Grothendieck、Yoneda 引理和所谓的[ 相对视角（relative point of view）]
段落一：
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, \(\rightarrow X\). Apparently that's called the relative point of view.
我偶尔听人说，Grothendieck 的一个重要洞见是：当我们对某个对象 \(X\) 感兴趣时，与其直接研究 \(X\) 本身，不如研究指向它的态射，即 \(\rightarrow X\)。显然，这种观点被称为 相对视角（relative point of view）。
段落二：
First question. How is that principle applied in practice? What are some concrete examples in mathematics where the relative point of view is useful?
第一个问题： 这个原则在实践中是如何运用的？有哪些数学中的具体例子说明了“相对视角”的实用性？
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.
维基百科提到了黎曼–罗赫定理，而一个数学堆栈（MSE）上的类似问题中也提到一个关于凝聚层（coherent sheaves）的定理。不幸的是，我目前还不懂代数几何。
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?
有没有一些更通俗、易于本科生理解的“相对视角”的应用？比如在线性代数、群论、环论、伽罗瓦理论，甚至基础的范畴论中？
What are (some of) the most important theorems that feature the relative point of view?
有哪些重要的定理体现了这种“相对视角”？
段落三：
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.
我最近学到了范畴论中的 Yoneda 引理（我知道这个引理的表述并能证明它）。我知道它可以用于证明：当两个对象拥有相同的泛性质（universal property）时，它们是同构的。
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?
在 Awodey 的范畴论教材中有一个具体应用：在结构良好的范畴中，
这个证明很优雅，我同意。但这好像并不配得上许多人对 Yoneda 引理的高度评价，对吗？
Maybe a more concrete application in non-category theory would help me to get convinced of the contrary.
或许在非范畴论领域中找到一个更具体的应用，能让我改变这种看法。
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.)
例如，我在维基百科（和其他地方）看到 Grothendieck 在他著名的著作 EGA（Éléments de Géométrie Algébrique） 中使用了 Yoneda 引理（而这本书好像大家都在谈论）。
实际上，Grothendieck 的另一个洞见似乎就是：Yoneda 是有用的。
段落四：
Second question. So what were Grothendieck's main applications of the Yoneda lemma in algebraic geometry?
第二个问题： 那么，Grothendieck 在代数几何中是如何主要应用 Yoneda 引理的？
(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.)
（和第一个问题不同，在这个问题中，我只想大致知道 Grothendieck 用 Yoneda 引理证明了什么样的结论——不需要细节，因为我已经理解了 Yoneda 引理的某个应用。）
段落五：
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.)
第三个问题： 第二个问题和第一个问题有关系吗？换句话说，“相对视角” 和 “Yoneda 引理” 之间是否存在某种联系？
（至少那个维基百科页面提到了 Yoneda 引理。）
总结
这段话可以被视为对以下三个问题的追问：
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