Catching up on my mail, I found this saved.
John: care to chat a bit more how you use topos theory?
We've tried to get someone to chat about category theory, either at
wedtech or in the complex, but thus far no one has felt up to the
task. Not sure why. Possibly just too elusive? Or possibly evolving
too quickly?
-- Owen
On Aug 13, 2008, at 1:58 PM, John F. Kennison wrote:
Carl,
I have only skimmed parts of Goldblatt's book. It did look like it
was trying to do the hard job of giving the important concepts of
topos theory, along with the basic technical details. (it is easier
to assume that the readers know category theory and also know how to
digest a book that only gives a formal approach to a subject.)
Probably the best way to digest a new topic is to see how it applies
to a particular problem, that is of interest. When I get organized
(right now I am, teaching an intensive two-week course for incoming
students who need a brushing up on pre-calculus, or pre-pre-calculus-
and I also have a 12-year old granddaughter, who likes math, suduko
and monopoly visiting until Tues.) --I might try to explain how I
use topos theory to break a dynamic system into its cyclic parts.
John
On 8/13/08 3:36 PM, "Carl Tollander" <[EMAIL PROTECTED]> wrote:
John,
How do you feel about Goldblatt's book on Topoi? I've been working
through it slooowly and like it so far, but I'm not sure whether it is
leaving important things out. In particular, if you need something to
understand the exposition, say, sheaves, then he goes back and tells
you
just enough about sheaves instead of referring you elsewhere.
Like this:
http://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260/ref=pd_bbs_2?ie=UTF8=books=1218653851=8-2
<http://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260/ref=pd_bbs_2?ie=UTF8&s=books&qid=1218653851&sr=8-2
><http://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260/ref=pd_bbs_2?ie=UTF8&s=books&qid=1218653851&sr=8-2
>
When I try to talk to non-math-centric folks about Category Theory, I
usually start off with Derek Wise's "Stuff with Structure, having
certain Properties" based explanation, but usually people think its
such
an advanced topic that such a starting point couldn't possibly be that
straightforward. If they do buy into that, however, you can give
them a
feel for n-Cats and natural transformations. At that point they start
thinking what they could do with them and there's (maybe) enough
motivation to backfill in with the formal definitions. It's harder I
think to go with the formal stuff first (I know it is for me) if there
isn't much formal math background to relate to.
I think category theory (particularly for us as it relates to
complexity) represents a cultural change and so the initial
explanations
we seek have to resonate broadly at that level if we are going to
set a
foundation for not fooling ourselves (and our clients) when things get
more formal.
And I always liked the idea of using "stuff" as a technical term.
Carl
John F. Kennison wrote:
Further thoughts on categories and their applications.
References: Toposes. Theories and Triples can be found at Michael
Barr's home page, www.math.mcgill.ca/barr/. The notes suggested by
Jochen, below, are a good starting point.
Applications: There are a lot of different types of categories and
categorical constructions. So there are, potentially, lots of
possible
applications. It is probably best to have a team approach, with at
least one expert in the area of the intended application and at least
one expert in category theory. But all experts have to learn
something
of the language, basic results, concepts of both fields, then they
can
see if one set of ideas can map onto another.
This sort of provides an answer to Nick's question. One can benefit
(or perhaps enjoy) a field of abstract mathematics if the underlying
concepts can be made intuitively clear with a minimum of technical
complexity.
Specifically can categories relate to questions of metaphor and
analogy? Rosen in "Life Itself" belabors an approach to metaphor
which
strikes me as heavy-handed yet not comprehensive enough. Is there a
better connection? --I think that's a good question.
--John
On 8/12/08 3:39 PM, "Jochen Fromm" <[EMAIL PROTECTED]> wrote:
I wonder if category theory can be applied
to model metaphors and analogies? Or perhaps
gene regulatory networks?
The following slides seem to be suitable for folks
with a good undergrad math background:
"Category Theory for Beginners"
http://www.cs.toronto.edu/~sme/presentations/cat101.pdf
<http://www.cs.toronto.edu/%7Esme/presentations/cat101.pdf><http://www.cs.toronto.edu/%7Esme/presentations/cat101.pdf
>
-J.
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============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
