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varkor
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Existence of nontrivial categories in which every object is atomic
Thanks. I had spotted that having an initial object made the category pointed, but completely overlooked that this implied triviality.
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Existence of nontrivial categories in which every object is atomic
@TimCampion: I would also be interested to know if there are any such examples, if there are no nonposetal examples :) Maybe I'll cheekily update my question.
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Existence of nontrivial categories in which every object is atomic
Thank you, this is already very strong evidence that atomocity of every object is an extremely restrictive assumption. In other words, every tiny object must be "superinitial" (assuming enough coproducts).
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Existence of nontrivial categories in which every object is atomic
@MaximeRamzi: thanks, I have clarified that I meant "internally tiny"; I overlooked the distinction between the two notions.
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Adjunctions with respect to profunctors
I've discovered the same question was asked here.
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What are _all_ of the exactness properties enjoyed by stable $\infty$-categories?
This is a beautiful idea, and I very much look forward to reading the paper when it is ready. Is there any relation to the fact that (unenriched) limits can be constructed using large colimits (e.g. Proposition 12.8 of The Joy of Cats)?
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EM functor from monads to adjunctions
You may also be interested in this recent talk, which presents a novel perspective on the structure–semantics adjunction which has similar inspiration to your questions.
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EM functor from monads to adjunctions
Pumplün's article is freely available on a certain repository of scientific papers :)
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EM functor from monads to adjunctions
Unfortunately I still don't have time to expand, but Auderset's Adjonctions et monades au niveau des 2-catégories gives a fairly explicit derivation of the Eilenberg–Moore and Kleisli constructions from the 2-categorical perspective. Pumplün's is a different (albeit related) construction, but more relevant to the structure–semantics adjunction, because it also characterises the functors between Eilenberg–Moore categories and between Kleisli categories appropriately, which Auderset's does not.
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How does Gabriel–Ulmer duality extend to (limit, colimit) sketches?
A duality relative to a limit doctrine gives a general answer to "To what extent can Gabriel–Ulmer duality be generalised to different classes of limits and colimits?". It's not in the terminology of sketches, but that should be a straightforward conversion. The enriched version will follow from Accessibility and presentability in 2-categories.
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Algebraically-free monadicity theorem
@VladimirSotirov: that's an interesting observation, thanks! (Presumably this is something you spotted and isn't written down in the literature anywhere?)
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Algebraically-free monadicity theorem
Thanks, these are helpful observations!
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