Yoneda lemma

米田引理

Category Theory · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For small $C$, presheaf $P$, and $A\in C$, evaluation at $1_A$ gives $\operatorname{Nat}(\operatorname{Hom}(-,A),P)\cong P(A)$ naturally.

对于小范畴 C、预层 P 和对象 A，在 1_A 处取值给出 Nat(Hom(-,A),P) 与 P(A) 的自然双射。

Example

For the poset $0<1<2$ and constant presheaf $\{0,1\}$, exactly two transformations $h_1\to P$ exist.

Usage and distinctions

The direction of the hom-functor must match the variance of P.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Tom Leinster — Basic Category Theory (author arXiv edition)§4.2, Theorem 4.2.1; §4.3, Corollary 4.3.7 · Checked 2026-10-14

Translation sources

1. 李文威 — 代数学方法，第一卷（作者源码）第2章，「可表函子」，定理 prop:Yoneda-lemma 与「米田引理」段 · Checked 2026-10-14

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