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Added alternative terminology “local right adjoint” and “strongly cartesian monad” from Berger-Mellies-Weber. They claim the former “has become the more accepted terminology” than “parametric right adjoint”; does anyone know other references to support this? (I think it’s certainly more logical, in that it fits with the general principle of “local” meaning “on slice categories” — not to be confused with the different general principle of “local” meaning “in hom-objects”.)
Sorry, that was me. Rewriting:
In section 2 on Properties, can someone explain the need for accessibility hypotheses? I thought that I was able to prove that for a Grothendieck topos , if preserves connected limits, then it is a parametric right adjoint. The idea being that the induced functor preserves all limits, and since a Grothendieck topos is a cototal category, this is enough to ensure that it is a right adjoint.
Does any interesting kind of (co)monad arise through compositions of these?
I don’t know!
But I did just notice that the “generic morphisms” in p.r.a. theory are precisely the universal family of arrows in a multi-adjoint. And that means the converse is true too: if a functor has a left multi-adjoint and its domain has a terminal object, then it is a parametric right adjoint. So the two notions are really almost exactly the same.
Are the left multi-adjoints of polynomial functors given a special name?
Presumably this is all general abstract and will appear homotopified, etc. E.g., if Tambara functors are a kind of polynomial functor, is there a more general p.r.a. form?
At polynomial (∞,1)-functor there’s mention of a local right adjoint functor, which would redirect here. Or should we have a special -p.r.a. page?
I don’t know of a special name.
Unless we have something particular to say about the -version, we may as well redirect it here.
Re #5, I see Diers worked on ’multimonads’, Multimonads and multimonadic categories.
Any functor which has a left multiadjoint generates a multimonad on B.
Added
In database theory p.r.a.s between copresheaf categories, known as data migration functor, are treated in
- {#Spivak10} David Spivak, Functorial Data Migration, (arXiv:1009.1166)
In Section 3 of the current version of this page, it is asserted (just before the first Proposition of that section) that if is any monad, then the Kleisli category admits a (generic, free) orthogonal factorization system. I’m dubious of this – after a quick search, the closest I could find to a reference is some discussion after Scholium 2.7 in Weber’s Familial 2-functors and parametric right adjoints, where he says that Berger showed this in A cellular nerve for higher categories, under the assumption that is parameteric right adjoint. But even then, Weber doesn’t give a more precise reference than that, and I haven’t been able to track it down in that paper.
So (1) what is the correct statement here and (2) what is a precise reference?
Just to clarify, it asks for to be any monad for which every morphism admits a generic factorisation, which seems a reasonably strong condition. Another reference for these factorisation systems is Monads with arities and their associated theories, which might make the situation clearer.
Oh, of course you’re right, thanks. So the statements seem to match up now, and that seems like a great lead for a reference.
Mention conflict of terminology with local adjunctions.
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