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

$\displaystyle A$ in $\displaystyle \operatorname{Ob}(\mathcal{C})\mapsto |A|=$ the underlying set of $\displaystyle A,$
$\displaystyle u$ in $\displaystyle \operatorname{Fl}(\mathcal{C})\mapsto |u|=$ the morphism, $\displaystyle u$ , as a map of sets.

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