÿes 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 classiÿcation 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.