Re: Reference requested

[F William Lawvere <wlawvere@buffalo.edu>]

To may@math.uchicago.edu, categories
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Reply-To: F William Lawvere <wlawvere@buffalo.edu>

Probably it was not coined but borrowed, from general topology.
For at least 50 years, the two, words 
chaotic
codiscrete
were alternate terminology for a certain kind of space.

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).

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.

Bill Lawvere

> Date: Wed, 28 Sep 2011 20:35:02 -0500
> From: may@math.uchicago.edu
> CC: categories@mta.ca
> Subject: categories: Reference requested
> 
> I have a reference question. Who first coined the term
> ``chaotic category'' for a groupoid with a unique morphism
> between each pair of object, and in what context? It is a
> ridiculously elementary concept, but one that is extremely
> useful in work on equivariant bundle theory that is needed
> for equivariant infinite loop space theory and equivariant
> algebraic K-theory.
> 
> Peter May
> 
> 




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