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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeDec 30th 2018

    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”.)

    diff, v8, current

    • CommentRowNumber2.
    • CommentAuthorGuest
    • CommentTimeMay 2nd 2020
    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 $E$, if $T: E \to Set$ preserves connected limits, then it is a parametric right adjoint. The idea being that the induced functor $E \to Set/T(1)$ preserves all limits, and since a Grothendieck topos is a cototal category, this is enough to ensure that it is a right adjoint.
    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 2nd 2020

    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 EE, if T:ESetT: E \to Set preserves connected limits, then it is a parametric right adjoint. The idea being that the induced functor ESet/T(1)E \to Set/T(1) preserves all limits, and since a Grothendieck topos is a cototal category, this is enough to ensure that it is a right adjoint.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeMar 7th 2021

    Added a remark that any parametric right adjoint has a left multi-adjoint. I haven’t seen this in the literature anywhere; has anyone else?

    diff, v9, current

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 8th 2021

    Does any interesting kind of (co)monad arise through compositions of these?

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMar 8th 2021

    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.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 8th 2021

    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?

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 8th 2021

    At polynomial (∞,1)-functor there’s mention of a local right adjoint functor, which would redirect here. Or should we have a special (,1)(\infty, 1)-p.r.a. page?

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeMar 8th 2021

    I don’t know of a special name.

    Unless we have something particular to say about the \infty-version, we may as well redirect it here.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 10th 2021

    Re #5, I see Diers worked on ’multimonads’, Multimonads and multimonadic categories.

    Any functor U:ABU : A \to B which has a left multiadjoint generates a multimonad on B.

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 16th 2021

    Added

    In database theory p.r.a.s between copresheaf categories, known as data migration functor, are treated in

    diff, v11, current

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeApr 26th 2021

    The legs of a two-sided discrete fibration are not necessarily individually discrete.

    diff, v12, current

    • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 15th 2021

    I removed accessibility hypotheses in accordance with an earlier comment.

    diff, v13, current

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