Re: Reference requested

["Fred E.J. Linton" <fejlinton@usa.net>]

Bill recalls:

> For at least 50 years, the two, words 
> chaotic
> codiscrete
> were alternate terminology for a certain kind of space.

My memory rather matches instead what I see in (3.2(d)) of Willard, 
that the topology with only the whole space and the empty set "open"
is called either 'trivial' or 'indiscrete'.

In my experience I've never encountered 'chaotic' as the
adjective used for that attribute -- indeed, 'chaotic' would have 
conflicted rather badly with the Chaos Theory arising out of René 
Thom's Catastrophe Theory of the early '60s or so -- and 'codiscrete' 
strikes me as what only a categorist hoping (as we many of us 
long did) to systematize terminology into dual camps of 'properties' 
and 'coproperties' (in the model of adjoint/coadjoint, limit/colimit, 
terminal/coterminal, etc.) could have come up with -- not a term any
self-respecting point-set topologist would have thought to use :-) .

'Codiscrete' of course does, for just that reason, have its merits,
as Bill points out:

> I prefer "codiscrete" since it clearly indicates something
> opposite to discrete, the precise sense of oppositeness being
> that of inclusions adjoint to the same uniting functor, often called
> the "underlying". (In higher dimensions, coskeletal and skeletal
> are similar identical opposites, with "truncation" as uniter).
> 
> In fact every groupoid is a colimit of codiscrete ones, indeed
> groupoids form a reflective subcategory of the topos that classifies 
> Boolean algebras, and the latter has a site consisting of codiscrete
> groupoids. (The generic Boolean algebra 2^( ) has as its natural
> geometric realization the infinite-dimensional sphere, containing 
> the ordinary interval as a generating distributive lattice).

And I object not one whit to any of that :-) .

> More recently, "chaotic" has come to have a different meaning, 
> although one also involving a right adjoint. If f:X->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=pi0(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.

This again suggests that 'chaotic' might not be the best choice of
adjective for that indiscrete/codiscrete topology, or the analogous
type of category, or groupoid, or topological category or groupoid.

Cheers, -- Fred



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