[Inquiry] Re: Category Theory

[Jon Awbrey]

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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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