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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 10th 2011
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAre there some interesting things to extract from this paper, accessible [here](http://ncatlab.org/nlab/show/cohesive+topos#references_80), 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 $M$, a knowledge of all the special maps $chaotic(K) \to M$ 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 $discrete(K) = codiscrete(K)$ for all $K$ 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 $M$ in which variation rather than cohesion is most important, the 'same' four adjoints relating it to $K$ are usually called rather orbits $\dashv$ stationary $\dashv$ equilibria $\dashv$ chaotic, where the last only sometimes exists. Is that 'variation' pointing to $(\infty, 1)$-toposes?

   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 $M$, a knowledge of all the special maps $chaotic(K) \to M$ 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 $discrete(K) = codiscrete(K)$ for all $K$ 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 $M$ in which variation rather than cohesion is most important, the ’same’ four adjoints relating it to $K$ are usually called rather orbits $\dashv$ stationary $\dashv$ equilibria $\dashv$ chaotic, where the last only sometimes exists.

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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 10th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 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 $n$-groupoids generated by the $n$-simplex, for any $n$, 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?

   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 $n$-groupoids generated by the $n$-simplex, for any $n$, 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?

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 10th 2011
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI 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:$X \to Y$ 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 $X$ to the cofree action $Y^T$ is epimorphic. A classic "symbolic" example has $Y=\pi_0(X)$, i.e. the observation recorded by $f$ is merely of which component we are passing through, but almost any $T$-sequence of such is obtained by a sufficiently clever choice of initial state in $X$. Back in the old days, the cafe used to [discuss](http://golem.ph.utexas.edu/category/2007/01/quantization_and_cohomology_we_10.html) 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?

   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:$X \to Y$ 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 $X$ to the cofree action $Y^T$ is epimorphic. A classic “symbolic” example has $Y=\pi_0(X)$, i.e. the observation recorded by $f$ is merely of which component we are passing through, but almost any $T$-sequence of such is obtained by a sufficiently clever choice of initial state in $X$.

   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?