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Morgan Rogers (he/him) (Nov 19 2025 at 17:31):Recall the notion of [[adhesive category]]. Any topos is adhesive. I noticed today that while Set is not coadhesive, it is the case that pullback squares of epimorphisms in Set are pushouts-- the failure is that these pullback squares are not stable under pushout.
Let's call categories where this happens "costicky" (or perhaps someone knows of an established term for the corresponding weakening of adhesive).
Which categories are costicky? Are regular categories costicky, for instance?
Graham Manuell (Nov 20 2025 at 05:30):I've been working on co(quasi)adhesive categories recently. One interesting result is that under certain conditions it implies protomodularity. We also have a few nice examples. I'm not sure about your weaker condition though.
Morgan Rogers (he/him) (Nov 20 2025 at 08:31):What does the "(quasi)" indicate?
Morgan Rogers (he/him) (Nov 20 2025 at 09:48):Ah I found in Lack and Sobicinski that it's the variant obtained by replacing "mono" with "regular mono" everywhere in the definition of adhesive. That makes sense.
Morgan Rogers (he/him) (Nov 20 2025 at 14:21):So it turns out that this property hold in any regular category: not only are pullbacks of epis epis, the squares are pushout squares.
Maybe this is in Borceux somewhere?
Morgan Rogers (he/him) (Nov 20 2025 at 14:31):(deleted)