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Yoneda's lemma concerns functors from a fixed category to the category of sets, . If is a locally small category (i.e. the hom-sets are actual sets and not proper classes), then each object of gives rise to a natural functor to called a hom-functor. This functor is denoted: . The ( covariant) hom-functor sends to the set of morphisms and sends ...
For example, universal objects being unique up to unique isomorphism can be thought of as an application of the Yoneda lemma. If nothing else, the Yoneda lemma gives us the Yoneda embedding, which eventually leads to the functor of points which Dylan alludes to above.
- Some elaboration on Dylan Moreland's comment is in order. Consider the gadget $\\text{GL}_n(-)$. What sort of gadget is this, exactly? To every comm...
- I think one of the classical examples of the Yoneda lemma in action was Serre's observation that "cohomology operations" (i.e. natural transformati...
- This answer is much more naive than the others, but I'll post it anway. The most standard application of Yoneda's Lemma seems to be the uniqueness,...
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Sep 14, 2017 · The Yoneda lemma gives us surjectivity. To see this, set F = hom(−,Y) F = hom ( −, Y) in the statement of the lemma. Then we have a bijection. Nat(hom(−,X),hom(−,Y)) ≅hom(X,Y). N a t ( hom ( −, X), hom ( −, Y)) ≅ hom ( X, Y). Now suppose η: hom(−,X) → hom(−,Y) η: hom ( −, X) → hom ( −, Y) is any natural transformation.
The Yoneda Lemma Review Emily Saunders September 2020 1 The Yoneda Lemma First we start by recalling the de niton of the Yoneda functor hA=h Aor Hom C( ;A)=Hom C(A; ) of an object A2C. De nition 1.1. Given an object A2Cwith Csmall, the contravariant Yoneda functor hA: Cop! Setis de ned on objects by, hA(X) = Hom C(X;A)
Yoneda tells us that X0(A) ˙ X(A) naturally in A; in other words, X0˙ X. If Yoneda were false then starting from a single presheaf X, we could build an infinite sequence X;X0;X00;:::of new presheaves, potentially all di erent. But in reality, the situation is very simple: they are all the same. The proof of the Yoneda lemma is the longest ...
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."
1. 2 SHU-NAN JUSTIN CHANG. introductory paper to category theory provides the Yoneda Lemma as a stepping stone which must be understood before grappling with more complex concepts in category theory. This paper assumes basic knowledge of some fundamental algebraic objects, such as rings, groups, modules over rings, etc.
