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Fred E.J. Linton | 1 Oct 2011 19:46
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Re: Reference requested

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
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jpradines | 2 Oct 2011 16:48
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Re: Reference requested

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
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Sergei SOLOVIEV | 2 Oct 2011 23:10
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diagrams in computer algebra

Question to the list:

I am very much interested, does there exist any practically usable
system for the "computer algebra style" work with diagrams. I.e. not
just write down the diagrams (like XyPic) but for example for
diagram chasing,
verification of commutativity in any interesting classes of categories
with structure etc. As I remember there were some systems permitting
computation of some limits in very limited classes of categories,
also some heavy attempts of formalisation in some general
purpose proof assistants
but I do not know about anything else.

Regards to all

Sergei Soloviev

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Ellis D. Cooper | 3 Oct 2011 03:48

Natural Functorial Categorical Intuition

Many thanks for responses to my initial post on the Subject.
It was partly motivated by the seeming variety of kinds of intuition
in the cited
references.

Also, I am utterly in awe of categorical intuitions codified, for example, by
adjoint pairs of functors in geometry, algebra, and logic. Until recently
I only dreamed of some day having such an intuition. I think I now have one,
and would like to know if you agree that it is specifically a
categorical intuition.
It is a separate question whether there is a rigorous explication and proof.

Define a (two-dimensional) shape to be a smooth injection of the circle
into the plane. By the Jordan Curve Theorem a shape has an exterior.
Given a point in the exterior -- call it a viewpoint -- there exists
a finite set
of intersections of the lines through the viewpoint -- call them
sightlines -- which are either tangent
to or pass through an inflection point (with respect to an orthogonal
coordinate system in which the sightline is one coordinate).

For a sufficiently remote viewpoint (maybe infinitely far away) there
exist exactly
two of these sightlines tangent to the shape between which
the angle is less than pi radians, and such that all other of these sightlines
are contained within the sweep of one of the two to the other.

I am guessing that if a cover relation (as in Hasse diagrams of
finite posets) is
defined to be an acyclic irreflexive relation, then the category of
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Fred E.J. Linton | 3 Oct 2011 08:06
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Re: Reference requested

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
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Christopher Townsend | 3 Oct 2011 16:09

Characterizing the adjunction that arises given a groupoid G in a cartesian category C

If G = (G_1 => G_0) is a groupoid in a cartesian category C then there

is an adjunction between [G,C] and C provided that C has coequalizers.

Here [G,C] is notation for the category of objects over G_0 with G_1

actions. The right adjoint takes an object X to (G_0xX,mxId) where m is

the groupoid G's composition.

I'd appreciate any guidance on what has been noted/published about the

following lemma whose proof I believe follows easily once you realise

that the left adjoint sends (G_1,m) to G_0:

Lemma: If L-!R is an adjunction between cartesian categories D and C

(L:D->C and R:C->D), then this adjunction is equivalent to the

adjunction [G,C]->C just described for some groupoid G internal to C if

and only if there exists W an object of D such that the pullback functor

W*:D->D/W is monadic and the functor D/W->C/LW induced by L is an

equivalence.

I feel it is a very neat characterization of the situation where an
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Michael J Healy | 3 Oct 2011 20:02
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Re: diagrams in computer algebra

Sergei,

My colleagues and I have been looking for something like this for a project.  We need to be able to specify
small categories as the completions of finite graphs we are given, extend these by specifying
commutative diagrams, pullbacks, etc, of interest, then define functors generated from graph
homomorphisms, and take colimits of diagrams in Cat, etc etc.  We haven't found anything that does all
this.  So, we're programming it in Haskell---one of our grad students knows the language.   We'll be happy to
share our experience and will probably make the code available.  It's a work in progress.

Best regards,
Mike Healy

On Oct 2, 2011, at 3:10 PM, Sergei SOLOVIEV wrote:

> Question to the list:
> 
> I am very much interested, does there exist any practically usable
> system for the "computer algebra style" work with diagrams. I.e. not
> just write down the diagrams (like XyPic) but for example for
> diagram chasing,
> verification of commutativity in any interesting classes of categories
> with structure etc. As I remember there were some systems permitting
> computation of some limits in very limited classes of categories,
> also some heavy attempts of formalisation in some general
> purpose proof assistants
> but I do not know about anything else.
> 
> Regards to all
> 
> Sergei Soloviev
(Continue reading)

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Alex Simpson | 4 Oct 2011 16:17
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5th Scottish Category Theory Seminar

SCOTTISH CATEGORY THEORY SEMINAR
http://personal.cis.strath.ac.uk/~ng/sct.html

***  Fifth Meeting ***
***  Informatics Forum, University of Edinburgh  ***
***  Friday 25 November 2011, 2.00-5.40pm  ***

The fifth meeting of the Scottish Category Theory Seminar will feature
talks by the following speakers.

   - Anders Kock (Aarhus University)
   - Paul-Andre Mellies (CNRS, Universite Paris 7 - Denis Diderot)
   - Tom Leinster (University of Glasgow)

The meeting is open to all. There is no registration, but, if you
intend to come, it would be helpful (but is not essential) to let us
know by email: scotcats <at> cis.strath.ac.uk

This meeting is generously supported by the Glasgow Mathematical
Journal Trust.

Alex Simpson (local organizer)

(ScotCats organizers: Neil Ghani, Tom Leinster, AS)

--

-- 
Alex Simpson, LFCS, School of Informatics, Univ. of Edinburgh, UK
Email: Alex.Simpson <at> ed.ac.uk             Tel: +44 (0)131 650 5113
Web: http://homepages.inf.ed.ac.uk/als   Fax: +44 (0)131 651 1426
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Eduardo J. Dubuc | 4 Oct 2011 18:04
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Re: diagrams in computer algebra

I paste and copy some old posting to this list that I have kept. It
seems to me that they are relevant for your discussion.

e.d.

> I'd like to announce a selection of Web-based category theory
> demonstrations that I've put up at
> http://www.j-paine.org/cgi-bin/webcats/webcats.php . The page contains a
> number of buttons such as "generate and demonstrate an equaliser" and
> "generate and demonstrate a limit": clicking on one will generate
> an example of the construct in the category of finite sets, and display it
> as a listing of its objects and arrows, and as a diagram. For limits and
> colimits, the demos generate a small random graph, convert it to a
> diagram, then compute and display its limit or colimit.
>
> Comments would be very welcome. The demos are a bit of an experiment: I
> had some categorical algorithms lying around from other work, and thought
> it would be interesting to connect them to the Web.

> "If you could commission a computer demonstration of any categorical idea,
> what would you ask for? Could such demonstrations have helped you, or your
> students, learn tricky ideas? And, would you be willing to share the
> visualisations and metaphors that you have devised to explain these ideas
> to yourself or others?"
>
> We've started a thread on this topic at the n-Category Cafe',
> http://golem.ph.utexas.edu/category/2009/04/graphical_category_theory_demo.html
> . Anybody interested in using computers to demonstrate concepts from
> category theory, do please have a look there. I've just added an
> explanation of the techniques available for delivering demonstrations over
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Barney Hilken | 4 Oct 2011 18:38

Sheaf of sites?

If (C,J) is a site, and F is a sheaf of sites on (C,J), in the sense that F(X) is a site for each object X of C and
F(f) is a morphism of sites for each arrow f of C, are there any straightforward, useful conditions under
which F forms an internal site in Sh(C,J), or is it usually necessary to construct the powerobject and work
from the definition?

Thanks, 

Barney.

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Gmane