advanced_tools:category_theory:yoneda_lemma
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
advanced_tools:category_theory:yoneda_lemma [2017/11/09 09:16] jakobadmin [Student] |
advanced_tools:category_theory:yoneda_lemma [2017/11/09 09:22] (current) jakobadmin [Why is it interesting?] |
||
|---|---|---|---|
| Line 6: | Line 6: | ||
| + | It is important for physics, because it allows us to make the ideas of [[advanced_notions:relational_physics|relational physics]] precise. | ||
| <tabbox Layman> | <tabbox Layman> | ||
| Line 19: | Line 19: | ||
| <blockquote>[T]he collection of all ways to relate an object to other objects is isomorphic to the object itself.<cite>https://news.ycombinator.com/item?id=7715277</cite></blockquote> | <blockquote>[T]he collection of all ways to relate an object to other objects is isomorphic to the object itself.<cite>https://news.ycombinator.com/item?id=7715277</cite></blockquote> | ||
| - | * http://www.math3ma.com/mathema/2017/9/6/the-yoneda-embedding | + | <blockquote>[A]n object is completely determined by its relationships to other objects. |
| - | * http://www.math3ma.com/mathema/2017/9/14/the-yoneda-lemma | + | <cite>http://www.math3ma.com/mathema/2017/9/14/the-yoneda-lemma</cite> |
| + | </blockquote> | ||
| <blockquote> One way to think of the Yoneda lemma is | <blockquote> One way to think of the Yoneda lemma is | ||
| Line 51: | Line 53: | ||
| The Yoneda lemma says that this "set with extra structure" knows | The Yoneda lemma says that this "set with extra structure" knows | ||
| - | everything you'd ever want to know about the object c. <cite>[[https://groups.google.com/forum/#!msg/sci.physics.research/6cET8VmcUZU/IH9Tpq6NLTsJ|John Baez]]</cite></blockquote> | + | everything you'd ever want to know about the object c. |
| + | |||
| + | Note that I got away with less than you might have thought I'd | ||
| + | need! I only considered the morphisms *from* c, not the morphisms | ||
| + | *to* c. In fact there is another version of the Yoneda lemma | ||
| + | that uses the morphisms *to* c instead. I believe this is | ||
| + | the one people usually talk about - but of course it doesn't | ||
| + | really matter. | ||
| + | |||
| + | <cite>[[https://groups.google.com/forum/#!msg/sci.physics.research/6cET8VmcUZU/IH9Tpq6NLTsJ|John Baez]]</cite></blockquote> | ||
| + | |||
| + | Great explanations can be found here: | ||
| + | * http://www.math3ma.com/mathema/2017/9/6/the-yoneda-embedding | ||
| + | * http://www.math3ma.com/mathema/2017/9/14/the-yoneda-lemma | ||
| <tabbox Researcher> | <tabbox Researcher> | ||
advanced_tools/category_theory/yoneda_lemma.1510215417.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
