ONT Re: Category Theory
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CAT. Note 5
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| Introduction (cont.)
|
| The notion of a monoid (a semigroup with identity)
| plays a central role in category theory. A monoid M
| may be described as a set M together with two functions:
|
| (2). !m! : M x M -> M
|
| !h! : 1 -> M
|
| such that the following two diagrams in !m! and !h! commute:
|
| (3).
| 1 x !m!
| M x M x M o------------------>o M x M
| | |
| | |
| | |
| !m! x 1 | | !m!
| | |
| | |
| v v
| M x M o------------------>o M
| !m!
|
| !h! x 1 M x M 1 x !h!
| 1 x M o------------------>o------------------>o M x 1
| | | |
| | | |
| | | |
| !q! | | !m! | !r!
| | | |
| | | |
| v v v
| M o===================o===================o M
| M
|
| Here 1 in 1 x !m! is the identity function M -> M, and 1 in 1 x M
| is the one-point set 1 = {0}, while !q! and !r! are the bijections
| of (1) above. To say that these diagrams commute means that the
| following composites are equal:
|
| !m! o (1 x !m!) = !m! o (!m! x 1)
|
| !m! o (!h! x 1) = !q!
|
| !m! o (1 x !h!) = !r!
|
| These diagrams may be rewritten with elements, writing the function !m! (say)
| as a product !m!(x, y) = x y for x, y in M and replacing the function !h! on
| the one-point set 1 = {0} by its (only) value, an element !h!(0) = u in M.
| The diagrams above then become:
|
| <x, y, z> o|----------------->o <x, yz>
| - -
| | |
| | |
| | |
| | |
| | |
| v v
| <xy, z> o|----------------->o (xy)z = x(yz)
|
| <0, x> o|----------------->o <u, x>
| - -
| | |
| | |
| | |
| | |
| | |
| v v
| x o===================o u x
|
| <x, u> o<-----------------|o <x, 0>
| - -
| | |
| | |
| | |
| | |
| | |
| v v
| x u o===================o x
|
| They are exactly the familiar axioms on a monoid, that
| the multiplication be associative an have an element u
| as left and right identity.
|
| This indicates, conversely, how algebraic identities
| may be expressed by commutative diagrams.
|
| Mac Lane, 'Cat Work Math', pp. 2-3.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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