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 beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality education elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory k-theory kan lie lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar 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-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory 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
    • CommentTimeDec 19th 2017

    Created a stubby poset-valued set.

    • CommentRowNumber2.
    • CommentAuthormaxsnew
    • CommentTimeDec 19th 2017
    • (edited Dec 19th 2017)

    Looks like a comma double category from FF to PP. Objects are functions FAPF A \to P and horizontal morphisms are squares from some FR:FAFBF R : F A \nrightarrow F B to P\leq_P along those functions. Then you should have vertical arrows which are of the form FfF f and make a commutative triangle and some kind of squares. I’m not very familiar with models of linear logic though so I don’t know if those notion already has a name for coherence spaces.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeDec 19th 2017

    That was exactly my thought on reading it!! There ought to be a general theory of when comma double categories inherit structure from their inputs. Comma double categories have also been used recently to induce structure on decorated cospans, and I believe that some forms of the Dialectica construction can also be represented as comma double categories. However, I don’t have time to think about this any more right now.

    • CommentRowNumber4.
    • CommentAuthormaxsnew
    • CommentTimeFeb 9th 2018

    Just found mention of “HH-valued sets” on the tripos page. Are they the same? They look very similar.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 10th 2018

    They do look somewhat similar, but I don’t think they’re the same. There are no symmetry or transitivity requirements, and the morphisms between them are different.

    • CommentRowNumber6.
    • CommentAuthormaxsnew
    • CommentTimeFeb 24th 2018

    I’ve been looking at the relationship between polynomial monads and multicategories, and the construction of the double category of T-multicategories is the same as the construction here. You have a double category of polynomials, then pick a specific polynomial monad TT, then you take the “slice” over TT to get a double category where the horizontal morphisms are TT-graphs, and then a TT-multicategory is a monad/monoid in that (virtual?) double category. This one looks very similar: a poset is just a monoid/monad in the double category of functions and relations, the only difference is it’s a comma over that monoid because of FF (might be interesting to look at the monoids in this setting).

    But neither of them actually looks like it fits the definition of comma double category (described for instance here) because a monoid doesn’t consitute a subcategory of the double category it’s in, so I’m not sure how to connect it to that general notion.

    • CommentRowNumber7.
    • CommentAuthormaxsnew
    • CommentTimeFeb 24th 2018

    Oh! A monad is just a lax morphism from the unit double category. Duh! So it is the same

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 25th 2018

    Ah, neat. Yes, the general sort of comma double category is a comma of an oplax functor over a lax functor, so you can put a monad on the bottom (and, if you want, a comonad on the top). I think that’s a new universal property of the double category of T-spans that I’ve never seen before; a pity it only works when T is polynomial.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 25th 2018

    Someone ought to write a paper about comma double categories, or at least an nlab page.

    • CommentRowNumber10.
    • CommentAuthormaxsnew
    • CommentTimeFeb 25th 2018

    I’ll write up a page this week. Would have helped me avoid my mistake above.