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nLab > Latest Changes: idempotent adjunction

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  1.  
    • CommentRowNumber1.
    • CommentAuthorbarakat
    • CommentTimeAug 20th 2014
    • (edited Aug 20th 2014)
    I made a small change in idempotent adjunction. If I didn't miss something the words reflective and coreflective had to be replaced in one sentence.
    ---
    It then follows that F and G restrict to an equivalence of categories between the full images of F and of G (which are, respectively,

    a reflective subcategory of D and a coreflective subcategory of C)
    ->
    a coreflective subcategory of D and a reflective subcategory of C)
    ---

    I would even suggest to add:

    ---
    a coreflective subcategory of D and a reflective subcategory of C, both equivalent to the intermediate category E in 11)
    ---

    I don't know how to refer to 11. properly.
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2014
    • (edited Aug 20th 2014)

    Thanks, that seems right.

    I have made the typesetting of the factorization come out with the fully faithful functors displaying as hooked arrows. Also I have added mentioning that those full subcategories which are equivalent are he “EE”, from item 11.

    But something else is still wrong: the entry claims that an adjunction is idempotent when its induced monad and comonad is. This contradicts the statement at idempotent monad, which says that this is true (only) if one of the two adjoints is fully faithful.

    • CommentRowNumber3.
    • CommentAuthorbarakat
    • CommentTimeAug 20th 2014
    • (edited Aug 20th 2014)
    The hooks look nice.

    idempotent monad states correctly that there exits *an* adjunction where the right adjoint is ff, this is not necessarily the pair you started with.
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2014

    Oh, sure, right. Thanks.

  5. Added another example: comma and cocomma.

    Jade Master

    diff, v17, current

  6. Wasn’t sure if the comma construction was a left or right adjoint so I edited my example to no longer claim this.

    Jade Master

    diff, v17, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 4th 2021

    I have added this original reference for the characterization of idempotent adjunctions stated here:

    Also I have adjusted the formatting and cross-linking of the only reference which used to be offered (now here).

    diff, v21, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 4th 2021

    also pointer to:

    diff, v21, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeOct 5th 2021

    added pointer to:

    diff, v23, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 5th 2021
    • (edited Oct 5th 2021)

    I have given the list of examples more formatting and more hyperlinks.

    Then I have added the example

    TopSpAA⊥AA⟶Cdfflg⟵DtplgDifflgSp TopSp \underoverset{ \underset{ Cdfflg }{\longrightarrow} }{ \overset{ Dtplg }{\longleftarrow} }{\phantom{AA}\bot\phantom{AA}} DifflgSp

    (by !include-ing it).

    The earliest textbook reference for the full characterization of idempotent adjunctions that I have found so far is still Grandis 2021. Is there an earlier textbook that states the Proposition in citable form?

    diff, v23, current

1 to 10 of 10

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