# CategoryTheory

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In very brief...
A category C consists of:
• A set A of things a1, a2, etc
• ForEach? pair (a1, a2) of elements of A,
• a set Maps_a1,a2 of things. These things are viewed as "maps" "from a1 to a2", and sometimes described (and drawn) as arrows.
• Subject to a few constraints.  These require that every element a of A has one (distinguished) element in its "maps" set, Maps_a_a, called the "identity on a"; a requirement that if you have one arrow  f: a1-->a2, and another arrow g: a2-->a3, then you can combine them to obtain  f.g: a1-->a3; and about three other such restrictions.  They're all reasonably sensible.

For example:
• Let A be the set of all pages on ToothyWiki.  So A consists of {ToothyWiki, RecentChanges, CategoryCategory, MoonShadow, AlexChurchill, ...}
• For each pair (page1, page2) of elements of A,
• let Maps_page1_page2 be defined as "all sequences of links which start at page1 and end at page2".
• Each page  page1  has an identity element: the sequence of links saying "Click the title of page1, and then click the entry page1 in the resulting list".
• If you have an "arrow" s: page1-->page2  (i.e. a sequence of links starting at page1 which ends you up at page2), and another arrow t: page2-->page3, you can naturally combine them to end up with s.t: page1-->page3.
• The other requirements can be satisfied similarly.

More mathematical examples would be:
• Any partial order P can be viewed as a category: let the set A be the elements of P, and for each pair (p1,p2) in P, Maps_p1_p2 has one element iff p1 <= p2.
• You can have a category of "all matrices".  One way to do this is by letting your set A be the positive integers A = {1, 2, 3, ...}.  Then you let Maps_n_m be the set of all n-by-m matrices.  This is nice because composing arrows (in category terms) becomes multiplying matrices, and so on.

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
f1
|           |
|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

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