202505091505

Tags : Category Theory

Representable Functors Define Representing Objects

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We will not talk about a property being represented by a functor being universal, to do we get the following result using the Yoneda Lemma

Theorem

Consider a pair of object $x$ and $y$ in a locally small category:

• If the functors represented by $x$ and $y$ are isomorphic, then $x$ and $y$ are isomorphic.

The fully faithful embeddings $C\hookrightarrow {\text{Set}}^{C^{\text{op}}}$ and the other way round create isomorphisms.

$$C(x,-)\simeq F_x\simeq F_y\simeq C(y,-)$$

This by Yoneda Embedding we get $x\simeq y$. If the functors were contravariant, then a similar argument would have worked with $C(-,x)$ instead. This also holds true if $x$ and $y$ represent the same functor.

this also shows that a Representable Functors, defines its representing object object.

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References

• Yoneda Lemma
• Yoneda Embedding
• Representable Functors