advanced_tools:category_theory:yoneda_lemma
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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?] |
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| + | 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> | ||
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| <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 44: | Line 46: | ||
| consisting of all morphisms from c to c'. Second of all, it's | consisting of all morphisms from c to c'. Second of all, it's | ||
| a set with an "action of C". In other words, given an element | a set with an "action of C". In other words, given an element | ||
| - | f in hom(c,c'), and a morphism g: c' -> c'', we get an element | + | f in hom(c,c'), and a morphism g: c' -> c' ', we get an element |
| - | fg in hom(c,c''), just by composing f and g. | + | fg in hom(c,c' '), just by composing f and g. |
| A set with all this structure has a name: it's called a "functor | A set with all this structure has a name: it's called a "functor | ||
| 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.1510215376.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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