格罗腾迪克：相对视角 & Yoneda引理 & 凯莱定理

Grothendieck: the relative point of view & Yoneda Lemma & Cayley theorem

ai的回答

Your topic connects Grothendieck's relative point of view, the Yoneda Lemma, and Cayley's Theorem—three deep ideas from category theory and algebra that reflect a shift in how we understand mathematical structures. Let’s unpack these connections clearly and precisely.

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1. Cayley’s Theorem: Group as Actions

Cayley's theorem (from group theory) says:

Every group \(G\) is isomorphic to a subgroup of the symmetric group on \(G\), i.e., \(G \leq \text{Sym}(G)\).

In other words, every group is a group of symmetries (permutations) of some set (itself!). It embeds a group into the group of actions on a set—so we understand an abstract group by how it acts.

Key idea:
→ Understand an object by its action on other objects.

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2. Yoneda Lemma: Categories as Representable Functors

The Yoneda Lemma is the category-theoretic analog of Cayley’s theorem. It says:

For any (locally small) category \(\mathcal{C}\) and any object \(A \in \mathcal{C}\), the functor

\[h^A := \text{Hom}_{\mathcal{C}}(A, -): \mathcal{C} \to \mathbf{Set} \]

determines \(A\) up to isomorphism, and the natural transformations from \(h^A\) to any functor \(F\) are in bijection with the elements of \(F(A)\).

In other words, you can understand an object \(A\) by how it maps into other objects.

This is a vast generalization of Cayley’s idea:
→ An object is determined by its morphisms (its relations with other objects).对象的身份由它与其他对象的关系（态射）决定

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3. Grothendieck’s Relative Point of View

Grothendieck’s philosophy is that we should always understand objects relative to something else. For example:

• Instead of studying a scheme \(X\), study it relative to a base scheme \(S\): i.e., a morphism \(X \to S\).
• Instead of studying the absolute nature of a sheaf, study its behavior under pullback, i.e., how it varies over a base.
• Even in topos theory, objects are understood in terms of how they vary over a site.

This is sometimes called:

“Working fiberwise.”
Instead of studying things globally, study how they behave relative to a base, via morphisms.

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Connecting All Three: Philosophy of Externalization

These three ideas reflect the same shift:

Classical View	Modern View
Study an object internally	Study how it relates to other objects
Study absolute properties	Study relative/functorial properties
Identify an object by its structure	Identify an object by its morphisms

Principle	Expression
Cayley	A group is defined by its actions (representations).
Yoneda	An object is defined by its morphisms to other objects.
Grothendieck	An object is best understood by how it varies relative to a base.

They all push toward a relational, functorial, or relative understanding of mathematical objects. Instead of thinking of objects in isolation, we think in terms of contexts, maps, and interactions.

这些思想共同推动我们从研究“对象本身”转向研究“对象之间的结构关系”

Slogan Summary

• Cayley: Every group is a group of symmetries.
• Yoneda: Every object is determined by how it maps.
• Grothendieck: Every object should be studied relative to a base.

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References

1. Grothendieck's relative point of view and Yoneda lemma

2. What is the origin of the expression “Yoneda Lemma”?

3. Yoneda-Lemma as generalization of Cayley`s theorem?