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.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 10th 2011

    Are there some interesting things to extract from this paper, accessible here, for nLab?

    Interesting categories of Mengen–combinatorial or bornological in nature– can be found which are, in a certain sense, generated by the chaotic objects only, despite the fact that they contain objects with arbitrarily complicated higher connectivity properties.

    That is, for such special MM, a knowledge of all the special maps chaotic(K)Mchaotic(K) \to M suffices to determine the arbitrary object M completely. This phenomenon has been studied by topologists for over 50 years under the name ’simplicial complexes’.

    I’d never thought before of simplicial complexes as probings by codiscrete objects.

    There are also toposes in which there are few connected objects and in which discrete(K)=codiscrete(K)discrete(K) = codiscrete(K) for all KK less than a measurable cardinal, and yet ’codiscrete \neq discrete’ is the main feature in the sense that maps from codiscretes determine all objects: for example, the topos of bornological sets (in which linear algebra becomes functional analysis).

    Is that topos cohesive? Lawvere speaks in ’Volterra’s Functionals and Covariant Cohesion’ of bornology involving a notion of covariant cohesiveness.

    After mentioning the quadruple adjunction

    components \dashv discrete \dashv points \dashv codiscrete,

    he writes

    For a category MM in which variation rather than cohesion is most important, the ’same’ four adjoints relating it to KK are usually called rather orbits \dashv stationary \dashv equilibria \dashv chaotic, where the last only sometimes exists.

    Is that ’variation’ pointing to (,1)(\infty, 1)-toposes?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2011

    I’d never thought before of simplicial complexes as probings by codiscrete objects.

    I am wondering if that was meant to read “Kan simplicial complexes”, because in the cohesive topos sSet the codiscrete objects are the nn-groupoids generated by the nn-simplex, for any nn, I’d think.

    Is that ’variation’ pointing to (∞,1)-toposes?

    I am not sure what to make of this right now. Did you follow the recent discussion about the usage of “chaotic” for “indiscrere” over on the CatTheory mailing list, involving Lawvere?

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 10th 2011

    I did look at that a bit. I thought Lawvere was saying that ’chaotic’ and ’codiscrete’ used to be used indiscriminately, but now he wants a different use for it. Yes,

    More recently, “chaotic” has come to have a different meaning, although one also involving a right adjoint. If f:XYX \to Y is a map from a space equipped with an action of a monoid T to another space, then f is a chaotic observable if the induced equivariant map from XX to the cofree action Y TY^T is epimorphic. A classic “symbolic” example has Y=π 0(X)Y=\pi_0(X), i.e. the observation recorded by ff is merely of which component we are passing through, but almost any TT-sequence of such is obtained by a sufficiently clever choice of initial state in XX.

    Back in the old days, the cafe used to discuss things like the relation between statics and dynamics. In that quadruple of adjunctions,

    orbits \dashv stationary \dashv equilibria \dashv chaotic

    I suppose chaotic is a motion covering every point in a space, and stationary is a motion staying still at a point. Is it a dynamic version of the topological quadruple?