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 bundle 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 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.
    • CommentAuthorSam Staton
    • CommentTimeJul 29th 2019

    Page on double glueing

    v1, current

    • CommentRowNumber2.
    • CommentAuthorSam Staton
    • CommentTimeJul 29th 2019

    Reference

    v1, current

    • CommentRowNumber3.
    • CommentAuthorSam Staton
    • CommentTimeJul 29th 2019

    mention focuses

    v1, current

    • CommentRowNumber4.
    • CommentAuthorSam Staton
    • CommentTimeJul 30th 2019

    examples

    diff, v2, current

    • CommentRowNumber5.
    • CommentAuthorSam Staton
    • CommentTimeJul 30th 2019

    added example of quantum causal structure

    diff, v3, current

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeOct 13th 2019

    I finally got around to having a look at the Hyland-Schalk paper. In trying to make abstract sense of it, I realized that (unless I’m mistaken) their input datum for a double gluing construction — a lax symmetric monoidal functor L:CEL:C\to E and a functor K:CE opK:C\to E^{op} together with “contractions” L(R)K(RS)K(S)L(R) \otimes K(R\otimes S) \to K(S) (section 4.2.1) — is just a lax symmetric monoidal functor CChu(E,1)C \to Chu(E,1). This explains their observation in section 4.3.1 that when CC is *\ast-autonomous any lax symmetric monoidal L:CEL:C\to E extends canonically to a KK with contraction, by Pavlovic’s observation that ChuChu is right adjoint to the forgetful functor from *\ast-autonomous categories to closed symmetric monoidal categories equipped with a chosen object (and morphisms that colaxly preserve the object, which is automatic when the chosen object in the codomain is terminal).

    However, so far I haven’t been able to reformulate the actual construction of the double-glued category nicely in terms of this functor CChu(E,1)C\to Chu(E,1) as input, which is very unsatisfying. Any ideas?

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeOct 13th 2019

    In particular, the ordinary gluing of (L,K):CChu(E,1)(L,K) : C\to Chu(E,1) would have, as objects, tuples consisting of an object RCR\in C, an object (U,X)Chu(E,1)(U,X)\in Chu(E,1) — which is to say two objects U,XEU,X\in E — and a morphism (U,X)(L,K)(R)(U,X) \to (L,K)(R) in Chu(E,1)Chu(E,1) — which is to say morphisms UL(R)U\to L(R) and K(R)XK(R) \to X in EE. This looks almost like the double gluing except that the morphism K(R)XK(R) \to X goes the wrong way! Unless I am confused somehow.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeOct 13th 2019
    • (edited Oct 13th 2019)

    Ah, I got it! It’s another comma double category. Regarding CC and Chu(E,1)Chu(E,1) as vertically discrete double categories, we can map them both into the double Chu construction hu(E,1)\mathbb{C}hu(E,1) (the inclusion of Chu(E,1)Chu(E,1) being an isomorphism onto the horizontal category). Then the double gluing category is the horizontal category of the comma object in double categories (and vertical transformations) of the cospan Chu(E,1)Chu(E,1)C \to \mathbb{C}hu(E,1) \leftarrow Chu(E,1). And to encode the monoidal structure, we can use double polycategories (or multicategories) instead.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeOct 14th 2019

    Added the general case, in terms of comma double polycategories.

    diff, v5, current