
The Yoneda Lemma (2017) - ghosthamlet
http://hkopp.github.io/2017/11/the-yoneda-lemma
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atq2119
These kinds of articles would be infinitely more helpful if they spelled out
_how_ the specific examples listed in the beginning can be derived as a
consequence of the more general result. As it is, it's still mostly a bunch of
general abstract nonsense, as one of my math professors liked to call it.

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kevinventullo
The Yoneda Lemma basically says that if you take your favorite object, and
just consider morphisms to (or from) that object, that tells you everything
you need to know.

So for example, if I'm in the category of groups, and I look at Z/3, then what
are morphisms from Z/3 to a given group G? Well those are just the 3-torsion
elements of G (elements such that g^3 = identity). That is, the image of 1
(mod 3) must be such an element, and conversely such an element determines a
morphism by sending 1 (mod 3) there.

Yoneda says this actually characterizes the group Z/3\. The language used is
that Z/3 "represents" the functor taking a group to the set of its 3-torsion
elements.

This can be useful when the object you're trying to characterize is more
complicated than the functor it represents.

In algebraic number theory, the functor taking a field k to the set of pairs
(E,p) where E is an elliptic curve and p is a point of order 593229 defined
over k, _is representable_ by some equation, but the equation would be opaque
and maybe not so useful.

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laretluval
Here's another article, with more pictures and examples.

[https://www.math3ma.com/blog/the-yoneda-
lemma](https://www.math3ma.com/blog/the-yoneda-lemma)

