advanced_tools:category_theory:yoneda_lemma
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advanced_tools:category_theory:yoneda_lemma [2017/11/09 09:13] 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> | ||
| 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> | ||
| + | <blockquote>[A]n object is completely determined by its relationships to other objects. | ||
| + | <cite>http://www.math3ma.com/mathema/2017/9/14/the-yoneda-lemma</cite> | ||
| + | </blockquote> | ||
| + | |||
| + | |||
| + | <blockquote> One way to think of the Yoneda lemma is | ||
| + | precisely this: that the objects of any category can be | ||
| + | interpreted as sets with extra structure. Think about this | ||
| + | a minute. We have an abstract category C and we wish to | ||
| + | associate to each object of C some set equipped with extra | ||
| + | structure. Moreover, we want to do this in a way which | ||
| + | completely records everything there is to know about this | ||
| + | object. How can we do it? | ||
| + | |||
| + | Well, the only interesting thing about an object in a category | ||
| + | is its morphisms to and from other objects, and how these compose | ||
| + | with *other* morphisms. This principle should be our guide. | ||
| + | |||
| + | So, what should we do? Simple: associate to the object c the | ||
| + | set of all morphisms from c to other objects in C! Let's call | ||
| + | this set hom(c,-). | ||
| + | |||
| + | Of course, this is more than a mere set: it's a set with extra | ||
| + | structure. First of all, it's a set made of lots of little | ||
| + | subsets for each object c' in C, we get a subset hom(c,c'), | ||
| + | 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 | ||
| + | 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. | ||
| + | |||
| + | A set with all this structure has a name: it's called a "functor | ||
| + | from C to Set". | ||
| + | |||
| + | The Yoneda lemma says that this "set with extra structure" knows | ||
| + | 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/2017/9/6/the-yoneda-embedding | ||
| * http://www.math3ma.com/mathema/2017/9/14/the-yoneda-lemma | * http://www.math3ma.com/mathema/2017/9/14/the-yoneda-lemma | ||
advanced_tools/category_theory/yoneda_lemma.1510215216.txt.gz ยท Last modified: 2017/12/04 08:01 (external edit)
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