Representable functor
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In category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.
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Definition
Let <math>\mathcal C<math> be an arbitrary category and the <math>\mathbf{Set}<math> be the category of sets. For each object <math>A<math> in <math>\mathcal C<math> we define a functor <math>\mathrm{Hom}_{\mathcal C}(A,\;\cdot\;):\mathcal C\rightarrow\mathbf{Set}<math> as follows:
- <math>\mathrm{Hom}_{\mathcal C}(A,\;\cdot\;)<math> maps each object <math>X<math> in <math>\mathcal C<math> to the set of morphisms <math>\mathrm{Hom}_{\mathcal C}(A,X)<math>
- <math>\mathrm{Hom}_{\mathcal C}(A,\;\cdot\;)<math> maps each morphism <math>f:X \rightarrow Y<math> to the function <math>\mathrm{Hom}_{\mathcal C}(A,X)\rightarrow\mathrm{Hom}_{\mathcal C}(A,Y)<math> given by <math>g\mapsto f\circ g<math>.
An arbitrary functor <math>F:\mathcal C\rightarrow\mathbf{Set}<math> is said to be 'represented by a pair', <math>(A,\phi)<math>, where <math>A<math> is an object of <math>\mathcal C<math> and <math>\phi<math> is in <math>F(A)<math>, if there is a natural isomorphism <math>\Phi:\mathrm{Hom}_{\mathcal C}(A,\;\cdot\;)\rightarrow F<math>, given by the consistent family of bijections <math>\Phi_X:\mathrm{Hom}_{\mathcal C}(A,X)\rightarrow F(X)<math>, such that
- <math>\Phi_X(u)=(Fu)(\phi)<math> for all <math>u<math> in <math>\mathrm{Hom}_{\mathcal C}(A,X)<math>.
It is also common in this case to say that <math>F<math> is 'representable'. Note that <math>\phi=\Phi_A(\mathrm{id}_A)<math>.
A dual set of definitions and statements apply to contravariant functors. Let <math>\mathcal C<math> be an arbitrary category. For each object <math>A<math> in <math>\mathcal C<math> we define a contravariant functor <math>\mathrm{Hom}_{\mathcal C}(\;\cdot\;,A):\mathcal C\rightarrow\mathbf{Set}<math> as follows:
- <math>\mathrm{Hom}_{\mathcal C}(\;\cdot\;,A)<math> maps each object <math>X<math> in <math>\mathcal C<math> to the set of morphisms <math>\mathrm{Hom}_{\mathcal C}(X, A)<math>
- <math>\mathrm{Hom}_{\mathcal C}(\;\cdot\;,A)<math> maps each morphism <math>f:X\rightarrow Y<math> to the function <math>\mathrm{Hom}_{\mathcal C}(Y,A) \rightarrow\mathrm{Hom}_{\mathcal C}(X,A)<math> given by <math>g\mapsto g\circ f<math>.
An arbitrary contravariant functor <math>G:\mathcal C\rightarrow\mathbf{Set}<math> is said to be represented by a pair, <math>(A,\phi)<math>, where <math>A<math> is an object in <math>\mathcal C<math> and <math>\phi<math> is in <math>G(A)<math>, if there is a natural isomorphism <math>\Phi:\mathrm{Hom}_{\mathcal C}(\;\cdot\;,A) \rightarrow G<math>, given by the consistent family of bijections <math>\Phi_X:\mathrm{Hom}_{\mathcal C}(X,A)\rightarrow G(X)<math>, such that
- <math>\Phi_X(u)=(Gu)(\phi)<math> for all <math>u<math> in <math>\mathrm{Hom}_{\mathcal C}(X,A)<math>.
Note again that <math>\phi=\Phi_A(\mathrm{id}_A)<math>.
Uniqueness
The representing pair <math>(A,\phi)<math> is unique in the following sense. If <math>(A_1,\phi_1)<math> and <math>(A_2,\phi_2)<math> represent the same functor, then there exists one and only one isomorphism from <math>A_1<math> to <math>A_2<math> so that <math>\phi_1<math> in <math>F(A_1)<math> maps to <math>\phi_2<math> in <math>F(A_2)<math>. This is because we have the isomorphisms <math>\Phi_1:\mathrm{Hom}_{\mathcal C}(A_1,\;\cdot\;)\rightarrow F<math> and <math>\Phi_2:\mathrm{Hom}_{\mathcal C}(A_2,\;\cdot\;)\rightarrow F<math> and so we have an isomorphism <math>\Phi_2^{-1}\circ\Phi_1:\mathrm{Hom}_{\mathcal C}(A_1,\;\cdot\;)\rightarrow\mathrm{Hom}_{\mathcal C}(A_2,\;\cdot\;)<math>. By the Yoneda lemma, <math>A_1<math> is isomorphic to <math>A_2<math> via the isomorphism determined by <math>\Phi_1<math> and <math>\Phi_2<math>, and this maps <math>\phi_1<math> to <math>\phi_2<math>. Uniqueness follows as everything is determined by <math>\phi_1<math> and <math>\phi_2<math>.
Examples
- Let <math>\mathcal C=\mathbf{Set}<math> and consider the contravariant functor <math>P:\mathbf{Set}\rightarrow\mathbf{Set}<math> given by <math>P(S)=<math> the power set of <math>S<math>, and if <math>\theta:S\rightarrow T<math> is a map of sets, the morphism <math>P(\theta):P(T)\rightarrow P(S)<math> is the map that sends every subset <math>V\subset T<math> to its inverse image, <math>\theta^{-1}(V)<math>, in <math>S<math>. To represent this functor we need a set, <math>Q<math>, and a <math>\phi<math> in <math>P(Q)<math>, that is, in some subset of <math>Q<math>, so that <math>\mathrm{Hom}_{\mathbf{Set}}(B,Q)<math> is isomorphic to <math>P(B)<math> via <math>\Phi_B(u)=P(u)(\phi)<math>.
Now, we know that <math>P(u):P(Q)\rightarrow P(B)<math> is the map that sends a subset, <math>S<math>, of <math>Q<math> to its inverse image, <math>u^{-1}(S)<math>, a subset of <math>B<math>. So, <math>P(u)(\phi)<math> is the inverse image of our chosen <math>\phi<math>. Take <math>Q=\{0,1\}<math> and <math>\phi=\{1\}<math>. Then subsets of <math>B<math> are exactly of the form <math>u^{-1}(1)<math> for the various <math>u<math> in <math>\mathrm{Hom}_{\mathbf{Set}}(B,Q)<math>, which are thus characteristic functions.
- Let <math>\mathcal C=\mathbf{Rng}<math>, the category of rings and let <math>F:\mathbf{Rng}\rightarrow\mathbf{Set}<math> be the forgetful functor. To represent this we need a ring, <math>P<math>, and an element <math>\phi<math> in <math>P<math>, so that for all rings <math>B<math>, <math>\mathrm{Hom}_{\mathbf{Rng}}(P,B)<math> is isomorphic to <math>|B|<math> via
- <math>(u<math> in <math>\mathrm{Hom}_{\mathbf{Rng}}(P,B))\mapsto(u(\phi)<math> in <math>|B|)<math>.
Take <math>P=\mathbb Z[T]<math>, the polynomial ring in one variable with integer coefficients, and <math>\phi=T<math>. Then any ring homomorphism <math>u<math> in <math>\mathrm{Hom}_{\mathbf{Rng}}(\mathbb Z[T],B)<math> is uniquely determined by <math>u(T)=b<math>, where any <math>b<math> in <math>|B|<math> can be used.
Representability and adjoints
The following result shows the relationship between representability of a functor and adjointness.
Proposition: A functor, <math>G:\mathcal C\rightarrow\mathcal D<math>, has a left adjoint if and only if, for every <math>A<math> in <math>\mathcal C<math>, the functor from <math>\mathcal D<math> to <math>\mathrm{Set}<math> mapping <math>B<math> to <math>\mathrm{Hom}_{\mathcal C}(A,G(B))<math> is representable. If <math>(F(A),\phi)<math> represents this functor then <math>F<math> is the object part of a left-adjoint of <math>G<math> for which the isomorphism <math>\Phi_B<math> is functorial in <math>B<math> and yields the adjointness.