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

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

\mathcal{C}

be any category based on sets, e.g. groups - sets of elements - or topological spaces - sets of 'points'. As usual, write

\mathrm{Ob}(\mathcal{C})

for the objects of

\mathcal{C}

and write

\mathrm{Fl}(\mathcal{C})

for the morphisms of the same. Consider the rule:

A

\mathrm{Ob}(\mathcal{C})\mapsto |A|=

the underlying set of

A,

u

\mathrm{Fl}(\mathcal{C})\mapsto |u|=

the morphism,

u

, as a map of sets.

The functor

|\;\;|

is then the forgetful functor from

\mathcal{C}

\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