This shows you the differences between two versions of the page.
Both sides previous revision Previous revision | Next revision Both sides next revision | ||
advanced_tools:category_theory:monoids [2017/11/10 09:41] jakobadmin [Student] |
advanced_tools:category_theory:monoids [2017/11/10 09:41] jakobadmin [Student] |
||
---|---|---|---|
Line 13: | Line 13: | ||
<tabbox Student> | <tabbox Student> | ||
- | <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. | ||