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What about proper co/counter variancy of the Yoneda embedding?

$\operatorname{Hom}(C,-)$ or $\operatorname{Hom}(-,C)$?

The Wikipedia seems to say something different than my books.

Please explain it in details, stressing why it behaves this way.

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"Contravariance", not "counter variancy". Neither of the functors you name is the Yoneda embedding. The covariant Yoneda embedding is the functor $C \mapsto \textrm{Hom}(-, C)$. The contravariant Yoneda embedding is the functor $C \mapsto \textrm{Hom}(C, -)$. –  Zhen Lin Jun 15 '12 at 17:56
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I don't understand the question. –  Qiaochu Yuan Jun 15 '12 at 22:49

1 Answer 1

up vote 4 down vote accepted

In more detail. Let $\mathcal{C}$ be a locally small category: then for each object $c$ of $\mathcal{C}$, there is a representable presheaf $H_c = \mathcal{C}(-, c) : \mathcal{C}^\textrm{op} \to \textbf{Set}$ and a representable copresheaf $H^c = \mathcal{C}(c, -) : \mathcal{C} \to \textbf{Set}$. These extend to two different functors: $H_\bullet : \mathcal{C} \to [\mathcal{C}^\textrm{op}, \textbf{Set}]$ and $H^\bullet : \mathcal{C}^\textrm{op} \to [\mathcal{C}, \textbf{Set}]$. (What's really happening here is we are currying the bifunctor $\mathcal{C}(-, -) : \mathcal{C}^\textrm{op} \times \mathcal{C} \to \textbf{Set}$ in two different ways.)

The Yoneda lemma says that, for each presheaf $F : \mathcal{C}^\textrm{op} \to \textbf{Set}$, there is a bijection $$\textrm{Nat}(H_c, F) \cong F(c)$$ that is natural in both $c$ and $F$. By duality, for each copresheaf $G : \mathcal{C} \to \textbf{Set}$, there is a bijection $$\textrm{Nat}(H^c, G) \cong G(c)$$ that is natural in both $c$ and $G$. This immediately implies that $H_\bullet$ and $H^\bullet$ are fully faithful functors.


Now, allow me to evangelise a little about which of $H_\bullet$ and $H^\bullet$ is the "true" Yoneda embedding. In my opinion, it is the "covariant" Yoneda embedding $H_\bullet : \mathcal{C} \to [\mathcal{C}^\textrm{op}, \textbf{Set}]$ that is more fundamental. As I explain here, when $\mathcal{C}$ is small, the embedding $H_\bullet : \mathcal{C} \to [\mathcal{C}^\textrm{op}, \textbf{Set}]$ is the universal functor making $[\mathcal{C}^\textrm{op}, \textbf{Set}]$ the free cocompletion of $\mathcal{C}$. Although one could argue that $H^\bullet : \mathcal{C} \to [\mathcal{C}, \textbf{Set}]^\textrm{op}$ has a similar universal property, the fact of the matter is that $[\mathcal{C}^\textrm{op}, \textbf{Set}]$ has many of the same properties that $\textbf{Set}$ has (in the sense of being a topos), whereas $[\mathcal{C}, \textbf{Set}]^\textrm{op}$ is a rather less familiar category, in terms of intuition. (For example, what is $\textbf{Set}^\textrm{op}$? It is the category of complete atomic boolean algebras and continuous homomorphisms, but I'd like to think that we understand $\textbf{Set}$ better than $\textbf{Set}^\textrm{op}$.)

Moreover, it is more common for a category to have "wrong" or "missing" colimits (say, for example, the category $\textbf{Aff} = \textbf{CRing}^\textrm{op}$) than it is for a category to have missing limits. By passing to the category $[\mathcal{C}^\textrm{op}, \textbf{Set}]$ via the covariant Yoneda embedding, we get the opportunity to change the "wrong" colimits to something more desirable; whereas passing to the category $[\mathcal{C}, \textbf{Set}]^\textrm{op}$ via the contravariant Yoneda embedding preserves all colimits and instead destroys the limits which were already "correct"!

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Sorry for a stupid question: You define but not use $H_\bullet$ and $H^\bullet$. Why do you defined these? –  porton Jun 16 '12 at 11:25

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