ONT Category Theory
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CAT. Category Theory
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CAT. Note 1
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| Excerpts from 'Categories for the Working Mathematician' by Saunders Mac Lane
|
| Introduction
|
| Category theory starts with the observation that many properties of
| mathematical systems can be unified and simplified by a presentation
| with diagrams of arrows. Each arrow f : X -> Y represents a function;
| that is, a set X, a set Y, and a rule x ~> f x which assigns to each
| element x in X an element f x in Y; whenever possible we write f x
| and not f(x), omitting unneccessary parentheses. A typical diagram
| of sets and functions is:
|
| Y
| o
| ^ \
| / \
| f / \ g
| / \
| / v
| o---------->o
| X h Z
|
| It is commutative when h is h = g o f, where g o f is the usual composite
| function g o f : X -> Z, defined by x ~> g(f x). The same diagrams apply
| in other mathematical contexts; thus in the "category" of all topological
| spaces, the letters X, Y, and Z represent topological spaces while f, g,
| and h stand for continuous maps. Again, in the "category" of all groups,
| X, Y, and Z stand for groups, f, g, and h for homomorphisms.
|
| Mac Lane, 'Cat Work Math', p. 1.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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