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 k-theory lie-theory limit 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 simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject 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).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 2nd 2018

    A comparatively long and technical section “From hom-functors to units and counits” (on adjoint functors) was sitting inside the Idea-section of adjunction. It seemed plainly misplaced there, and distracting attention from what should be the content of this entry, as opposed to the entry adjoint functor. So I have moved it now to where it seems to belong: inside the Examples-section.

    diff, v55, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 9th 2019

    The diagrams in Direct Definition do not seem to be rendering properly. (Not really my observation; I was alerted in a chat room at MO).

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeApr 9th 2019

    Probably need to be re-done with tikz.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2019

    added pointer to

    • Ralf Hinze, Generic Programming with Adjunctions, In: J. Gibbons (ed.) Generic and Indexed Programming Lecture Notes in Computer Science, vol 7470. Springer 2012 (pdf, slides doi:10.1007/978-3-642-32202-0_2)

    diff, v59, current

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 26th 2019

    Added

    • Jeremy Gibbons, Fritz Henglein, Ralf Hinze, Nicolas Wu, Relational Algebra by Way of Adjunctions, pdf

    diff, v60, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2019

    Thanks. I have added the publication data, too.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 26th 2019

    Hmm, but maybe these should be in adjoint functor. We’re running this page adjunction as an abstract 2-categorical concept, although ’adjoint functors’ makes up a large chunk of the page as an example.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2019

    Yeah, I have already done some harmonizing of the list of References to both entries. That’s just the way it is now. At least each References section is now alerting the reader to also look at the other.

    • CommentRowNumber9.
    • CommentAuthorJason Gross
    • CommentTimeMar 31st 2021
    The two equations in "Direct definition" show up as an error: "Unexpected text node: ' '" I'm not sure how to fix this...
    • CommentRowNumber10.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 31st 2021
    • (edited Mar 31st 2021)

    We should rewrite them in Tikz or xymatrix. I believe Urs already did so in one form at triangle identities, and I recall having done so myself in a different form somewhere, though I do not remember where just now; we can just copy the code over. (Edit: the ones I wrote are at left adjoint. I will fix this myself tomorrow if no-one gets to it before me.)

  1. Fixing a couple of diagrams.

    diff, v64, current

  2. The ’Examples’ section on this page is rather a mess. Apart from the fact that the contents are mostly not really examples, it would be good if some industrious reader can re-work the material into some structure (proposition, proof, remark, example etc).

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeOct 15th 2021

    Added (here) a brief remark on adjunctions in Ho 2(Cat )Ho_2(Cat_{\infty}) to the list of examples.

    diff, v66, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeOct 15th 2021

    Following Richard in #12 I have tried to bring the old material under “Examples – Adjoint functors” (here) into some shape: I have added Proposition/Example/Proof/Remark-environments and slightly adjusted the text here and there. (One could still improve the “Proof” to make the converse implication more explicit.)

    diff, v66, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeOct 15th 2021

    Also added pointer to the proof of this that I once wrote at adjoint functor here.

    diff, v66, current

    • CommentRowNumber16.
    • CommentAuthorvarkor
    • CommentTimeMar 31st 2023

    Mention characterisation in terms of comma categories.

    diff, v69, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2023

    This paragraph should probably rather go the page adjoint functor. I have copied it to there.

    • CommentRowNumber18.
    • CommentAuthorvarkor
    • CommentTimeMar 31st 2023

    Ah, I thought the page adjunction was looking barer than I remembered; I didn’t recall that there were two pages.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2023

    Right, it can be confusing. So I have added a disambiguation line to the very top of the entry, prominently pointing to the page adjoint functors.

    Also moved the paragraph on slice categories to there.

    diff, v70, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2023

    added (here) mentioning of the example of enriched adjoint functors

    diff, v72, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2023

    tried to clean up the Definition-section (here)

    in particular I added the form of the zig-zag law as an equality of 2-morphisms

    and I moved the equivalent expression in terms of string diagrams to right after that inside the definition.

    diff, v73, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2023

    also tried to streamline the Idea-section (here)

    The entry is still lacking an “original reference” for the notion of adjunction in general 2-categories, if such exists. Just in order to have at least one reference earlier that Johnson & Yau (2020), I have now added pointer to:

    diff, v73, current

    • CommentRowNumber23.
    • CommentAuthorvarkor
    • CommentTimeApr 29th 2023

    Added the original reference (as far as I’m aware).

    diff, v74, current

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2023
    • (edited Apr 29th 2023)

    Thanks!

    I have added more details, such as a link to the article:

    • Max Kelly, §2 in: Adjunction for enriched categories, in: Reports of the Midwest Category Seminar III, Lecture Notes in Mathematics 106, Springer (1969) [doi:10.1007/BFb0059145]

    diff, v75, current

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2023

    added pointer to:

    diff, v76, current

    • CommentRowNumber26.
    • CommentAuthorvarkor
    • CommentTimeJun 2nd 2023

    Added an earlier reference for adjunctions in a 2-category.

    diff, v78, current

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeJun 2nd 2023

    Great! I have added the page number (pp. 762).

    diff, v79, current

  3. Corrected diagrams for the zigzag law (there is no id functor C -> D!)

    Baptiste Loreau

    diff, v80, current

  4. Fixed previous edit so that it has the intended effect.

    Baptiste Loreau

    diff, v80, current

    • CommentRowNumber30.
    • CommentAuthorvarkor
    • CommentTimeJan 19th 2024

    Added two classical 2-categorical references.

    diff, v82, current

  5. Fix link

    Empress of Light

    diff, v83, current