Dear Organlearners,
Nick Arnett <listbot@mccmedia.com> writes:
>Could you recommend a good introduction to category theory,
>either a book or a web site, perhaps both? The things you
>describe here seem to fit quite a few patterns in the world today.
Greetings Nick,
As to how many patterns in the world of today category theory helps to
systemise lies solely in the eyes of the beholder. Some will sit upright,
feeling the goose flesh developing. But others will object to the exterme
level of abstraction involved, complaining that it is devoid of any
practice.
As for web sites, I do not expect much. The reason is that Category Theory
(CT) is rich in graphical diagrams. These diagrams consists of
objects/sets ("beings") and arrows/functors ("becomings"). The arrows
cannot be represented by text, but are represented by directed line
segments, almost like vectors. (However, there are vast differences
between arrows and vectors so that we should not ever confuse them.) To
load every diagram with JPG or BMP images will take ages and bucks.
Further more, I have a very limited budget for web browsing so that I have
not searched for a CT websites.
In our library there are a dozen or so books on CT, apart from the many
mathematical journals with CT papers in them. Unfortunately, these books
are aimed at the very advanced level. It is only our Master and Doctorate
students in mathematics who have to study topics from CT. These books are
intended for them. There is no under-graduate course at our university in
mathematics or philosophy with CT as topic. Hence no books have been
bought for an introductory level. I am not a proffesional mathematician so
that I do not know what is available from publishers.
There is one book which I can recommend. It begins at an introductory
level, but increases the pace so that after the first ten chapters
(two hundred pages) the pedal has to be stepped in deeply. The first
three chapters is not partcular heavy. The book is:
R Goldblatt 1979. Topoi. The categorial analysis of logic.
North Holland Publ Co, New York.
[Host's Note: Unfortunately, this book is out of print according to
Amazon.com ...Rick]
Consider the definition of any category in CT.
A category CAT is a SYSTEM which
(1) contains objects (sets) and arrows (functors)
(2) they commute, i.e. ....object-arrow-object-arrow....
(3) they satisfy the identity diagram
(4) they satisfy the associative diagram
Nick, the following notes might interest you:
(a) CT is primarily concerned with systems. It is pity that Systems
Thinking in Operational Research and Management Science has not yet given
any attention to CT. This is the result of apartheid between academical
subjects.
(b) The system CAT is the simplest of all mathematical systems. Thus it
forms the background against which all other mathematical activities takes
place. It is like the 4D space-time background against which the
relativistic dynamics of Einstein takes place.
(c) All other mathematical systems emerge form CAT by bringing in
additional axioms constructively. Conditions (3) and (4) may be considered
as the defining axioms of CAT. To bring in any third or additional axioms
which please one's intuition and test one's construtive skills, the
essentiality friutfulness is required. See my Essentiality -
"connect-beget" (fruitfulness) LO18750
<http://www.learning-org.com/98.07/0206.html>
(d) Condition (1) points to a the sub-level of CAT. At the sublevel
"beings" (objects or sets) and "becomings" are distinguished. No CT
activities happen at the sub-level. Thus CT is not analytic and
reductionistic as so many other theories.
(e) Condition (2) points to the base-level of CAT. The commuting of
diagrams means that objects have to connect through arrows and that arrows
have to connect through objects. In other words, apartheid between
objects and arrows are not allowed. The most simplest diagram is
"arrow-object". Such "arrow-object" diagrams act as qualifiers. At the
next level we have "object-arrow-object" and "arrow-object-arrow"
diagrams. The concept of a function, namely "input-act-output" lies in
this level. Likewise the concept of set-membership, i.e
"element-ismemberof-set". This base-level of CAT is made possible by
liveness. See my Essentiality - "becoming-being" (liveness) LO17651
<http://www.learning-org.com/98.04/0036.html>
(f) Conditions (3) and (4) point to a super-level of CAT. The
sub-level and the base-level can always be distinguished in
this super-level. Condition (3) is made possible by sureness.
See my
Essentiality - "identitity-categoricity" (sureness) LO17823
<http://www.learning-org.com/98.04/0207.html>
Condition (4) is made possible by wholeness. See my
Essentiality - "associativity-monadicity" (wholeness) LO18276
<http://www.learning-org.com/98.06/0040.html>
The super-level of CAT can become extremely complex, as
complex as one can wish for.
Jon Krispin, if you are reading, I hope to have wet your appetite also.
The behaviourism of psychology is closer to CT thinking than one would
expect. Just as a behaviourist is sensitive to "becoming" (which the
behaviourist calls behaviour), the mathematical categorist is also
sensitive to "becoming" (which is called arrow or functor). Action
research is another topic close to CT thinking. Languages are also open to
CAT constructions because the nouns may be thought of as objects (sets),
the verbs as arrows (functors) and the predicates (adjectives and adverbs)
as arrow-object and object-arrow qualifiers. Once we begin to seek
applications for CT thinking, nature and culture have more examples than
what we ever could cope with. I believe that CT thinking will become much
more common in the next century, even more common than SET thinking which
children in primary school nowadays have to cope with.
I have kept this last paragraph for Rick. Rick, I will send you under a
separate message as file attachment a bitmap illustrating (i) the Axiom of
Identity and (ii) the Axiom of Associaitivity as CAT diagrams. With your
permission, please make it available on the LO server and give our fellow
learners the URL. Thank you very much.
Best wishes
--At de Lange <amdelange@gold.up.ac.za> Snailmail: A M de Lange Gold Fields Computer Centre Faculty of Science - University of Pretoria Pretoria 0001 - Rep of South Africa
Learning-org -- Hosted by Rick Karash <rkarash@karash.com> Public Dialog on Learning Organizations -- <http://www.learning-org.com>