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nLab > Latest Changes: multi-adjoint

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  1.  
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeMar 7th 2021

    Created page.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 10th 2021

    Added to references

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2021

    slightly touched the formatting, e.g. moving remarks on the definition from a subsection at the end of the entry to remarks following the definition

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 10th 2021

    There should be some Isbell duality connection with Osmond’s ideas, no?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2021

    added pointer to

    diff, v6, current

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 10th 2021

    In Multimonads and multimonadic categories, Diers writes

    We use the notion of multiadjunction developed in [4]

    [4] is

    • Some spectra relative to functors, to appear in J. Pure Appl. Algebra.
    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 10th 2021

    Where Diers then refers us back to

    • Y. Diers, Spectres et localisations relatifs a un foncteur, C.R. Acad. Sci. Paris. Serie A 287 (1978), 985-988.
    • Y. Diers, Familles universelles de morphismes, Ann. Sot. Sci. Bruxelles 93111 (1979) 175-195.
    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 10th 2021

    Added a reference to Diers spectrum.

    diff, v8, current

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 11th 2021

    Added a section on multi-monads

    Any functor U:A→BU: A\to B which has a left multi-adjoint generates a multi-monad on BB. Categories AA which can be reconstructed from this multi-monad are called multi-monadic (Diers 80).

    Multi-monadic categories on SetSet can be characterized in the following way: they are regular, with connected limits, with coequalizers of coequalizable pairs, their equivalence relations are effective, their forgetful functors preserve coequalizers of equivalence relations and reflect isomorphisms. Unlike monadic categories they need not have products. Examples include local rings, fields, inner spaces, locally compact spaces, locally compact groups, and complete ordered sets. (Diers 80, p.153)

    Seems like important ideas, these ’multi-’ constructs. Why do we hear so little about them?

    diff, v10, current

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 11th 2021

    How about some right multi-adjoints and multi-comonads - any nice cases?

    • CommentRowNumber11.
    • CommentAuthoranuyts
    • CommentTimeFeb 3rd 2023

    Characterize multi-adjoints using isomorphism of Homs.

    diff, v11, current

    • CommentRowNumber12.
    • CommentAuthoranuyts
    • CommentTimeFeb 3rd 2023

    Dont use align*

    diff, v11, current

    • CommentRowNumber13.
    • CommentAuthoranuyts
    • CommentTimeFeb 3rd 2023

    Multi-adjoints are relative adjoints.

    diff, v12, current

    • CommentRowNumber14.
    • CommentAuthoranuyts
    • CommentTimeFeb 3rd 2023

    Proof for parametric right adjoint.

    diff, v12, current

    • CommentRowNumber15.
    • CommentAuthoranuyts
    • CommentTimeFeb 3rd 2023

    Formatting.

    diff, v12, current

    • CommentRowNumber16.
    • CommentAuthormaxsnew
    • CommentTimeFeb 5th 2023

    mention equivalence to parametric right adjoints in the idea section.

    diff, v13, current

    • CommentRowNumber17.
    • CommentAuthoranuyts
    • CommentTimeFeb 27th 2023

    Reexplain Fam(D).

    diff, v14, current

    • CommentRowNumber18.
    • CommentAuthorvarkor
    • CommentTimeMar 16th 2023

    Link to multimonad.

    diff, v15, current

    • CommentRowNumber19.
    • CommentAuthorvarkor
    • CommentTimeMar 16th 2023

    Add another characterisation of multiadjoints.

    diff, v15, current

    • CommentRowNumber20.
    • CommentAuthorvarkor
    • CommentTimeApr 29th 2023
    • (edited Apr 29th 2023)

    It looks like the “abstract polynomial functors” of Street’s “Variation on a comprehensive theme” are exactly the functors with left multiadjoints.

    • CommentRowNumber21.
    • CommentAuthorvarkor
    • CommentTimeJun 21st 2023

    Add some more redirects.

    diff, v18, current

  22. Fixed the definition of Fam

    David Jaz

    diff, v21, current

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeNov 11th 2023
    • (edited Nov 11th 2023)

    Moved the lengthy definition/discussion of Fam(D)Fam(D) (here) out of the following theorem environment. Added links to (dual of) free coproduct completion, where this is all discussed further.

    diff, v22, current

  24. Linked new page for Field, the category of fields. Also linked those.

    diff, v24, current

    • CommentRowNumber25.
    • CommentAuthorvarkor
    • CommentTimeMay 22nd 2024

    Updated the relative adjunction terminology to disambiguate.

    diff, v25, current

    • CommentRowNumber26.
    • CommentAuthorTim_Porter
    • CommentTime6 days ago
    • (edited 6 days ago)

    They are due to Diers, and the previous wording results in an impression of doubt. Yves Diers talked on them at category theory meetings for many years in the 1970s and 1980s.

    diff, v27, current

    • CommentRowNumber27.
    • CommentAuthorTim_Porter
    • CommentTime6 days ago

    Changed the wording slightly.

    diff, v28, current

1 to 27 of 27

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