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jmc
jmc
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Let $C$ be a category, and $X$ an object in $C$. Then $h_{X} = \mathrm{Hom}(X, -)$$h_{X} = \textrm{Hom}(X, \_)$ is a functor from $C$ to $\mathrm{Set}$$\textrm{Set}$.

If for two objects $X,Y$ in $C$ the functors $h_{X}$ and $h_{Y}$ are naturally isomorphic, then so are $X$ and $Y$. This is called the Yoneda lemma.

In fancy categorical terms it says that the functor $h \colon C^{\textrm{opp}} \to \mathrm{Func}(C, \mathrm{Set})$$h \colon C^{\textrm{opp}} \to \textrm{Func}(C, \textrm{Set})$ is fully faithful. The slogan is "tell me who your friends are, and I will tell you who you are". So if you know all the arrows from (or to) an object, then you can determine the object up to isomorphism.

Let $C$ be a category, and $X$ an object in $C$. Then $h_{X} = \mathrm{Hom}(X, -)$ is a functor from $C$ to $\mathrm{Set}$.

If for two objects $X,Y$ in $C$ the functors $h_{X}$ and $h_{Y}$ are naturally isomorphic, then so are $X$ and $Y$. This is called the Yoneda lemma.

In fancy categorical terms it says that the functor $h \colon C^{\textrm{opp}} \to \mathrm{Func}(C, \mathrm{Set})$ is fully faithful. The slogan is "tell me who your friends are, and I will tell you who you are". So if you know all the arrows from (or to) an object, then you can determine the object up to isomorphism.

Let $C$ be a category, and $X$ an object in $C$. Then $h_{X} = \textrm{Hom}(X, \_)$ is a functor from $C$ to $\textrm{Set}$.

If for two objects $X,Y$ in $C$ the functors $h_{X}$ and $h_{Y}$ are naturally isomorphic, then so are $X$ and $Y$. This is called the Yoneda lemma.

In fancy categorical terms it says that the functor $h \colon C^{\textrm{opp}} \to \textrm{Func}(C, \textrm{Set})$ is fully faithful. The slogan is "tell me who your friends are, and I will tell you who you are". So if you know all the arrows from (or to) an object, then you can determine the object up to isomorphism.

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David White
David White
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Let $C$ be a category, and $X$ an object in $C$. Then $h_{X} = \mathrm{Hom}(X, \_)$$h_{X} = \mathrm{Hom}(X, -)$ is a functor from $C$ to $\mathrm{Set}$.

If for two objects $X,Y$ in $C$ the functors $h_{X}$ and $h_{Y}$ are naturally isomorphic, then so are $X$ and $Y$. This is called the Yoneda lemma.

In fancy categorical terms it says that the functor $h \colon C^{\textrm{opp}} \to \mathrm{Func}(C, \mathrm{Set})$ is fully faithful. The slogan is "tell me who your friends are, and I will tell you who you are". So if you know all the arrows from (or to) an object, then you can determine the object up to isomorphism.

Let $C$ be a category, and $X$ an object in $C$. Then $h_{X} = \mathrm{Hom}(X, \_)$ is a functor from $C$ to $\mathrm{Set}$.

If for two objects $X,Y$ in $C$ the functors $h_{X}$ and $h_{Y}$ are naturally isomorphic, then so are $X$ and $Y$. This is called the Yoneda lemma.

In fancy categorical terms it says that the functor $h \colon C^{\textrm{opp}} \to \mathrm{Func}(C, \mathrm{Set})$ is fully faithful. The slogan is "tell me who your friends are, and I will tell you who you are". So if you know all the arrows from (or to) an object, then you can determine the object up to isomorphism.

Let $C$ be a category, and $X$ an object in $C$. Then $h_{X} = \mathrm{Hom}(X, -)$ is a functor from $C$ to $\mathrm{Set}$.

If for two objects $X,Y$ in $C$ the functors $h_{X}$ and $h_{Y}$ are naturally isomorphic, then so are $X$ and $Y$. This is called the Yoneda lemma.

In fancy categorical terms it says that the functor $h \colon C^{\textrm{opp}} \to \mathrm{Func}(C, \mathrm{Set})$ is fully faithful. The slogan is "tell me who your friends are, and I will tell you who you are". So if you know all the arrows from (or to) an object, then you can determine the object up to isomorphism.

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jmc
jmc
  • 5.5k
  • 27
  • 60

Let $C$ be a category, and $X$ an object in $C$. Then $h_{X} = \mathrm{Hom}(X, \_)$ is a functor from $C$ to $\mathrm{Set}$.

If for two objects $X,Y$ in $C$ the functors $h_{X}$ and $h_{Y}$ are naturally isomorphic, then so are $X$ and $Y$. This is called the Yoneda lemma.

In fancy categorical terms it says that the functor $h \colon C^{\textrm{opp}} \to \mathrm{Func}(C, \mathrm{Set})$ is fully faithful. The slogan is "tell me who your friends are, and I will tell you who you are". So if you know all the arrows from (or to) an object, then you can determine the object up to isomorphism.