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.
    • CommentAuthorMike Shulman
    • CommentTimeJun 17th 2016

    Expanded dinatural transformation a little with examples and references.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 17th 2016

    I’m very glad you did, because it was through the Paré and Román reference that I relocated examples of dinatural transformations (due to Freyd) that I thought were forever lost to buried memory. I may copy them out later.

    • CommentRowNumber3.
    • CommentAuthorRodMcGuire
    • CommentTimeJun 17th 2016

    All I did in my edit was to fix the bad link in:

    Here is a blog post inspired by the above discussion that discusses these concepts in the context of the programming language Haskell:

    to

    Dinatural transformations and coends

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 16th 2020

    Added

    • Guy McCusker, Alessio Santamaria, A Calculus of Substitution for Dinatural Transformations, I, (arXiv:2007.07576)

    diff, v21, current

    • CommentRowNumber5.
    • CommentAuthorHurkyl
    • CommentTimeAug 15th 2020

    Added a yoneda-like correspondence between dinatural transformations and natural transformations between certain hom-sets.

    Also, the discussion in “dinaturality vs extranaturality” has some errors. In particular, in the “bending” calculation, G(x,y) F(y,x)G(x,y)^{F(y,x)} doesn’t make sense. Is it supposed to be D(F(y,x),G(x,y))D(F(y,x), G(x,y)) similar to the yoneda-like description I added?

    diff, v22, current

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeAug 15th 2020

    Yes, I think that’s what’s meant. In fact what’s going on here is basically the same as in your two-sided Yoneda lemma: the hom-profunctor is a representing object for extranatural transformations.

    I don’t see any other errors, what else do you have in mind?

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 24th 2020

    Fixed.

    diff, v23, current