Can someone explain the Yoneda Lemma to an applied mathematician?

I have trouble following the category-theoretic statement and proof of the Yoneda Lemma. Indeed, I followed a category theory course for 4-5 lectures (several years ago now) and felt like I understood everything until we covered the Yoneda Lemma, after which point I lost interest.

I guess what I’m asking for are some concrete examples of the Yoneda Lemma in action. For example, how does it apply to specific categories, like a category with one element, or the category Grp or Set? What results does it generalize? Is there a canonical route to understanding the statement of the Lemma?

If you need to assume knowledge, then assume I have a fairly rigorous education in pure/applied mathematics at the undergraduate level but no further.

Answer

Roughly speaking, the Yoneda lemma says that one can recover an object X up to isomorphism from knowledge of the hom-sets \text{Hom}(X, Y) for all other objects Y. Equivalently, one can recover an object X up to isomorphism from knowledge of the hom-sets \text{Hom}(Y, X).

As I have said before on math.SE, there is a meta-principle in category theory that to understand something for all categories, you should first understand it for posets, regarded as categories where a \le b if and only if there is a single arrow a \to b, and otherwise there are no morphisms from a to b. For posets, Yoneda’s lemma says that an object is determined up to isomorphism by the set of objects less than or equal to it (equivalently, the set of objects greater than or equal to it). In other words, it is determined by a Dedekind cut! In other other words, Yoneda’s lemma for posets says the following:

a \le b if and only if for all objects c we have c \le a \Rightarrow c \le b.

This is a surprisingly useful idea in real analysis. More generally it leads to the idea that Yoneda’s lemma, among other things, embeds a category into a certain “completion” of that category: in fact the standard contravariant Yoneda embedding embeds a category into its free cocompletion, the category given by “freely adjoining colimits.”

As far as examples go, it turns out that in many categories one can restrict attention to a few specific objects Y. For example:

  • In \text{Set} one can completely recover an object S from \text{Hom}(1, S).
  • In \text{Grp} one can completely recover an object G from \text{Hom}(\mathbb{Z}, G). This is because a homomorphism from \mathbb{Z} is freely determined by where it sends 1, and one can recover the multiplication because \mathbb{Z} is naturally a cogroup object, which is equivalent to the claim that \text{Hom}(\mathbb{Z}, G) naturally carries a group structure (that of G).
  • In \text{CRing} (the category of commutative rings) one can completely recover an object R from \text{Hom}(\mathbb{Z}[x], R). The story is similar to that above; it is explained in this blog post. Geometrically this means that one can completely recover an affine scheme \text{Spec } R from \text{Hom}(\text{Spec } R, \mathbb{A}^1(\mathbb{Z})), the ring of functions on it.
  • In \text{Top} (the category of topological spaces) one can completely recover an object X from \text{Hom}(1, X) (the points of X) and \text{Hom}(X, \mathbb{S}), where \mathbb{S} is the Sierpinski space. The latter precisely gives you the open sets of X, and together with knowledge of the composition map \text{Hom}(1, X) \times \text{Hom}(X, \mathbb{S}) \to \text{Hom}(1, \mathbb{S}) you can recover the knowledge of which points sit inside which open sets.

Attribution
Source : Link , Question Author : Chris Taylor , Answer Author : Qiaochu Yuan

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