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 comma 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 finite 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 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 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.
    • CommentAuthorJohn Baez
    • CommentTimeMar 31st 2011
    • (edited Mar 31st 2011)

    I made a little addition to opposite category, pointing out some amusing nuances regarding the opposite of a VV-enriched category when VV is merely braided. This remark could surely be clarified, but I think you’ll get the idea.

    (In case you’re wondering why I did this, it’s because I needed a reference for “opposite category” in a blog entry I’m writing.)

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeMar 31st 2011

    Thanks! I added some remarks about dual objects in Prof.

    Are there actually more than two ways? I seem to remember thinking about this but I don’t remember the answer. I mean, obviously you could try twisting the two objects around each other arbitrarily many times, but would the resulting composition still be associative?

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 8th 2019
    • (edited Aug 8th 2019)

    This article does not mention opposite functors (F^op: C^op→D^op if F:C→D) or opposite natural transformations (t^op: G^op→F^op if t:F→G).

    These are often denoted by the same letter as the original functor and natural transformations, without the superscript. But it may be desirable to distinguish them for the same reason that we distinguish C and C^op. For instance, a natural transformation F^op→G^op and a natural transformation F→G are very different things.

    Shall we add a section about this? Or perhaps create separate articles opposite functor and opposite natural transformation?

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 8th 2019

    I typically write F opF^{op} myself. I can’t remember the last time I had occasion to write t opt^{op}, but that also makes sense to me.

    My own inclination would be to add material to opposite category, but I wouldn’t object to creating other articles.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeAug 8th 2019

    Writing F opF^{op} and t opt^{op} makes extra sense if you regard () op(-)^{op} as the name of the entire “oppositization” 2-functor Cat coCatCat^{co} \to Cat. I’d also probably suggest adding a section here for now.

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 9th 2019

    Opposite functors, opposite natural transformations.

    diff, v25, current

    • CommentRowNumber7.
    • CommentAuthorvarkor
    • CommentTimeSep 28th 2022

    Add redirect for “op”.

    diff, v29, current

    • CommentRowNumber8.
    • CommentAuthorvarkor
    • CommentTimeSep 28th 2022

    Mention the relationship between opposite monoidal categories and opposite 2-categories.

    diff, v29, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2023

    added pointer to:

    diff, v30, current

    • CommentRowNumber10.
    • CommentAuthornonemenon
    • CommentTimeSep 6th 2023

    Is there a palatable description of the opposite category of the category of models of a general Lawvere theory? I am specifically interested in the dual category of Monoids, details of which seem completely absent in the literature. I would assume there are at least some folklore concerning of this natural question, in case anyone can give me some direction. Thank you.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 6th 2023

    Opposite categories of commutative monoids, at least, have been discussed as categories of generalized affine schemes. I have compiled some references here, but there are more.

    • CommentRowNumber12.
    • CommentAuthornonemenon
    • CommentTimeSep 7th 2023

    Thank you.