Wikipedia defines that a functor $F$ from a locally small category $\mathcal{C}$ to $\mathrm{Set}$ is representable if it is isomorphic to some $\mathrm{Hom}$ functor. On the other hand, nLab defines that a representation of a category $\mathcal{C}$ in a category $\mathcal{D}$ is just a functor $F \colon \mathcal{C} \to \mathcal{D}$. I would expect that a representation of $\mathcal{C}$ in $\mathrm{Set}$ is a representable functor, but this does not seem to be necessary. Would we then call it a "representable representation"? This seems a bit redundant.

Is there any relationship between a representation of a category and a representable functor? Is it just by accident that their names are so similar?

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    $\begingroup$ My guess is the two are unrelated notions. $\endgroup$ – Dustan Levenstein Mar 16 '18 at 12:20

Yes, it's mostly an accident I think.

A functor is representable when it can be represented by some object $X$, i.e. $F = \hom(X,-)$ doesn't contain more information than the object $X$ by the Yoneda lemma, so what you know about $X$ you know about $F$ and conversely. The object $X$ is a "representative" of $F$.

On the other hand, the terminology "representation of a category" comes from group theory, where a representation of a group $G$ is a morphism $G \to \operatorname{End}(V)$ for some vector space $V$. If you view $G$ as a category with one object, then a representation of $G$ is a functor $G \to \mathsf{Vect}$ from $G$ to the category of vector spaces ($V$ is the image of the unique object). Then, nLab being nLab, the most possible general definition is given: you can have representations that are not vector spaces, so the target category is any category $\mathcal{D}$; and you can consider representations of something other than a groupoid with one object (i.e. a group), so the source category can be any category $\mathcal{C}$. Then all you're left with is a functor. The only interest of this definition is the perspective shift, when you're working with some categories for which it makes sense to speak about "representations" rather than "functors".

At this level of generality, it could make sense to wonder when a "representation" (= a functor) is "representable". But if you think back to the original motivation, there aren't many representable functor $G \to \mathsf{Vect}$...

  • $\begingroup$ Thank you for your answer. As for the last comment / exercise: if I am not wrong, a representation $F(*)=V$ of a group $G$ on $\mathrm{Vect}$ is representable iff $V$ is in bijection with $\mathrm{hom}(*,*) \cong G$ in such a way that multiplication by $g$ corresponds to applying the map $F(g)$. In particular, if $G$ is finite and non-trivial and we are considering vector spaces over a field of characteristic $0$, then no representation is representable. Is this correct? $\endgroup$ – 57Jimmy Mar 16 '18 at 13:42

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