ONT Re: Category Theory
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CAT. Note 6
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| Introduction (cont.)
|
| The same process applies to other identities; for example, one may describe
| a group as a monoid M equipped with a function !z! : M -> M (of course, the
| function x ~> x^(-1)) such that the following diagram commutes:
|
| (4).
| !d! M x M 1 x !z!
| M o------------------>o------------------>o M x M
| | |
| | |
| | |
| | | !m!
| | |
| | |
| v v
| 1 o-------------------------------------->o M
|
| <x, x>
| x o|----------------->o------------------>o <x, x^(-1)>
| - -
| | |
| | |
| | |
| | |
| | |
| v v
| 0 o|------------------------------------->o u = x x^(-1)
|
| Here !d! : M -> M x M is the diagonal function x ~> <x, x> for x in M, while
| the unnamed vertical arrow M -> 1 = {0} is the evident (and unique) function
| from M to the one-point set. As indicated [in the element-mapping diagram],
| this diagram does state that !z! assigns to each element x in M an element
| x^(-1) which is a right inverse to x.
|
| This definition of a group by arrows !m!, !h!, and !z!
| in such commutative diagrams makes no explicit mention
| of group elements, so applies to other circumstances:
|
| If the letter 'M' stands for a topological space (not just a set) and the arrows
| are continuous maps (not just functions), then the conditions (3) and (4) define
| a topological group -- for they specify that M is a topological space with a
| binary operation !m! of multiplication which is continuous (simultaneously
| in its arguments) and which has a continuous right inverse, all satisfying
| the usual group axioms.
|
| Again, if the letter 'M' stands for a differentiable manifold (of class C^oo)
| while 1 is the one-point manifold and the arrows !m!, !h!, and !z! are smooth
| mappings of manifolds, then the diagrams (3) and (4) become the definition of
| a Lie group.
|
| Thus groups, topological groups, and Lie groups can
| all be described as "diagrammatic" groups in the
| respective categories of sets, of topological
| spaces, and of differentiable manifolds.
|
| Mac Lane, 'Cat Work Math', pp. 3-4.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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