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 bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics 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 finite 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 internal-categories k-theory lie lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number 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 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
    • CommentTimeNov 5th 2012

    I added some references to adjoint triple for the folklore theorem about fully faithful adjoint triples.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeNov 5th 2012
    • (edited Nov 5th 2012)

    Several lemmas concerning adjoint pairs and adjoint triples are included in

    Unfortunately there is a somewhat nonstandard usage of terminology continuous functor (and flatness there includes having right adjoint).

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeNov 5th 2012

    Feel free to add them to the page! (-:

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJul 8th 2015

    I added to adjoint triple a sketch proof that one of the adjunctions is an idempotent adjunction if and only if the other is, somewhat analogously to the situation with full-faithfulness.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 25th 2015

    added the beautiful observation by Dan Licata and Mike Shulman, that adjoint triples are equivalently adjunctions of adjunctions.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 25th 2015

    So is an adjoint quadruple, like we find in cohesion, an adjoint triple in adjunctions? Presumably yes, as an adjoint quadruple is a pair of adjoint triples, so could be arranged as two squares vertically.

    But then is it also an adjunction in the 2-category of adjoint triples? (Arranging the squares horizontally.)

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 25th 2015

    Maybe most entertainingly, an adjoint quadruple should be an adjunction of adjunctions of adjunctions. A third order duality.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 25th 2015

    So a cube.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeSep 25th 2015

    Thanks! I modified it a bit so that it doesn’t appear to be claiming that we discovered that fact.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 26th 2015

    This mode/adjoint logic is reminding me of Neel Krishnaswami’s comment.

    I wonder if Peirce’s gamma graph constructions have anything to teach us still. Given that Peirce provided you, Mike, with a diagrammatic notation for monoidal fibrations, it’s not impossible. For a small taste, on p. 24 of this we hear about his use of tinctures to depict “As far as is known, a Turk exists who is the husband of two different persons”.

    I see I’ve been promoting this for a while:

    And once you’ve sorted out Peirce’s beta system, I look forward to a category theoretic rendition of the gamma system.

    By the way, the reference [12] has a misspelt ’cirlce’.

    • CommentRowNumber11.
    • CommentAuthorThomas Holder
    • CommentTimeSep 26th 2015

    Dotting the i’s in the reference section: in [7] it’s officially ’F. W. Lawvere’ I think and for my taste the capital letters of Mac Lane in [10] and Meyer-Vietoris in [4] should be protected as well (ultimately this depends upon the exact titles of the papers cited though).

    Slightly more relevant: Are you aware of this arXiv-paper ? It has certain parallels with your approach.

    Another thing: what these guys (doi:S0304-3975(98)00359-4) are doing might turn out to be an instance of the adjoint logic with a quintessential localization as mode category or at least this might suggest a direction for possible applications of such a gadget.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeSep 26th 2015

    You guys need to be explicit that you are giving feedback on my paper with Dan! At first I thought you were talking about the references in the nLab article (which this discussion page is, after all, about). Thanks for the suggestions, I’ll look into them.

    • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 27th 2015

    On a similar issue of explicitness as #12, was your ’Cute!’ referring to Neel’s comment, Mike?

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeSep 27th 2015

    I guess; that was a long time ago. (-:

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJul 12th 2018
    • (edited Jul 12th 2018)

    Regarding the statement that adjunctions of adjunctions are adjoint triples:

    Why the need/desire to define the 2-morphisms in AdjAdj (p. 13) to be “conjugate pairs” of natural transformations?

    If we consider instead the 2-category of categories, left adjoint functors, and plain natural transformations between these. Then an adjunction in that 2-category is still/already an adjoint triple.

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeJul 12th 2018

    Those two 2-categories are biequivalent, so up to equivalence it doesn’t matter. And sometimes it’s useful to have both adjoints as given data, e.g. it corresponds more directly to what we see in the type theory.

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)