Let be a universe and let be a universe strictly greater than . Then a -small lex -cocomplete category is -presentable if and only if it is -compact in .
Hint
The “only if” part follows from the following facts/observations.
- -If is a -presentable lex -cocomplete category, then it is a lex localization of the category of -small presheaves on a -small lex category.
- -A lex localization of a -presentable lex -cocomplete category is a coinverter in .
- -The category of -small presheaves on a -small lex category is -compact in .
For the “if” part, observe that the -compact objects in are constructible under -small colimits from the free lex -cocomplete category over one object. Then the proof proceeds by induction.
- -The free lex -cocomplete category over one object is the category of -small presheaves on the free lex category over one object. Thus, it is -presentable.
- --presentable lex -cocomplete categories are closed under -small colimits in . This is proved by tracking the construction of colimits of -presentable lex -cocomplete categories given in Section 6.3.4 of [Lurie--2009-0000]. The key observation is that colimits of -presentable lex -cocomplete categories are decomposed as colimits of -small lex categories and lex localizations both of which are colimits in .