advanced_tools:category_theory:monoids
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advanced_tools:category_theory:monoids [2017/11/10 09:41] jakobadmin [Student] |
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| + | A monoid is a generalisation of [[advanced_tools:group_theory|group]], in that the | ||
| + | requirement that each element has an inverse is dropped. | ||
| - | <blockquote>what is a category with one object? It is a "monoid". The usual definition of a monoid is this: a set M with an associative binary product and a unit element 1 such that a1 = 1a = a for all a in S. | + | <blockquote>what is a category with one object? It is a "monoid". The usual definition of a monoid is this: a set M with an associative binary product and a unit element 1 such that a1 = 1a = a for all a in S. [...] |
| - | [...] | ||
| We tend to think of this ability to "undo" any process as a key aspect of symmetry. I.e., if we rotate a beer bottle, we can rotate it back so it was just as it was before. We don't tend to think of SMASHING the beer bottle as a symmetry, because it can't be undone. But while processes that can be undone are especially interesting, it's also nice to consider other ones... so for a full understanding of symmetry we should really study monoids as well as groups. | We tend to think of this ability to "undo" any process as a key aspect of symmetry. I.e., if we rotate a beer bottle, we can rotate it back so it was just as it was before. We don't tend to think of SMASHING the beer bottle as a symmetry, because it can't be undone. But while processes that can be undone are especially interesting, it's also nice to consider other ones... so for a full understanding of symmetry we should really study monoids as well as groups. | ||
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| [A] monoid is like a [[advanced_tools:group_theory|group]], but the "symmetries" no longer need be invertible; a category is like a monoid, but the "symmetries" no longer need to be composable! | [A] monoid is like a [[advanced_tools:group_theory|group]], but the "symmetries" no longer need be invertible; a category is like a monoid, but the "symmetries" no longer need to be composable! | ||
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| + | Note for physicists: the operation of "evolving initial data from one spacelike slice to another" is a good example of a "partially defined" process: it only applies to initial data on that particular spacelike slice. So dynamics in special relativity is most naturally described using groupoids. Only after pretending that all the spacelike slices are the same can we pretend we are using a group. It is very common to pretend that groupoids are groups, since groups are more familiar, but often insight is lost in the process. Also, one can only pretend a groupoid is a group if all its objects are isomorphic. Groupoids really are more general. | ||
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| <cite>http://math.ucr.edu/home/baez/week74.html</cite></blockquote> | <cite>http://math.ucr.edu/home/baez/week74.html</cite></blockquote> | ||
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