ONT Re: Category Theory
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CAT. Note 2
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| Introduction (cont.)
|
| Many properties of mathematical constructions may
| be represented by universal properties of diagrams.
| Consider the cartesian product X x Y of two sets,
| consisting as usual of all ordered pairs <x, y>
| of elements x in X and y in Y. The projections
| <x, y> ~> x, <x, y> ~> y of the product on its
| "axes" X and Y are functions p : X x Y -> X,
| q : X x Y -> Y. Any function h : W -> X x Y
| from a third set W is uniquely determined by
| its composites p o h and q o h. Conversely,
| given W and two functions f and g as in the
| diagram below, there is a unique function h
| which makes the diagram commute; namely,
| h w = <f w, g w> for each w in W.
|
| W
| o
| /|\
| / | \
| / | \
| / | \
| f / | \ g
| / | \
| / | \
| / | \
| v v v
| o<--------o-------->o
| X p XxY q Y
|
| Thus, given X and Y, <p, q> is "universal" among pairs of
| functions from some set to X and Y, because any other such
| pair <f, g> factors uniquely (via h) through the pair <p, q>.
| This property describes the cartesian product X x Y uniquely
| (up to a bijection); the same diagram, read in the category
| of topological spaces or of groups, describes uniquely the
| cartesian product of spaces or the direct product of groups.
|
| 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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