Not signed in (Sign In)

Start a new discussion

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 bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics 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 foundations functional-analysis functor galois-theory 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 internal-categories k-theory lie lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number 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 string string-theory 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
    • CommentTimeMay 15th 2012

    Created ternary frame, a class of models for substructural logic which are basically obtained from Day convolution for promonoidal posets. Presumably people who know things know this, but in a few minutes of looking I haven’t found anyone mentioning it. Girard’s “phase space” semantics for linear logic (no relation to phase spaces of physics) are just the special case of regarding a monoidal category as a promonoidal one.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeDec 19th 2017

    I cleaned up ternary frame a bit, including adding some references (none of which I’ve read myself yet!). I still don’t quite understand the “compatibility relation” from the promonoidal/Day perspective.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeMay 24th 2018

    Added a note about how regarding PCAs as ternary frames almost recovers the notion of implication from realizability.

    diff, v7, current

    • CommentRowNumber4.
    • CommentAuthorSam Staton
    • CommentTimeMay 24th 2018

    That’s a nice point. I suppose you say “almost” because it amounts to xPQx \Vdash P\multimap Q if for all yy with yQy \Vdash Q, either (xy)Q(xy)\Vdash Q or (xy)(xy) undefined. So it’s a kind of partial realizability?

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeMay 24th 2018

    Yeah, that’s what I had in mind by “almost”. I was wondering whether it could be made to coincide exactly by some trick, like introducing a formal bottom element \bot and defining xy=x y = \bot if it used to be undefined. Then xPQx \Vdash P\multimap Q iff either xPQx \Vdash P \to Q in the realizability sense or Q\bot \Vdash Q

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)