ONT Re: Category Theory
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CAT. Note 7
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| Introduction (concl.)
|
| Finally, an 'action' of a monoid <M, !m!, !h!> on a set S
| is defined to be a function !n! : M x S -> S such that the
| following two diagrams commute:
|
| 1 x !n!
| M x M x S o------------------>o M x S
| | |
| | |
| | |
| !m! x 1 | | !n!
| | |
| | |
| v v
| M x S o------------------>o S
| !n!
|
| !h! x 1
| 1 x S o------------------>o M x S
| \ |
| \ |
| \ |
| \ |
| !q! \ | !n!
| \ |
| \ |
| \ |
| \ |
| o
| S
|
| If we write !n!(x, s) = x * s to denote the
| result of the action of the monoid element x
| on the element s in S, these diagrams state
| just that:
|
| x * (y * s) = (x y) * s
|
| u * s = s
|
| for all x, y in M and all s in S. These are the
| usual conditions for the action of a monoid on a
| set, familiar especially in the case of a group
| acting on a set as a group of transformations.
| If we shift from the category of sets to the
| category of topological spaces, we get the
| usual continuous action of a topological
| monoid M on a topological space S. ...
|
| Mac Lane, 'Cat Work Math', p. 5.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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