You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
Nathanael Arkor (Mar 18 2022 at 13:35):If is a bijective-on-objects functor, then is monadic. Does anyone know where I can find a reference for this fact?
Zhen Lin Low (Mar 18 2022 at 14:54):I don't have a reference but this seems to be an easy application of the monadicity theorem: if you have a e.s.o. functor then precomposition is conservative; and in any case, precomposition has both a left adjoint and a right adjoint. But then a conservative functor that preserves coequalisers and has a left adjoint must be faithful, hence monadic.
Nathanael Arkor (Mar 18 2022 at 14:56):Yes, I agree, thank you. But it seems like a useful enough observation that it should be written down somewhere, and I would hope that it has been already: it seems more appropriate to give a reference than to reprove the statement in any paper in which it is needed.
Zhen Lin Low (Mar 18 2022 at 14:59):I'm pretty sure I've seen it before somewhere when I was reading about many-sorted algebraic theories.
Zhen Lin Low (Mar 18 2022 at 15:02):In fact, I wrote specifically about the b.o. case here (1.7.14), hah. Maybe it's in the arities for monads paper.
Morgan Rogers (he/him) (Mar 18 2022 at 20:58):It's already true when f is just surjective on objects up to retracts (this is how one gets the surjective part in the surjection-inclusion factorization system on essential geometric morphisms between presheaf toposes coming from functors between small categories)
So you could quote the relevant example from A4.2 in the Elephant?
Nathanael Arkor (Mar 19 2022 at 02:45):Ah yes, it is stated in Example A4.2.7, thank you!