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# equivalent definition of a representable functor

We provide an equivalent, motivating, way of defining a representable functor.

Let $\mathcal{C}$ be a category and $F:\mathcal{C}\rightarrow Set$ be a covariant functor and $A\in\mathcal{C}$. Then the following are equivalent

1. $\mathcal{C}(A,-)$ is naturally isomorphic to $F$ (or, isomorphic in the appropriate category of functors)

2. There exists an element $i\in F(A)$ such that for every $B\in\mathcal{C},r\in F(B)$ there exists a unique $f\in\mathcal{C}(A,B)$ such that $F(f)(i)=r$

To illustrate the significance of this, consider the category $\mathcal{C}=\bf{Vect}_{k}$ of vector spaces over a field $k$. For arbitrary vector spaces $V,W$ consider the functor $F:\mathcal{C}\rightarrow Set$ determined by

$F(U)=\operatorname{Bilin}(V\times W,U)$ |

Where this denotes the set of maps which are linear in both entries. This is a covariant functor in the obvious way. Then one may define $V\otimes W$ as the object which represents $F$ (if it exists). The significance of the result is it shows this is equivalent to the ’usual’ definition: there is a bilinear map $i:V\times W\rightarrow V\otimes W$ through which all bilinear maps from $V\times W$ (these are quantified by r in the theorem) factor uniquely. This is because $r:V\times W\rightarrow U$ factors through $i$ exactly when there is an $f\in\mathcal{C}(V\otimes W,U)$ such that $F(f)(i)=r$.

Such universal constructions can be shown to be functorial in the basic objects. For instance the tensor product may be shown to be a functor

${\bf{Vect}_{k}}\times{\bf{Vect}_{k}}\rightarrow{\bf{Vect}_{k}}$ |

To generalise this suppose that $\mathcal{D}$ is a category (roughly representing ${\bf{Vect}_{k}}\times{\bf{Vect}_{k}}$ in our case) and we have a functor

$F:\mathcal{D}^{{op}}\times\mathcal{C}\rightarrow Set$ |

such that $F(d,-):\mathcal{C}\rightarrow Set$ is isomorphic to $\mathcal{C}(G(d),-)$ for some object $G(d)$. Then one may show that $G$ extends to a functor in such a way that $F(-,-)$ is naturally isomorphic to $\mathcal{C}(G(-),-)$.

We may show further that if $F,F^{\prime}$ are isomorphic functors and $G,G^{\prime}$ are functors which represent them respectively, then there is a natural isomorphism between $G$ and $G^{\prime}$.

## Mathematics Subject Classification

18-00*no label found*

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