Processing math: 100%
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 definitions 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 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 nforum 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.
    • CommentAuthormetaweta
    • CommentTimeJul 14th 2016

    At “Definition as a 2-functor”, the article says

    For a proarrow H:BD and ordinary arrows f:AB and g:CD, we write H(g,f) for the composite D(g,1)HB(1,f); it is a proarrow from A to C. We also write UA, A(1,1), or simply A for the identity proarrow AA.

    In seems that the nLab has settled on the convention that a profunctor H:BD is a functor H:Dop×BSet. Therefore, in the expressions H(g,f) and B(1,f) the morphism f is in the covariant slot, which means that either f should be a morphism out of B or H should be a profunctor into A.

    What am I missing?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJul 14th 2016

    It seems right to me. The covariant slot of H is the category that it is “from” as a profunctor, and we are substituting a value of f into that slot; so it’s correct that H is a profunctor from B to something else, and that the values of f belong to B, i.e. that f is a morphism into B.

    • CommentRowNumber3.
    • CommentAuthormetaweta
    • CommentTimeJul 14th 2016
    • (edited Jul 14th 2016)

    In that case, D and g are the problem. Thinking of H as a set of heteromorphisms from B to D, we can precompose with f to get a set of heteromorphisms from A to D, but to postcompose, we need g:DC, not g:CD. That is,

    H(f:AB,g:DC)(h:BD)=ghf:AC.
    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJul 15th 2016

    f is not a morphism in B, it’s a functor with codomain B, and similarly g is not a morphism in D, it’s a functor with codomain D. We’re not describing the action of the functor H on its elements/heteromorphisms; we’re describing its restriction along a pair of functors.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 15th 2016

    It’s like the multiplication of three matrices (the outer two being special in deriving from functions), if that helps. Or the composition of relations.

    • CommentRowNumber6.
    • CommentAuthormetaweta
    • CommentTimeJul 15th 2016
    • (edited Jul 15th 2016)

    Mike: Then shouldn’t it be D(1,g)HB(1,f), that is, f followed by H followed by the taking the preimage under g? The expression D(g,1) doesn’t make sense; since B is the codomain of f, having g in the other slot means we’d want the domain of g. If we want the codomain of g, it needs to be in the same slot that f is.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJul 15th 2016

    The D in D(g,1) represents the hom-functor of D, which is a profunctor from D to D. Thus, g being a functor with codomain D, it can be plugged into either slot of this profunctor.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJul 15th 2016

    Although in the context of the quote in question, D(1,g) and D(g,1) are just notations for the two proarrows associated to the arrow g. The reason for using that notation is as I said: in the case of categories and profunctors, they are (x,y)HomD(x,g(y)) and (y,x)HomD(g(y),x).

    • CommentRowNumber9.
    • CommentAuthormetaweta
    • CommentTimeJul 15th 2016
    • (edited Jul 15th 2016)

    OK, so in that notation, both slots are covariant. I’ll add a note to that effect. Thanks!

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJul 15th 2016

    I think I have some idea of what you mean, but I don’t think “both slots are covariant” is really the correct way to say it. H is still a functor from Dop×B to Set, so in that sense the first slot is contravariant. And if the question is what kind of a functor H(g,f) is as a functor of g and f, then it’s a functor from K(A,B)op×K(C,D) to M(A,C), so in that sense also the first slot is contravariant.