# Grothendieck's relative point of view and Yoneda lemma

Occasionally I hear people saying that one of Grothendieck's big insights was that often when interested in an object $$X$$ it's better to study morphisms into that object, $$-\to X$$. Apparently that's called the relative point of view.

First question. How is that principle applied in practice? What are some concrete examples in mathematics where the relative point of view is useful?

Wikipedia mentions the Riemann–Roch theorem and a similar MSE question mentions a theorem about coherent sheaves. Unfortunately, I don't know any algebraic geometry yet. Are there more down-to-earth applications of the relative point of view that an undergraduate can understand, say, in linear algebra, group theory, ring theory, Galois theory, or maybe even in basic category theory?

What are (some of) the most important theorems that feature the relative point of view?

I recently heard about the Yoneda lemma in category theory (I know the statement and can prove it). I know that it can be used to prove that two objects are isomorphic whenever they have the same universal property. In Awodey's category theory book, there's a concrete application of that: in categories with enough structure, $$(A\times B)+(A\times C)\cong A\times (B+C)$$. That proof is elegant, I agree. But it doesn't live up with the praise many people give to the Yoneda lemma, does it?

Maybe a more concrete application in non-category theory would help me to get convinced of the contrary. For instance, I read on Wikipedia (and elsewhere) that Grothendieck used the Yoneda lemma in his famous book EGA (which a lot of people seem to talk about). (In fact, it seems this was another insight of him: that Yoneda is useful.)

Second question. So what were Grothendieck's main applications of the Yoneda lemma in algebraic geometry? (In contrast to the first question, here it suffices for me to just know roughly what kind of statement he proved with the Yoneda lemma---rather than understanding it in detail, because I already know one application of the Yoneda lemma.)

Third question. Is the second question related to the first one, i.e., is there a connection between the relative point of view and the Yoneda lemma? (At least the Wikipedia page linked above mentions the Yoneda lemma.)

• This is a hard question to attempt to respond to. One thing that may interest you is that Cayley's theorem ("every finite group is isomorphic to a subgroup of a symmetric group") can be seen as a very particular case of the Yoneda lemma and its consequence, the Yoneda embedding (which exhibits any small category as a subcategory of a certain functor category). May 4, 2022 at 23:38
• @TabesBridges Thanks! What other applications of Yoneda come to your mind? May 5, 2022 at 13:14
• Crosspost: mathoverflow.net/questions/421978/… May 8, 2022 at 15:00
• Does this answer your question? Grothendieck's "Relative" Point of View May 11, 2022 at 14:28