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위키데이터

• ID : Q320577

말뭉치

1. Almost identically, there’s a contravariant Yoneda Lemma, saying that for every contravariant functor .
2. I’ll come back tomorrow to try explaining what the Yoneda Lemma means.
3. It turns out to be a fairly direct application of the Yoneda Lemma, arguably the most important result in category theory.
4. For a locally small category, the Yoneda embedding is the functor sending an object to the contravariant functor and sending a morphism to the natural transformation given by composition.
5. The Yoneda lemma shows that an object in a category is determined up to isomorphism by the presheaf it represents.
6. The Yoneda lemma tells us, roughly speaking, that an object is determined by its generalized points.
7. The Yoneda lemma in this case says that a downward closed set contains if and only if it contains .
8. Before this, I didn’t have any relevant concrete examples to think about the Yoneda Lemma.
9. In a series of blog posts, I want to relay the example of affine group schemes and try to explain the Yoneda Lemma with this example.
10. I only assume that the reader has heard of the Yoneda Lemma and will briefly recall its statement in the text.
11. Conversely, it might also serve to provide some insight into the Yoneda Lemma.
12. Firstly, in CAT, the Yoneda embedding y X y_X exists only for locally small X X .
13. With this structure we can naturally state the Yoneda lemma in a 2-category.
14. So we have used the Yoneda lemma as a definition of the hom-functors A ( a , 1 ) A(a,1) ; the axiom asserts their existence.
15. This dual statement is also sometimes known as the Yoneda lemma.
16. If you can write that much down, the Yoneda Lemma says that you’ve got a “space” with geometry to work with.
17. What it means for a variety to be modular comes from these Galois actions, and none of it would be possible without the Yoneda Lemma shaping how we think about spaces!
18. Yoneda lemma says that every category can be thought as a full subcategory of generalized algebras over generalized signatures.
19. Welcome to our third and final installment on the Yoneda lemma!
20. The Yoneda lemma gives us surjectivity.
21. The Yoneda lemma is sometimes described as a generalization of Cayley's theorem from group theory.
22. At those links, you'll notice that there's a third classic corollary of the Yoneda lemma, which we did not cover in this series.
23. The Yoneda lemma stands out in this respect as a sweeping statement about categories in general with little or no precedent in other branches of mathematics.
24. The Yoneda lemma tells us that a natural transformation between a hom-functor and any other functor F is completely determined by specifying the value of its single component at just one point!
25. So let’s review the naturality condition between the two functors involved in the Yoneda lemma.
26. And here’s where the magic of the Yoneda lemma happens: g can be viewed as a point p' in the set C(a, a) .
27. Furthermore, we compare our notion with the notion of category left-tensored over M , and prove a version of Yoneda lemma in this context.
28. We apply the Yoneda lemma to the study of correspondences of enriched (for instance, higher) ∞-categories.
29. The Yoneda lemma is usually the first serious challenge, because to understand it, you have to be able to juggle several things in your mind at once.
30. That, by the way, answers our other question about the dependence on the choice of A in the Yoneda embedding.
31. But in practice it’s more convenient to skip the middle man and define natural transformations in the Yoneda lemma as going directly from these morphisms to F(X).
32. The Yoneda lemma tells us exactly how to construct such natural transformations.
33. Now the Yoneda lemma becomes the following observation.
34. The Yoneda lemma has the following direct consequences.
35. The assumption of naturality is necessary for the Yoneda lemma to hold.
36. The Yoneda lemma is effectively the reason that Isbell conjugation exists.
37. The Yoneda embedding essentially states that for every (locally small) category, objects in that category can be represented by presheaves, in a full and faithful manner.
38. {C} \to \mathbf {Sets} } the following formulas are all formulations of the Yoneda lemma.
39. As stated above, the Yoneda lemma may be considered as a vast generalization of Cayley's theorem from group theory.
40. In order to comprehend the Yoneda embedding, the more elaborate categorical notions of representable functors are needed.
41. The Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object, which allows the embedding of any category into a category of functors defined on that category.

메타데이터

위키데이터

• ID : Q320577

Spacy 패턴 목록

• [{'LOWER': 'yoneda'}, {'LEMMA': 'lemma'}]
• [{'LOWER': 'yoneda'}, {'LEMMA': 'embed'}]