ONT Re: Category Theory
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CAT. Note 3
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| Introduction (cont.)
|
| W
| o
| /|\
| / | \
| / | \
| / | \
| f / | \ g
| / | \
| / | \
| / | \
| v v v
| o<--------o-------->o
| X p XxY q Y
|
| Adjointness is another expression for these universal properties. If we write
| hom(W, X) for the set of all functions f : W -> X and hom(<U, V>, <X, Y>) for
| the set of all pairs of functions f : U -> X, g : V -> Y, the correspondence
| h ~> <p h, q h> = <f, g> indicated in the diagram above is a bijection:
|
| hom(W, X x Y) ~=~ hom(<W, W>, <X, Y>).
|
| This bijection is "natural" in the sense (to be made more precise later)
| that it is defined in "the same way" for all sets W and for all pairs of
| sets <X, Y> (and it is likewise "natural" when interpreted for topological
| spaces or for groups). This natural bijection involves two constructions
| on sets: The construction W ~> W, W which sends each set to the diagonal
| pair !D!W = <W, W>, and the construction <X, Y> ~> X x Y which sends each
| pair of sets to its cartesian product. Given the bijection above, we say
| that the construction X x Y is a 'right adjoint' to the construction !D!,
| and that !D! is left adjoint to the product. Adjoints, as we shall see,
| occur throughout mathematics.
|
| Mac Lane, 'Cat Work Math', pp. 1-2.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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