Not signed in (Sign In)

Start a new discussion

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 beauty bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory 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 galois-theory 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 k-theory lie lie-theory limits linear linear-algebra locale localization logic mathematics 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 sheaves simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory 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).
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeJan 28th 2010

    Created idempotent adjunction. Does anyone know a published reference for this notion?

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJan 29th 2010

    This terminology, while has advantages, worries me, as does not correspond to idempotent monad but rather to the intersection of conditions for the idempotent monad and idempotent comonad. Interested to see how this will develop.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJan 30th 2010

    Not sure what you mean. The terminology exists, I didn't invent it.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeJan 31st 2010

    If I understood correctly: The monad constructed out of idempotent adjunction is itself idempotent but not necessarily the converse. The same for a comonad. But for converse one needs both conditions (on the monad and on the comonad) simultanaeoulsy. If one would like to have the terminology in complete correspondence with the properties of associated monads of comonads, then the first half could be called idempotent adjunction, second coidempotent adjunction and the strong case you quote as biidempotent adjunction.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 1st 2010

    No, the converse also holds. Either one of the induced monad or comonad being idempotent implies that the other also is and that the adjunction is idempotent.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeFeb 1st 2010

    Thanks, then I have some major gap in the understanding the subject.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeFeb 1st 2010
    • (edited Feb 1st 2010)

    I can't get it. look idempotent monad is equivalent to the counit of the adjunction to be isomorphism, i.e. we have a reflective subcategory. Idempotent comonad means that we have a coreflective subcategory (the other adjoint). The two are not the same unless we are in a Frobenius situation. What am I getting wrong ?

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 1st 2010

    An adjunction being an idempotent monad (i.e. a reflection) is equivalent to the counit being an isomorphism and the adjunction being monadic. A non-monadic adjunction can still induce an idempotent monad, and when it does it also induces an idempotent comonad and is an idempotent adjunction.

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeFeb 1st 2010

    Oh great, this explains what I overlooked! Neat.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)