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Stream: learning: questions

Topic: cancellation for monadic adjunctions

view this post on Zulip Nathanael Arkor (Apr 29 2023 at 14:58):

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In this situation, if r′r' and r′rr' r are monadic, then so is rr. Does anyone know a standard reference for this fact? I only found it in some lecture notes of Velebil (which references a paper of Bourn that I couldn't find), but the result must be much older than that.

view this post on Zulip Tom Hirschowitz (Apr 29 2023 at 16:55):

Borceux (vol 2), Corollary 4.5.7 (when CC has coequalisers)?

view this post on Zulip Nathanael Arkor (Apr 29 2023 at 17:10):

That's almost what I'm looking for, thanks, but I actually do need the version without the coequaliser assumption.

view this post on Zulip John Baez (Apr 29 2023 at 17:36):

I hadn't known this fact, thanks! I just knew the (more famous?) bad news, that if rr and r′r' are monadic then r′rr'r might not be.

(I mention this in case some amateurs out there are wondering....)

view this post on Zulip Ralph Sarkis (Jun 06 2023 at 09:19):

Is this result also true for strict monadicity ?

view this post on Zulip Nathanael Arkor (Jun 06 2023 at 10:11):

Yes; it's stated as Corollary 5.6 of Relative monadicity.

view this post on Zulip Ralph Sarkis (Mar 20 2024 at 19:55):

@Nathanael Arkor I don't know if you are still looking for that paper, but it is in CT 1991 on page 55.

view this post on Zulip Nathanael Arkor (Mar 20 2024 at 21:16):

@Ralph Sarkis: thanks – I managed to obtain a copy in the end, but this is useful to know!