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Representable Functor

In Category Theory, a representable functor is a functor which is naturally isomorphic to the Hom-Set functor from a particular element.

More explicitly, let \(C\) be locally small category and \(F\) be a set-valued functor on \(C\). A representation for \(F\) is a choice of \(c \in C\) together with a natural isomorphism \(C(c, -) \cong F\). If such a natural isomorphism exists, then \(F\) is said to be representable, and \(c\) is said to represent \(F\).

Examples of Representable Functors

  1. Let \(\star: C \to \text{Set}\) be the constant functor, which maps every object in \(C\) to the singleton set \(\{1\}\). \(C\) has an initial object if and only if \(\star\) is representable, and a terminal object if and only if its contravariant twin is representable.
  2. The identity functor on the category of sets is represented by any singleton set. This is to say that for any set \(X\), there is a natural isomorphism \(\text{Set}(\{1\}, X) \cong X\), which defines a bijection between elements \(x \in X\) and functions \(x: 1 \to X\), taking the singleton element of \(x\).
  3. The forgetful functor \(U: \text{Grp} \to \text{Set}\) is represented by the group \(\mathbb{Z}\). That is to say, for any group \(G\), there is a natural transformation \(\text{Grp}(\mathbb{Z}, G) \cong UG\) which associates every element \(g \in UG\) to the homomorphism \(\mathbb{Z} \to G\) which maps \(1\) to \(G\).
  4. Likewise, the forgetful functor \(U: \text{Top} \to \text{Set}\) is represented by the singleton space, and the functor \(\text{ob}: \text{Cat} \to \text{Set}\) which takes categories to the set of their elements is represented by the terminal category \(1\), which has one element.
  5. The Sierpinski Space \(S\) represents the open set functor. The natural bijection associates continuous functions \(X \to S\) with the preimages open sets.
  6. The functor \(\text{Hom}(- \times A, B): \text{Set}^{\text{op}} \to \text{Set}\) which sends a set \(X\) to the set of functions \(X \times A \to B\) is represented by the set \(B^A\) of functions from \(A\) to \(B\). I.e. there is a natural bijection between functions \(X \times A \to B\) and functions \(X \to B^A\). Computer scientists call this currying.

The Yoneda Lemma

One of the most prevalent uses of the idea of a representable functor is in the Yoneda Lemma, which states that a set-valued functor \(F\) on a locally-small category \(C\) obeys $$ \text{Hom}(C(c, -), F) \cong Fc $$ for all \(c \in C\).

Initial Elements, Terminal Elements, and the Category of Elements

A covariant set-valued functor is representable if and only if its category of elements has an initial object, and a contravariant set-valued functor (a pre-sheaf) is representable if and only if its category of elements has a terminal object.