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 comma 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 finite 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 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 Campion
    • CommentTimeFeb 3rd 2019

    Fixed a broken link to Jardine’s lectures.

    This article references Jardine’s lectures for a cubical subdivision functor, but I could not find it in this source. Is cubical subdivision described elsewhere?

    diff, v4, current

    • CommentRowNumber2.
    • CommentAuthorRichard Williamson
    • CommentTimeFeb 4th 2019
    • (edited Feb 4th 2019)

    Seems to be in section 5 of those lectures?

    That approach doesn’t seem especially elegant, though. It is enough just to specify what the subdivision functor does to 1\square_{\leq 1}, which can be done very explicitly. One gets everything else (including a right adjoint) from the universal properties. I think I described this once in some ancient nForum posts :-). Edit: think I found the nForum posts, see from #12 here.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2019
    • (edited Feb 4th 2019)

    So I changed the relevant sentence to a more proper citation:

    A cubical subdivision functor sdsd is discussed in Jardine 02, Section 5.

    diff, v6, current

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 5th 2019

    Updated link to Cisinski’s Astérisque monograph, this page used to point to his Paris 13 website, now defunct.

    diff, v7, current

    • CommentRowNumber5.
    • CommentAuthorTim Campion
    • CommentTimeFeb 5th 2019

    Ah yes – I think when I initially posted here the link was to a different paper of Jardine’s (there are two of them in the references). I eventually tracked it down and fixed the reference, but forgot to mention it. Sorry for the wild goose chase!

  1. Removed old discussion in query box:

    +– {: .query} David Corfield: Is this cubical set the same as Pratt is talking about on p. 13 here?

    “…the duality of bipointed sets, sets with two distinguished elements, and Boolean algebras without top or bottom. Contemplation of this duality, which Bill Lawvere suggested to me in a phone conversation as a simple construction of the theory of cubical sets”.

    If so, given that the real interval is a final coalgebra on bipointed sets, is there some dual to it in cubical sets?

    Also shouldn’t we have something on this page about Grandis’s use of cubical sets in directed algebraic topology, e.g., p. 3 ?

    Todd Trimble: The passage from Pratt’s paper is a bit brief, but my impression is that they are discussing the Lawvere algebraic theory of two constants, which is a cartesian prop, and which contains more figures than the pro given by the monoidal category of cubes. In particular, there are diagonal maps in the cartesian prop which aren’t present in the category of cubes in the sense here (and which aren’t reflected as far as I can tell by cubical sets with connection). Perhaps we need some disambiguation then?

    And please correct me if I’m wrong, but I believe the interval as final coalgebra is a coalgebra for the join-square endofunctor xxxx \mapsto x \vee x acting on the category of bipointed sets (where the two points are distinct). The condition that the two points are distinct is non-algebraic, so I can’t see a clear connection which would point to something dual in cubical sets in Pratt’s sense. But maybe there’s more going on than meets my eyes.

    David Corfield: Thanks Todd. I think you’re right about Pratt’s work, see example 5 here. If his usage is at all prevalent, we should disambiguate. So, next question, which are Grandis’s cubical sets? He seems to be able to do some remarkable things with them, e.g., the link to noncommutative spaces in section 3 of this.

    Todd Trimble: I believe Grandis is talking about cubes as we are here. His paper with Luca Mauri gives a rather thorough introduction to various categories of cubes (including, e.g., cubes with connections). The cartesian version doesn’t appear in that paper; I am guessing that most (all?) people who consider the category of cubes as Pratt does are in very close contact with Lawvere. The only items I found through google on this are one’s with Pratt’s name attached. But just to be on the safe side, I’ll write a brief note of disambiguation. =–

    David

    diff, v8, current

    • CommentRowNumber7.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 15th 2024

    Removed a silly comma

    diff, v9, current