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Ralph Sarkis (Apr 06 2022 at 15:27):I am starting to learn categorical logic/algebra and I was wondering how special was in the statement of completeness of algebraic theories (e.g. 1.1.29 here). The proof seems to say that isn't more special than usual (i.e. it is the category where we enrich everything and Yoneda lemma is great). So I thought maybe there is a characterization of what types of theories (algebraic, regular, etc...) are complete with respect to models (I even thought this could be true for any type of theory, but nlab says that it does not work for infinitary theories). I found an earlier discussion here that lead to this comment which sounds great.
Morgan Rogers (he/him) said:
but non-trivial completeness results such as "true in all set-models iff provable in the theory" are recovered as soon as there are 'enough' Set-models for the corresponding functors to be jointly conservative.
Can someone point me to a formal version of this comment?
Morgan Rogers (he/him) (Apr 06 2022 at 15:30):Yep, look to section D1.5 of Johnstone's Sketches of an Elephant.
Morgan Rogers (he/him) (Apr 06 2022 at 15:32):Regular theories have enough set-models constructively, coherent theories have enough set-models non-constructively, and general geometric theories can fail to have enough set-models.
Ralph Sarkis (Apr 06 2022 at 16:24):Thank you! This chapter had many many answers to my questions.