Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: idempotent adjunction

Bottom of Page

1 to 10 of 10

  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

  1.