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    • 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?