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Hayato Nasu (Jun 27 2024 at 13:56):Hey there! Here are the slides from the talk I gave two days back. I'm eager to discuss the topic I didn't get to cover in the talk, and I'd love to hear your thoughts on it too.
In the talk, I said that the double category of M-relations is unit-pure and Cauchy if and only if , which is equivalent to (1) . On the other hand, Pavlović proved in the bicategorical context that this is equivalent to (2) ([Pav95] in the references). So, with the two long proofs of his and ours, we obtained (1)(2) for any stable factorization system on any category with finite limits, which he conjectured to be false in the paper. We also got direct proof of this equivalence, (which was a week ago!) Does anybody know this was already shown in any literature?
Link:
https://hayatonasu.github.io/hayatonasu/Talks/DCR-CT2024.pdf
Zoltan A. Kocsis (Z.A.K.) (Jun 27 2024 at 14:02):I don't see the slides, is that a problem on my end or are they not linked?
Hayato Nasu (Jun 27 2024 at 14:33):Thank you for letting me know! I fixed it.
Chad Nester (Jun 28 2024 at 07:48):In Kelly's 1991 paper "A Note on Relations Relative to a Factorization System", in the paragraph following proposition 2.2, he claims to have established that in any category with a stable factorization system the strong epimorphisms coincide with the regular ones. This would imply that (1)(2), so it might be worth looking at!
Chad Nester (Jun 28 2024 at 07:49):(He may also be assuming that the category in question has finite limits, although I guess this shouldn't be a problem).
Hayato Nasu (Jun 28 2024 at 15:58):Thank you for your comment. Actually, by a stable factorization system, he means a proper one. (The right class is included in the class of the monomorphisms). In that case, the situation gets much simpler, which is also mentioned in the paper by Pavlović. I, for one, am interested in the cases where the factorization systems are not proper.
Chad Nester (Jun 29 2024 at 07:16):I see! Clearly I need to read more carefully. Sorry about that.
Hayato Nasu (Jan 29 2025 at 03:07):I completed my master's thesis on connecting categorical logic and double category theory and passed the exam! The thesis is available here: link. I posted it on arXiv two weeks ago, but some overlaps with previous papers seem to keep it on hold. I just wanted to share this with the community as soon as I can. Please feel free to ask questions and share your comments!
I should note one thing here. In Theorem 2.4.8., I presented a proof of the fact that the loose bicategory of a cartesian equipment is a cartesian bicategory, but @Nathanael Arkor told me that the statement was already proven by @Evan Patterson in his paper "Transposing cartesian and other structure in double categories". So I made a revision and added a few comments on the thesis. I need to study more to say something about it, but I am very curious how his proof is related or possibly reduces to mine.
Evan Patterson (Jan 29 2025 at 04:22):Hi, and congrats on completing your master's thesis! Your type theory for virtual double categories is intriguing.
Based on my glance at your Theorem 2.4.8, your proof does look on the surface rather different than mine, more concrete and calculational, and using the string diagram calculus for double categories no less. So as far as I'm concerned, it's great to have another, independent proof of the result in the literature.