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nLab > Latest Changes: sliced adjoint functors -- section

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2021

    this is a bare little section, to be !include-ed as a Properties-subsection at slice category and adjoint (∞,1)-functor (where two copies of this same section used to be all along) and also at slice (∞,1)-category and adjoint functor (where, for completeness, the same should be recorded, too, but wasn’t until now)

    It would also be good to expand a little here, for instance by adding a pointer to a 1-category textbook account (this is probably in Borceux, but I haven’t checked yet), or, of course, by adding some indication of the proof.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2021
    • (edited Jul 15th 2021)

    I have added the proof of the slice adjunction

    𝒟 /L(c)⊥⟶R /c⟵L /c𝒞 /c \mathcal{D}_{/L(c)} \underoverset {\underset{\;\;R_{/c}\;\;}{\longrightarrow}} {\overset{\;\;L_{/c}\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/c}

    in 1-category theory – and in tikzcd diagrams.

    The typesetting looks quite neat in my local pdf rendering. On the nnLab it currently comes out not quite as neat since:

    (1) the font size of the tikzcd diagrams does not match that of the ambient text,

    (2) tikzcd diagrams can’t be put inline here, or even just next to each others in a line.

    But I guess it’s okay.

    diff, v4, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2021

    added statement and proof also of the other slicing

    𝒟 /b⊥⟶R /b⟵L /b𝒞 /R(b) \mathcal{D}_{/b} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/R(b)}

    What’s a canonical reference for this? This ought to be in Borceux somewhere, but I haven’t looked for it yet.

    diff, v5, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2021

    I have added the remark (here) that the left adjoint of the “2nd form” of the sliced adjunction sends slice morphisms to their adjuncts.

    diff, v6, current

  5. Previous edit added the joint generalisation of both ways of slicing, which is proven in HTT. This edit fixes some typos.

    Jan Steinebrunner

    diff, v9, current

  6. I’m very confused about how to typeset the adjunctions, sorry… Should be fixed now

    Jan Steinebrunner

    diff, v9, current

1 to 6 of 6

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