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Nicola Gambino (Apr 18 2024 at 05:50):On page 58 of "Accessible Categories: The Foundations of Categorical Model Theory", by M. Makkai and R. Paré, it is mentioned that there is a characterisation of the class of sketches S whose categories are models Mod(S) are exactly the categories of models of complete theories (in the sense of Model Theory), with elementary embeddings as morphisms. This is followed by a reference to "[M]', which however does not seem to be in the book's references. Does anyone know where this could be found?
Ivan Di Liberti (Apr 18 2024 at 11:12):I guess this must be some paper of Makkai, but I could not tell which one. In LPAC this is attributed to Rosicky's PhD thesis, see the discussion at page 239. In private conversations with Rosicky, he always this result to be his, and large portions of this theory to be developed under the name of modelable categories by Lair.
Kevin Carlson (Arlin) (Apr 18 2024 at 17:05):Do you know what the characterization is, though, Ivan?
Nicola Gambino (Apr 20 2024 at 19:45):I looked on page 239 of "Locally presentable and accessible categories", but I do not see any (direct) mention of complete theories.
Ivan Di Liberti (Apr 24 2024 at 21:10):I am sorry, I misread the question and I concentrated on the part about "elementary embeddings" more than about "complete theories". Don't you think that must be some kind of typo, or imprecise phrasing, though? I can hardly imagine a full characterization of sketches corresponding to complete theories.
Ivan Di Liberti (Apr 24 2024 at 21:13):This MO answer comes to the same conclusion.
dusko (Apr 25 2024 at 00:14):Nicola Gambino said:
On page 58 of "Accessible Categories: The Foundations of Categorical Model Theory", by M. Makkai and R. Paré, it is mentioned that there is a characterisation of the class of sketches S whose categories are models Mod(S) are exactly the categories of models of complete theories (in the sense of Model Theory), with elementary embeddings as morphisms. This is followed by a reference to "[M]', which however does not seem to be in the book's references. Does anyone know where this could be found?
for makkai, the whole point of categorical model theory was to replace the abstract concept of completeness, as a flight through empty space from models to theories, by concrete representation theorems. his first claim at a characterization of completeness was the stone duality for first order logic. it is i think in advances in math, maybe in the late 80s. he turned to sketches as a more communicable language for that idea. the completeness theorem for generalized sketches was the subject of his seminar in the mid-90s. it was a sustained elaboration of barr's representation construction in the framework of sketches. i am sure that he wrote it all up, but i am not sure that it got accepted by either of the communities. the sketches community was just an ongoing argument whether this proof from the early 80s was the same as that proof from the late 80s, and whether it was stolen or independent. and the model theoretic community was... well, some people seemed genuinely religious and saying that there was a category was always a sacrilege. so i can imagine that makkai just kept his manuscript in his desk. there was a genuine objection that he was proving completeness by deriving syntax from semantics. i only have my own seminar notes which i am not sure i completely understood even at the time.
but the upshot seems to be that the model theoretic formalization of the notion of completeness was already a hack (the word used at the time would have been "theology") and that categorical model theory should provide provably universal concepts, falsifiable through functorial semantics, when adjunctions do not yield dualities.
sorry if this does not help much, but that is how much i remember.
Nicola Gambino (Apr 25 2024 at 08:36):Thank you, Ivan and Dusko for your replies.
@Ivan Di Liberti I would say that the MathOverflow rather focuses on the same aspect (i.e. elementary embeddings instead of completeness) rather than comes to same conclusion.
@dusko Makkai has a series of papers on "Generalized sketches as a framework for completeness theorems", so I will look there as well.
By the way, there is a 1982 paper by Lascar in the Journal of Symbolic Logic entitled "On the category of models of a complete theory". At the time, Lascar was at MGill and Makkai is acknowledged in the Introduction.