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 nlab noncommutative noncommutative-geometry number-theory object 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.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 26th 2011

    This is a ’latest changes’, but for the Café rather than the backroom! Can David C (or someone) fix the link that does not work to Steve Awodey’s paper (It should be http://www.andrew.cmu.edu/user/awodey/preprints/FoS4.phil.pdf).

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 26th 2011

    Thanks for the alert. I have fixed it now.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 26th 2011

    Thanks. You constructed (most of) those pages on algebraic and geometric models for modal logic, Tim. I wonder what can be said about the relation between them. There’s the idea of converting a Kripke frame into a topology, using the Alexandroff topology.

    I had a word with Steve Awodey, who didn’t sound overly convinced by the coalgebraic approach. It’s a pretty big movement though. A lot of people would have to be wrong if it’s not going anywhere important.

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 26th 2011

    My feeling when looking around in that area I got the feeling that what was missing was the intuitionist or constructivist viewpoint, as everything used BAOs. There was also my ’intuition’ that there should be a higher dimensional viewpoint that would really be needed here… depending on the applications. But these comments are a bit off the top of my head so may not be that pertinent. (Again on the off chance that it may be useful, don’t forget Eric Goubault and his brother wrote on the interconnections amongst these sorts of models for S4 logics, and there is the Leeds group that tries to use them to develop a spatial language (in some sense).) I must dash.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 27th 2011

    I have an inkling that ionads may be good for modelling first-order modal logic. I’m having difficulty, though, picturing the interior operator Set XSet XSet^X \to Set^X. Is there a nice simple example to keep in mind?

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 28th 2011
    • (edited Feb 28th 2011)

    I am only an amateur in the modal stuff. I will try to look at ionads (and really like the name). I always felt that the treatment in Kracht’s book was one that should be expanded greatly exactly in the sort of directions that you need. I would also repeat my point about the paper listed here from HHA (J. Goubault-Larrecq and É. Goubault. On the Geometry of Intuitionistic S4 Proofs. Homology, Homotopy and Applications 5(2), pages 137-209, 2003, available here) . The use of the Decalage type operator to model the Box in S4 seemed to me to be excellent and the general philosophical and methodological points in that paper are also very important. (I liked the style as well, since to give a DCPO and topological approach only to say ’ that is not really what is going on and here is the secret’ is great.)

    Although their approach is not coalgebraic and so does not directly give you an answer, and it is trying to do something different, but I think it may be more higher dimensional, more n-lab related and in fact side step some of the problems with the coalgebraic approach (once pushed a bit further and adapted towards the first order context). As I said, I am not expert in the modal side, but on the simplicial side, that paper is very pleasing to me.

    • CommentRowNumber7.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 28th 2011

    There is a set of lecture notes (I only just saw them so have not read them) at here. There is a fibred category look to this and the coalgebraic viewpoint does not seem rich enough to handle that (uninformed opinion <- not to be given too much weight!!!!:-))