Re: Reference requested

Re: Reference requested

[F William Lawvere]

To may@math.uchicago.edu, categories
Sender: categories@mta.ca
Precedence: bulk
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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Re: Reference requested

[Fred E.J. Linton]

I recently reported that

> ... I sought Search-engine advice regarding the use of the 
> 'chaotic topological space' lingo, and came up with ...

... surprisingly little. Early, early this morning I tried that again,
but enclosing the search string in double-quotation marks. Ah, now I
found two other references, worth citing, perhaps:

1) Volker Runde's 2005 Springer Universitext, isbn=038725790X, 
A Taste of Topology, includes a passage (page 72), beginning 

"Let (X,TX) be a chaotic topological space (ie, TX = {∅,X}), 
let (Y,TY) be a Hausdorff space, and let f: X → Y be continuous"
  
and deducing such f must be constant; links (to Google Books and a PDF):

[long url omitted by moderator],

ftp://210.45.114.81/math/2007_07_06/Universitext/V.Runde%20A%20Taste%20of%20Topology.pdf


(no hint, though, how 'standard' Runde thought his use of "chaotic" here was
:-{ ); and

2) Mat{´ı}as Menni's 2000 Edinburgh PhD thesis {Exact Completions and
Toposes} makes
multiple mention of chaotic structures, with frequent citations of Bill
Lawvere's 
interest in such things. All of Chapter 7 is about "Chaotic Situations", with
Section 
7.1 focused in particular on "Chaotic Objects"; and Section 8.4 returns to
"Chaotic Situations". A hint of the flavor is given in the Introduction
already:

"In 1999, Longley introduced a typed version of the notion of a partial 
combinatory algebra in [68] and described how to build a category of
assemblies 
Ass(A) over a [sic] such a structure A. Shortly after, Lietz and Streicher
showed 
that the ex/reg completion of Ass(A) is a topos if and only if the typed
structure 
A is equivalent, in a suitable sense, to an untyped structure. Their proof
uses 
the notion of a generic mono (a mono τ such that every other mono arises as a

pullback of τ along a not necessarily unique map) and of the constant-objects

embedding of Set into the category Ass(A) which they see as an inclusion of 
codiscrete objects. Related to this, it should be mentioned that Lawvere had 
already advocated for a conceptual use of codiscrete or chaotic objects in 
other areas of mathematics (see for example [59, 55, 61, 63])."

No surprise, then, to see the right adjoint ∇ to the forgetful functor Set
-> Top
described (p. 23) as follows: " ... the functor ∇: Set -> Top assigns to
each set 
S the “chaotic” topological space with underlying set S and, as open sets,

only S itself and the empty set."

Cf.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.22.9817&rep=rep1&type=pdf

---

BTW, those four Lawvere references are these:

[55] F. W. Lawvere. Toposes generated by codiscrete objects in combinatorial
topology and functional analysis. Notes for colloquium lectures given at
North Ryde, New South Wales, Australia on April 18, 1989 and at Madison
USA, on December 1, 1989.

[59] F. W. Lawvere. Categories of spaces may not be generalized spaces as
exemplified by directed graphs. Revista colombiana de matem´aticas,
20:179–
186, 1986.

[61] F. W. Lawvere. Some thoughts on the future of category theory. In
Proceed-
ings of Category Theory 1990, Como, Italy, volume 1488 of Lecture notes
in mathematics, pages 1–13. Springer-Verlag, 1991.

[63] F. W. Lawvere. Unit and identity of opposites in calculus and physics.
Applied categorical structures, 4:167–174, 1996.

---

All told, eight hits, all either these two, or references to them, or
search-database errors :-) . Not very heavy evidence in favor of "chaotic".

So: cheers -- and back to [co-|in-]discrete, I fear :-) , -- Fred 



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Re: Reference requested

[jpradines]

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


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Re: Reference requested

[Fred E.J. Linton]

Peter May, in re the Subject: categories: Re: Reference requested, wrote

> ... I prefer `chaotic' to `indiscrete' not just because
> of the `coarse' implications of the latter, but because
> indiscrete spaces are boring, `null or banal', whereas
> chaotic categories have genuinely significant applications. ...

Be that as it may, I sought Search-engine advice regarding the use of the  
'chaotic topological space' lingo, and came up with the following 'hits',
of which only the first reflects, in an afterthought, Peter's usage,
while the others all envision something rather quite different:

1) From  http://en.wikipedia.org/wiki/Grothendieck_topology : 

The discrete and indiscrete topologies

Let C be any category. To define the discrete topology, we declare all sieves
to be covering sieves. If C has all fibered products, this is equivalent to
declaring all families to be covering families. To define the indiscrete
topology, we declare only the sieves of the form Hom(−, X) to be covering
sieves. The indiscrete topology is also known as the biggest or chaotic
topology, and it is generated by the pretopology which has only isomorphisms
for covering families. A sheaf on the indiscrete site is the same thing as a
presheaf.

Other uses of 'chaotic', having nothing to do with indiscreteness,
predominate:

2) From http://www.math.uh.edu/~hjm/pdf26%284%29/03chara.pdf ,
reflecting the content of 

ON GENERALIZED RIGIDITY
by JANUSZ J. CHARATONIK
from Houston Journal of Mathematics (© 2000 University of Houston)
Volume 26, No. 4, 2000 :

A nondegenerate topological space X is said to be:

(a) chaotic if for any two distinct points p and q of X there exists an open
neighbourhood U of p and an open neighbourhood V of q such that no open
subset of U is homeomorphic to any open subset of V ; ... [snip] ...

3)  CHAOTIC GROUP ACTIONS
www.math.zju.edu.cn/amjcu/B/200301/030108.pdf

... no chaotic group actions on any topological space with free arc. ...
... topological space which admits a chaotic group action but admits ...

4)  CHAOTIC POLYNOMIALS IN SPACES OF CONTINUOUS AND ...
personales.upv.es/almimon/Preprint%20Aron-Miralles.pdf

... show that there exist chaotic homogeneous polynomials of degree m ≥ 2.
...

So I'd imagine 'chaotic', for 'indiscrete', is best dropped, and either
'indiscrete', 'codiscrete', or 'trivial' be used instead.

NB: while it's true that the trivial (indiscrete) topology on a set X 
is initial, in the sense that, as a collection of subsets of X, it's the 
smallest that's a topology on X, the indiscrete topological space on X
is terminal among all topological spaces on X and mappings that restrict
to the identity on X; the trivial (indiscrete) pre-order on X is likewise
terminal, in the sense that, as a subset of X x X, it's the largest.

A connected pre-ordered groupoid (i.e., indiscrete category), being
equivalent to the terminal category 1, has the property that, for each
category X, it admits exactly one isomorphism class of functor from X,
but while that may make it 2-terminal or [( co | op ) lax-] terminal, 
I'd still probably prefer to avoid such ... umm ... terminalogy :-) .

Cheers, -- Fred

PS: I re-emphasize: of all the hits I found, only one amongst the first
two dozen -- the first cited above -- spoke of the trivial topology as 
the chaotic topology; ALL the others used 'chaotic' in some other way, 
DESPITE the search having been explicitly for [ chaotic topological space  ]. 
And there were "about 175,000 results" all told :-) . -- F.

PPS: Typos? Perhaps; please forgive, I couldn't spiel-chuck. -- F.



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Re: Reference requested

[F William Lawvere]

Fred, you caught me ! Indeed my memory of JL Kelley
was blurred by wishful thinking. Hence, dear friends, please be
so generous as to ignore the attempt at historical justification
and instead further elaborate the rational arguments as Fred has.
Bill Lawvere

On Fri 09/30/11  5:19 PM , "Fred E.J. Linton" fejlinton@usa.net sent:
> 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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Re: Reference requested

[Fred E.J. Linton]

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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Re: Reference requested

[Peter May]

Thanks everybody for comments, although I guess the use
goes so far back into antiquity that the request for an original
reference is unanswerable.  For context, with two young
collaborators (Bertrand Guillou and Mona Merling), I
have a draft in progress tentatively entitled ``Chaotic
categories and equivariant classifying spaces''.

I prefer `chaotic' to `indiscrete' not just because
of the `coarse' implications of the latter, but because
indiscrete spaces are boring, `null or banal', whereas
chaotic categories have genuinely significant applications.
They are quite surprisingly central to the theory of universal
bundles, equivariant or not.

Via the (product-preserving) classifying space construction
from categories (especially categories internal to spaces)
to spaces, they provide a rich source of contractible spaces
that can very easily be given interesting additional structure.
That is just what one wants when constructing universal bundles.

More fun, it is just what one wants to construct an E infinity
operad of G-categories that defines `genuine' symmetric
monoidal G-categories (which are not merely symmetric
monoidal categories on which a group G acts in the obvious
`naive' way).   These which give rise to `genuine' G-spectra.
Genuine G-spectra that define equivariant algebraic K-theory
arise in precisely this way.    All starting from chaotic trivialities.


Cheers,

Peter


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Re: Reference requested

[David Roberts]

Or codiscrete groupoid?

David

On 29 September 2011 23:11, Ronnie Brown
<ronnie.profbrown@btinternet.com> wrote:
> Why not use the term `indiscrete groupoid' for the functor that gives a
> right adjoint to the  functor Ob: Groupoids \to Sets?   The left adjoint
> is then of course the `discrete groupoid'.  This agrees with the
> terminology for discrete and indiscrete topologies.
>
> I confess to have used different  terminology in various places.
>
> Of course one use of these notions is to show that the functor Ob
> preserves limits and colimits, which is  a start on constructing them.
...

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Re: Reference requested

[jpradines]

The use of a lot of terminologies stemming from topology for describing 
purely algebraic properties seems to be widespread  and fashionable among  an 
important part of the community of categorists.
This may be a convenient source of intuition and analogies by giving a 
topological or geometrical fragrance to such algebraic concepts.
However the considerable drawback is that this habit is a source of 
unsolvable clashes for people who are currently using topological or more 
specially Lie groupoids, more generally structured (in Ehresmann's sense), 
i. e. internal, groupoids, who are obliged to create alternative 
terminologies.
For the special case of the duet discrete/undiscrete (or indiscrete, or 
sometimes coarse) I'm personally using presently null/banal (there are a lot 
of different terminologies used by various authors).
(As to the term "chaotic", I prefer to avoid comments, being afraid to 
perturb the beautifully non chaotic weather we are presently enjoying in our 
region).

Jean Pradines

----- Message d'origine ----- 
De : "Ronnie Brown" <ronnie.profbrown@btinternet.com>
À : "Peter May" <may@math.uchicago.edu>
Cc : <categories@mta.ca>
Envoyé : jeudi 29 septembre 2011 15:41
Objet : categories: Re: Reference requested


> Why not use the term `indiscrete groupoid' for the functor that gives a
> right adjoint to the  functor Ob: Groupoids \to Sets?   The left adjoint
> is then of course the `discrete groupoid'.  This agrees with the
> terminology for discrete and indiscrete topologies.
>
> I confess to have used different  terminology in various places.
>
> Of course one use of these notions is to show that the functor Ob
> preserves limits and colimits, which is  a start on constructing them.
>
> It is not surprising that this concept occurs widely. In groupoids there
> is a notion of covering morphism and the universal cover of a group G
> is of course an indiscrete groupoid G'; this groupoid is by no means
> `trivial' since it comes equipped with a covering morphism  p: G' \to
> G.  This approach to covering space theory is given in my book `Topology
> and groupoids'.
>
> Ronnie
>
>

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Re: Reference requested

[Ronnie Brown]

Why not use the term `indiscrete groupoid' for the functor that gives a
right adjoint to the  functor Ob: Groupoids \to Sets?   The left adjoint
is then of course the `discrete groupoid'.  This agrees with the
terminology for discrete and indiscrete topologies.

I confess to have used different  terminology in various places.

Of course one use of these notions is to show that the functor Ob
preserves limits and colimits, which is  a start on constructing them.

It is not surprising that this concept occurs widely. In groupoids there
is a notion of covering morphism and the universal cover of a group G
is of course an indiscrete groupoid G'; this groupoid is by no means
`trivial' since it comes equipped with a covering morphism  p: G' \to
G.  This approach to covering space theory is given in my book `Topology
and groupoids'.

Ronnie



Ronnie

On 29/09/2011 02:35, Peter May wrote:
> 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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Reference requested

[Peter May]

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






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