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Stream: theory: category theory

Topic: terminology for functor that is almost monadic

Nathanael Arkor (Jul 07 2021 at 15:29):

Is there a name for a functor that satisfies all of Beck's monadicity conditions, except (necessarily) having a left adjoint? I feel I've encountered such terminology before, but now I can't find a reference.

Ivan Di Liberti (Jul 07 2021 at 19:25):

I've never encountered the terminology, and I know only few examples of this behaviour. Maybe I would use the terminology "virtually monadic"?

dusko (Jul 10 2021 at 05:21):

Nathanael Arkor said:

Is there a name for a functor that satisfies all of Beck's monadicity conditions, except (necessarily) having a left adjoint? I feel I've encountered such terminology before, but now I can't find a reference.

the conditions of effective descent predate monadicity, and were not expressed in terms of beck's imaginative split coequalizers, but the fact that effective descent is equivalent to monadicity without the left adjoint is in my paper in the Como proceedings, springer LNM 14??. the manuscript should also be on my web page under math.

dusko (Jul 10 2021 at 05:26):

((i am not sure how clearly i remember the contents of that paper, and i am not sure that i want to refresh the memory, but i remember the form: it was typeset on Apple Mark II using MacWrite which became MS Word and MacDraw which didn't become anything, but Allie Brosh's Solutions and Other Problems carries some of the spirit.))

it's here at the bottom: http://dusko.org/maths-and-computation/

Nathanael Arkor (Jul 10 2021 at 11:22):

but the fact that effective descent is equivalent to monadicity without the left adjoint is in my paper in the Como proceedings

Thanks @dusko – I had in my mind that it was likely related to descent but couldn't find the right term. On the nLab, it says that a functor of descent type has a left adjoint (and being of effective descent type is even stronger), so it seems this may still not be quite what I'm looking for. But it's a useful pointer, thanks.

dusko (Jul 11 2021 at 01:59):

Nathanael Arkor said:

but the fact that effective descent is equivalent to monadicity without the left adjoint is in my paper in the Como proceedings

Thanks dusko – I had in my mind that it was likely related to descent but couldn't find the right term. On the nLab, it says that a functor of descent type has a left adjoint (and being of effective descent type is even stronger), so it seems this may still not be quite what I'm looking for. But it's a useful pointer, thanks.

haha nLab is useful, but i am not sure how reasonable it is to take it into the authoritative source for such things. grothendieck defined fibrations (was it in SGA1) to specify functorial descent. a fibration has a left adjoint iff the fibres have an initial object. that property is obviously irrelevant for descent. so requiring that a functor of descent type has a left adjoint does not make much sense...

maybe someone was reading papers about descent, and noticed that in topos theory benabou-roubaud's characterization of descent by monadicity is most often used. but that is like requiring that all triangles are are right, because then we can apply the pythagoras theorem. descent is much more general than descending from categories of algebras, and predates topos theory by some 150 years.

Nathanael Arkor (Jul 11 2021 at 14:25):

@dusko: what you say makes a lot of sense. Unfortunately, it appears that functors of descent type have come to mean premonadic functors and functors of effective descent type have come to mean monadic functors, including the left adjoint condition, in the categorical literature. For one example, see Kelly–Power's Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads, but I found many more papers using the same convention (in Berger's On some aspects of the theory of monads, both terms appear explicitly).

Chad Nester (Jul 11 2021 at 15:48):

You might take the opportunity to coin “almostnadic” :thinking:

Nathanael Arkor (Jul 11 2021 at 16:08):

In several papers I have read sentences along the lines of "A functor satisfying [so and so] is monadic if it has a left adjoint.", so it does surprise me there is not an existing term. I may coin one, though I shall have to consider whether I want a punny term :big_smile:

Jon Awbrey (Jul 11 2021 at 18:25):

I think the Latin would suggest penmonadic.

Ian Coley (Jul 11 2021 at 20:23):

Jon Awbrey said:

I think the Latin would suggest penmonadic.

I don't think `pen' as a prefix has survived anywhere but peninsula, penumbra, and penultimate.

Mike Shulman (Jul 11 2021 at 20:48):

Perhaps worth considering "promonadic", by analogy with "profunctor", "promonoidal category", and so on, where the structure "exists but is not representable"?

Ian Coley (Jul 11 2021 at 20:50):

Is a promonad a monoid in the category of endo-profunctors? :upside_down:

Mike Shulman (Jul 11 2021 at 21:22):

That's certainly one possible meaning. I don't think it's very common though.

John Baez (Jul 11 2021 at 22:48):

The opposite of monadic is lessnadic.