Synthetic topos theory
[004Z] Exercise

Let UU be a universe and let VV be a universe strictly greater than UU. Then a VV-small lex UU-cocomplete category CC is UU-presentable if and only if it is UU-compact in LexCocomp(U,V)\mathord {\textnormal {\textsf {LexCocomp}}}(U,V).

Hint

The “only if” part follows from the following facts/observations.

  • -If CC is a UU-presentable lex UU-cocomplete category, then it is a lex localization of the category of UU-small presheaves on a UU-small lex category.
  • -A lex localization of a UU-presentable lex UU-cocomplete category is a coinverter in LexCocomp(U,V)\mathord {\textnormal {\textsf {LexCocomp}}}(U,V).
  • -The category of UU-small presheaves on a UU-small lex category is UU-compact in LexCocomp(U,V)\mathord {\textnormal {\textsf {LexCocomp}}}(U,V).

For the “if” part, observe that the UU-compact objects in LexCocomp(U,V)\mathord {\textnormal {\textsf {LexCocomp}}}(U,V) are constructible under UU-small colimits from the free lex UU-cocomplete category over one object. Then the proof proceeds by induction.

  • -The free lex UU-cocomplete category over one object is the category of UU-small presheaves on the free lex category over one object. Thus, it is UU-presentable.
  • -UU-presentable lex UU-cocomplete categories are closed under UU-small colimits in LexCocomp(U,V)\mathord {\textnormal {\textsf {LexCocomp}}}(U,V). This is proved by tracking the construction of colimits of UU-presentable lex UU-cocomplete categories given in Section 6.3.4 of [Lurie--2009-0000]. The key observation is that colimits of UU-presentable lex UU-cocomplete categories are decomposed as colimits of UU-small lex categories and lex localizations both of which are colimits in LexCocomp(U,V)\mathord {\textnormal {\textsf {LexCocomp}}}(U,V).