• Yoneda lemma states that an object is fully determined by its hom‑functor relationships. • Yoneda embedding maps each object X to functor hom(-,X), fully faithful. • Morphisms f: X→Y correspond bijectively to natural transformations hom(-,X)→hom(-,Y). • Each natural transformation uniquely arises from a single morphism, and vice versa. • This bijection extends beyond representable functors to all functors in the category. • The lemma provides a powerful tool for translating categorical problems into set‑theoretic ones.
Article Summaries:
- Summary
The article explains the Yoneda Lemma, a cornerstone of category theory. It begins by recalling that the Yoneda embedding sends an object (X) to the hom‑functor (\hom(-,X)), and that this embedding is fully faithful: morphisms (f:X\to Y) correspond bijectively to natural transformations (\hom(-,X)\to\hom(-,Y)). The main development is the general statement that for any functor (F:\mathcal{C}^{op}\to\mathbf{Set}) and any object (X), natural transformations (\hom(-,X)\to F) are in one‑to‑one correspondence with elements of (F(X)). The article outlines the construction of this bijection and emphasizes its significance: all natural transformations from a representable functor arise from evaluating (F) at the representing object.
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