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nLab > Latest Changes: ambidextrous adjunction

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 10th 2014

    started a stub for ambidextrous adjunction, but not much there yet

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 10th 2014
    • (edited Jan 10th 2014)

    also stubs for Frobenius monoid and Frobenius monad, just for completeness; added cross-links

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeJan 18th 2014

    Will any equivalence between the two adjoints do, or should it be compatible with the units and counits in some way?

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJan 18th 2014

    I don’t think there is any compatibility that you can ask for. The specified equivalence between the two adjoints should be regarded as part of the structure, i.e. being an ambijoint is not just a mere property of a functor.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 31st 2014
    • (edited Jan 31st 2014)

    I have added a paragraph Properties – Fiberwise characterization of ambid. Kan extension in order to record the corresponding result by Hopkins-Lurie.

    (Eventually there should maybe be a standalone section/entry on ambidextrous Kan extensions.)

    • CommentRowNumber6.
    • CommentAuthorvarkor
    • CommentTimeJul 26th 2021

    Mention “biadjoint” terminology.

    diff, v3, current

    • CommentRowNumber7.
    • CommentAuthorJohn Baez
    • CommentTimeJan 16th 2023

    Added reference to Qiaochu Yuan’s comment about the need to equip an ambidextrous adjunction with an equivalence between the left and right adjoints.

    diff, v6, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2023
    • (edited Mar 24th 2023)

    added cross-link with bireflective subcategory and pointer to:

    diff, v7, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2023
    • (edited Apr 25th 2023)

    added pointer to:

    diff, v9, current

    • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeApr 26th 2023
    • (edited Apr 26th 2023)

    Functor GG which has a left and right adjoint which are equivalent is said to be Frobenius functor.

    Connection to Hopf adjunctions

    • Harshit Yadav, Frobenius monoidal functors from (co)Hopf adjunctions, arXiv:2209.15606

    diff, v10, current

    • CommentRowNumber11.
    • CommentAuthorvarkor
    • CommentTimeMay 29th 2023

    Mention terminology “ambiadjunction” and add redirects.

    diff, v11, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeAug 10th 2023

    added pointer to the early article

    • Kiiti Morita, Adjoint pairs of functors and Frobenius extensions, Science Reports of the Tokyo Kyoiku Daigaku, Section A 9 202/208 (1965) 40-71 [jstor:43698658]

    where ambidextrous adjunctions were (first?) discussed under the name “strongly adjoint pair”

    diff, v12, current

    • CommentRowNumber13.
    • CommentAuthorvarkor
    • CommentTimeJul 17th 2024

    Mention that the stronger notion of ambidextrous adjunction (i.e. Frobenius adjunction) makes the notion a property rather than structure.

    diff, v16, current

    • CommentRowNumber14.
    • CommentAuthorncfavier
    • CommentTime6 days ago

    Added adjoint equivalences as an example.

    diff, v17, current

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