ec2-3-215-16-238.compute-1.amazonaws.com | ToothyWiki | RecentChanges | Login | Webcomic

In very brief...
A category C consists of:

For example:

More mathematical examples would be:

With this structure in place, it becomes truly astonishing what you can do with it.  Given that you can develop all of NumberTheory? simply using SetTheory?, this perhaps isn't surprising.

What is surprising is how pretty the pictures you get to draw in CategoryTheory are :)  AlexChurchill's "Essay" (literature review) for PartIIIMaths featured lots of pretty pictures, vaguely like this:

 A  ------>  B
 |           |
 |f2       g1|
 |           |
\|/         \|/
 '    g2     '
 D  ------>  E

You can then insist that the combination arrow f1.g1 (read as "f1-then-g1") is equal to the combination arrow f2.g2.  Or you can define /Equivalence?, and then say actually, the combination arrow f1.g1 doesn't have to be equal to f2.g2, just Equivalent to it.  And so on. 

And you can do startling amounts of really useful things with it.  It's really surprising how you can start seeing categories everywhere.  And once you've proved a lot of theorems about any category anywhere, you can then apply them to this random category off the street, and find a really useful consequence.

/Naturality? is a really big part of CategoryTheory.  It can appear that CategoryTheorists? are really sloppy with rigourousness, and just follow their intuition all the time.  They are actually being very rigourous; it just happens that the axioms of CategoryTheory are chosen in such a way to mean that lots of things do work out really naturally.  You get hundreds of equalities (or equivalences) just dropping out as the consequence of the axioms.  It's a really beautiful aspect of the theory.

AlexChurchill's Essay for PartIIIMaths was done on one aspect of /Naturality? in CategoryTheory, being the CoherenceTheorem?.  But I'll leave writing on that for another time.

Currently the bane of ChrisHowlett's PartIIIMaths life, much to his chagrin. It occurs to him that the course is, in fact, really quite pretty. It's just a shame he doesn't understand any of it.
That's a pity.  ISTR finding similar things in my time, though.  Feel free to talk to be about it, although I certainly make no guarantees as to how much sense I'll be able to make of it given the intervening 2 years.  --AlexChurchill

CategoryCategory?, CategoryMaths
Hehe.  PeterTaylor and myself had a discussion about this.  If anyone starts adding theories to the Wiki, and wants to categorise them, then CategoryTheory would be the appropriate category.  At that point, this could be added to the Category of Categories, and split in two somehow.  Until then, though, I think CategoryMaths is more appropriate  --AC

ec2-3-215-16-238.compute-1.amazonaws.com | ToothyWiki | RecentChanges | Login | Webcomic
Edit this page | View other revisions | Recently used referrers
Last edited November 18, 2003 4:11 pm (viewing revision 6, which is the newest) (diff)