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 R the underlying additive abelian group of 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 
be any category based on sets, e.g. groups - sets of elements - or topological spaces - sets of 'points'. As usual, write 
for the objects of and write 
for the morphisms of the same. Consider the rule:
- A in
the underlying set of A,
- u in
the morphism, u, as a map of sets.
The functor 
is then the forgetful functor from to 
, 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 
(the category of R-module) to has left adjoint F, with 
, the free R-module with basis X. For a more extensive list, see [Mac Lane].
References
- [Mac Lane] Categories for the Working Mathematician, Saunders Mac Lane, Springer Graduate Texts in Mathematics 5, 1997.
