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'representable functor'
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| Title of object: |
representable functor |
| Canonical Name: |
RepresentableFunctor |
| Type: |
Definition |
| Created on: |
2001-12-12 01:15:50 |
| Modified on: |
2003-09-06 03:42:54 |
| Classification: |
msc:18-00 |
| Keywords: |
category, functor, natural equivalence |
| Defines: |
represents |
Revision comment (for changes between this and next version):
| changes for correction #3761 |
Preamble:
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Content:
A contravariant functor $T : C \to {\bf Sets}$ between a category and the category of sets is {\em representable} if there is an object $X$ of $C$ such that $T$ is isomorphic to the functor $X^\bullet = {\rm Hom}(-,X)$.
Similarly, a covariant functor is $T$ called {\em representable} if it is isomorphic to $X_\bullet ={\rm Hom}(X,-)$.
We say that the object $X$ {\em represents} $T$. $X$ is unique up to canonical isomorphism.
A vast number of important objects in mathematics are defined as representing functors. For example, if $F:C\to D$ is any functor, then the adjoint $G:D\to C$ (if it exists) can be defined as follows. For $Y$ in $D$, $G(Y)$ is the object of $C$ representing the functor $X\mapsto {\rm Hom}(F(X),Y)$ if $G$ is right adjoint to $F$ or $X\mapsto {\rm Hom}(Y,F(X))$ if $G$ is left adjoint.
Thus, for example, if $R$ is a ring, then $N\otimes M$ represents the functor $L\mapsto {\rm Hom}_R(N,{\rm Hom}_R(M,L))$. |
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