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[parent] equivalent definition of a representable functor (Result)

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

$\displaystyle 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

$\displaystyle {\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

$\displaystyle 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$.



"equivalent definition of a representable functor" is owned by CWoo. [ full author list (2) | owner history (1) ]
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Cross-references: natural isomorphism, tensor product, universal, factor, bilinear map, represents, object, obvious, maps, field, vector spaces, category of functors, isomorphic, the following are equivalent, covariant functor, category, representable functor, equivalent
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This is version 6 of equivalent definition of a representable functor, born on 2006-03-30, modified 2007-08-03.
Object id is 7788, canonical name is EquivalentDefinitionOfARepresentableFunctor.
Accessed 1036 times total.

Classification:
AMS MSC: 18-00 (Category theory; homological algebra :: General reference works )
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