Re: Reference requested

["jpradines" <jpradines@wanadoo.fr>]

I was very happy and relieved when reading Fred Linton's message underneath
and then Bill's agreement, since, after reading Ross Street's message about the
use of the term "chaotic" by Bourbaki and French culture, I had started
getting serious doubts about the state of my own memory.
Indeed everybody can check that, from Bourbaki's first edition of Topology to the final
one  (seriously reshaped by Grothendieck, I think), published in 1971 and
74, though the indiscrete topology is the first example given for a
topology, followed by the discrete one, it is (rather curiously) not given a
name.
However, at least from the fifties, and certainly before, the French
terminology "topologie grossie`re" was universally accepted and used as the
unique one by the whole community of French mathematicians, at all levels, from education
to research. I am not aware that things have changed, though it is nowadays
difficult to find a text written in French by a young French native speaker
which would not be a poor translation from Frenglish of a poor translation
from French to Frenglish (a consequence of the fact that French language is no
longer taught at French schools, an open secret ; well, you understand that I am not a
candidate to the Ministry of Education).
  Of course (!) the use of the English term "coarse", which is perhaps the
translation in down-to-earth language of the French "grossier" was much less successful, and probably
used essentially in texts which were more or less translated from French or deriving from French culture. It
seems to be more or less abandonned nowadays and to have got some very special quite different and technical meaning (in 
connection with the study of Gromov metrics).
It seems also that the English-speaking classical topologists are nowadays generally pleased with the neologism/pun pair offered by in(or un-)discrete/indiscreet, which cannot have any French equivalent, for orthographic reasons (just one possible orthograph and meaning for "indiscret").

As to the term "chaotic", I have exactly the same (absence of) experience as
Fred (perhaps it was used by Grothendieck's school in the sixties, but I am
not familiar with that, and don't remember to have heard of it) and think it
is evoking unavoidably the field of dynamical systems (where however it is
probably used without a very precise and unique definition). I'm sure I'll
never use such a term for naming the very simple notion which is called "pair
groupoid" in the book by Kirill on Lie groupoids ; this latter terminology (which
however I don't like very much, see further) is presently accepted by many authors,
and I'm surprised that none of them came to the fore in that discussion.
It should be noted however that the geometer van Est sometimes used the term "blackhole" for naming the space of leaves of a foliation when its topology (in the elementary sense, i.e. the quotient one) is the indiscrete one; I suppose this was evocating the very chaotic behaviour of the leaves in such a case.

However all the preceding (purely linguistic) considerations are very secondary in my opinion
and I turn now in more details to my main point which was expressed in my previous message but found no echo.
First of all I claim that I perfectly agree with Peter May about the basic importance of what I prefer to call banal groupoids (in some internal contexts I am currently using, they don't always exist, an a quest for suitable substitutes may be an important motivation). But I don't think this justifies to look for terrific names, which anyway can reflect but a tiny part of these important properties. 
The term "banal" should not be regarded as pejorative. Think to mathematical objects like 0, 1, 2: they are probably among the most banal, or even silly, objects from a naïve point of view, though they are bearing a lot of very rich but highly degenerate structures, and they are perhaps encapsulating some basic processes for explaining the transition from nothingless to being (!); this would not be a reason for trying to give them mysterious names borrowed from philosophy or theology (!).

More precisely the adjunction property which is intended to be stressed by introducing duets such as discrete/co-discrete, is just a part of an iceberg of some interesting adjunction strings (with different lengths) arising from various forgetful functors.
But the structure of these strings are strongly depending on the "structure" you are forgetting and are quite different when dealing with the algebraic structure of groupoid or with a topological structure; in the latter case you may get the pi_0 functor (connected components) as arising in that way in such a string.
Now when dealing with topological (and particularly smooth) groupoids (internal groupoids in the category Top or Dif), subtle interesting interactions and clashes arise between the two strings: in a certain sense it may be said than a suitable pi_1 functor emerges as a certain kind of internal pi_0, in a way that includes notably the correct pi_1 concept (Poincare' functor) for spaces of leaves of a foliation; the latter was introduced by different geometers by means of various ad hoc constructions (Haefliger, van Est, Paul ver Ecke); it turns out that these constructions may be derived from a general algebraic construction described in the book by Gabriel-Zisman, which introduces an adjoint for the "nerve functor" from groupoids to simplicial sets. This approach yields immediately a van Kampen theorem by preservation of colimits, according to Ronnie's presentation.
This whole stuff was described in a CRAS Note (Paris) which I published in 1989 (cosigned with my irakian student Alta'ai) (t. 309, Se'rie I, p. 503-506). Unhappily neither Haefliger nor van Est understood a single word of this Note (which indeed they tried to get rejected, unsuccessfully), since they are not at all categorically minded. The main result was formulated in a different way, using what I called simplified calculus of fractions for Morita equivalences (which cannot be derived as a special case from the classical Gabriel-Zisman calculus of fractions) in a paper published in les Cahiers de Topologie in the same year, nowadays available with minor corrections and important added comments in arXiv:0803.4209v1. In fact my secret hope is that such a construction (from pi_0 to pi_1) might be a first step for an inductive process leading to higher homotopy in the spirit of Ronnie and Higgins.

Clearly the use of the same word "discrete", originated from (elementary) Topology to describe adjointness properties which are quite different for groupoids and for topological spaces, but are able to lead to fundamental interactions when studied for topological groupoids, make it strictly impossible to perform that type of studies, which I regard as very fecund. More generally further comments about the widespread misuse of topological terminology in the purely algebraic framework of abstract groupoids may be found in  arXiv:0711.1608v1, a paper written on the occasion of Ehresmann's birthday 100th anniversary. 
 

----- Message d'origine ----- 
De : "Fred E.J. Linton" <fejlinton@usa.net>
À : <categories@mta.ca>
Cc : "F. William Lawvere" <wlawvere@buffalo.edu>; "Peter May"
<may@math.uchicago.edu>
Envoyé : vendredi 30 septembre 2011 23:19
Objet : categories: Re: Reference requested


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


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]