Forgetful functor

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A forgetful functor is a type of functor in mathematics. The nomenclature is suggestive of such a functor's behaviour: given some algebraic object as input, some or all of the object's structure is 'forgotten' in the output. For an algebraic structure of a given signature, this may be expressed by curtailing the signature in some way: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the underlying set of a structure; this is in fact the most common case.

For example, the forgetful functor from the category of rings to the category of abelian groups assigns to each ring <math>R<math> the underlying additive abelian group of <math>R<math>. To each morphism of rings is assigned the same function considered merely as a morphism of addition between the underlying groups.

A common subclass of forgetful functors is as follows. Let <math>\mathcal{C}<math> be any category based on sets, e.g. groups - sets of elements - or topological spaces - sets of 'points'. As usual, write <math>\mathrm{Ob}(\mathcal{C})<math> for the objects of <math>\mathcal{C}<math> and write <math>\mathrm{Fl}(\mathcal{C})<math> for the morphisms of the same. Consider the rule:

<math>A<math> in <math>\mathrm{Ob}(\mathcal{C})\mapsto |A|=<math> the underlying set of <math>A,<math>
<math>u<math> in <math>\mathrm{Fl}(\mathcal{C})\mapsto |u|=<math> the morphism, <math>u<math>, as a map of sets.

The functor <math>|\;\;|<math> is then the forgetful functor from <math>\mathcal{C}<math> to <math>\mathbf{Set}<math>, the category of sets.

Forgetful functors are always faithful. Concrete categories have forgetful functors to the category of sets -- indeed they may be defined as those categories which admit a faithful functor to that category.

Forgetful functors tend to have left adjoints which are 'free' constructions. For example, the forgetful functor from <math>\mathbf{Mod}(R)<math> (the category of <math>R<math>-module) to <math>\mathbf{Set}<math> has left adjoint <math>F<math>, with <math>X\mapsto F(X)<math>, the free <math>R<math>-module with basis <math>X<math>. For a more extensive list, see [Mac Lane].

References

  • [Mac Lane] Saunders Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer-Verlag Berlin Heidelberg New York 1997. ISBN 0387984038
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