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Ambroise (Sep 11 2024 at 16:57):Is there any syntactic characterization of the generalised algebraic theories (or essentially algebraic theories, or any other syntactic specification for LP-cats) whose models are Grothendieck toposes?
Nathanael Arkor (Sep 11 2024 at 17:26):According to Theorem 3.1 of Di Liberti and Ramos González's Gabriel-Ulmer duality for topoi and its relation with site presentations, a LFP category is a Grothendieck topos if and only if its subcategory of finitely presentable objects is extensive and pro-exact, which reduces the problem to giving syntactic characterisations of these conditions. The paper Syntactic characterizations of various classes of locally presentable categories of Carboni, Pedicchio, and Rosický is also relevant, though I think they mean "syntax" in a rather weaker sense.
Ambroise (Sep 11 2024 at 17:28):Thanks for the references!
David Michael Roberts (Sep 12 2024 at 01:34):Extensivity seems not too hard to me, but having to specify it only for a subcategory feels like a little bit of a wrinkle. Or rather, figuring out how to identify the subcat of fin. pres. objects feels the tricky part, but one that is specified, extensivity of that feels not too hard.
Nathanael Arkor (Sep 12 2024 at 06:41):The point is that the syntax is specified by the subcategory of finitely presentable objects. In other words, this is the starting data, not the LFP category itself.
David Michael Roberts (Sep 12 2024 at 09:39):Ah, ok. That thought did occur to me, but I didn't realise that was the actual intention.