# Forgetful functor

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 [itex]R[itex] the underlying additive abelian group of [itex]R[itex]. 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 [itex]\mathcal{C}[itex] be any category based on sets, e.g. groups - sets of elements - or topological spaces - sets of 'points'. As usual, write [itex]\mathrm{Ob}(\mathcal{C})[itex] for the objects of [itex]\mathcal{C}[itex] and write [itex]\mathrm{Fl}(\mathcal{C})[itex] for the morphisms of the same. Consider the rule:

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

The functor [itex]|\;\;|[itex] is then the forgetful functor from [itex]\mathcal{C}[itex] to [itex]\mathbf{Set}[itex], 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 [itex]\mathbf{Mod}(R)[itex] (the category of [itex]R[itex]-module) to [itex]\mathbf{Set}[itex] has left adjoint [itex]F[itex], with [itex]X\mapsto F(X)[itex], the free [itex]R[itex]-module with basis [itex]X[itex]. 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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