Inmathematics, more specifically in the area ofcategory theory, aforgetful functor (also known as astripping functor) "forgets" or drops some or all of the input'sstructure orpropertiesbeforemapping to the output. For analgebraic structure of a givensignature, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, thefunctor is simply to take theunderlying set of a structure. Because many structures in mathematics consist of a set with an additional added structure, a forgetful functor that maps to the underlying set is the most common case.

As an example, there are several forgetful functors from thecategory of commutative rings. A (unital)ring, described in the language ofuniversal algebra, is an ordered tuple${\displaystyle (R,+,\times ,a,0,1)}$  satisfying certain axioms, where${\displaystyle +}$  and${\displaystyle \times }$  are binary functions on the set${\displaystyle R}$ ,${\displaystyle a}$  is a unary operation corresponding to additive inverse, and 0 and 1 are nullary operations giving the identities of the two binary operations. Deleting the 1 gives a forgetful functor to the category ofrings without unit; it simply "forgets" the unit. Deleting${\displaystyle \times }$  and 1 yields a functor to the category ofabelian groups, which assigns to each ring${\displaystyle R}$  the underlying additive abelian group of${\displaystyle R}$ . To eachmorphism of rings is assigned the samefunction considered merely as a morphism of addition between the underlying groups. Deleting all the operations gives the functor to the underlying set${\displaystyle R}$ .

It is beneficial to distinguish between forgetful functors that "forget structure" versus those that "forget properties". For example, in the above example of commutative rings, in addition to those functors that delete some of the operations, there are functors that forget some of the axioms. There is a functor from the categoryCRing toRing that forgets the axiom of commutativity, but keeps all the operations. Occasionally the object may include extra sets not defined strictly in terms of the underlying set (in this case, which part to consider the underlying set is a matter of taste, though this is rarely ambiguous in practice). For these objects, there are forgetful functors that forget the extra sets that are more general.

Most common objects studied in mathematics are constructed as underlying sets along with extra sets of structure on those sets (operations on the underlying set, privileged subsets of the underlying set, etc.) which may satisfy some axioms. For these objects, a commonly considered forgetful functor is as follows.Let${\displaystyle \mathcal{C}}$  be any category based onsets, e.g.groups—sets of elements—ortopological spaces—sets of 'points'. As usual, write${\displaystyle \mathrm{Ob}(\mathcal{C})}$  for theobjects of${\displaystyle \mathcal{C}}$  and write${\displaystyle \mathrm{Fl}(\mathcal{C})}$  for the morphisms of the same. Consider the rule:

For all${\displaystyle A}$  in${\displaystyle \mathrm{Ob}(\mathcal{C}),A\mapsto |A|=}$  the underlying set of${\displaystyle A,}$ 

For all${\displaystyle u}$  in${\displaystyle \mathrm{Fl}(\mathcal{C}),u\mapsto |u|=}$  the morphism,${\displaystyle u}$ , as a map of sets.

The functor${\displaystyle |\cdot |}$  is then the forgetful functor from${\displaystyle \mathcal{C}}$  toSet, thecategory of sets.

Forgetful functors are almost alwaysfaithful.Concrete categories have forgetful functors to the category of sets—indeed they may bedefined as those categories that admit a faithful functor to that category.

Forgetful functors that only forget axioms are alwaysfully faithful, since every morphism that respects the structure between objects that satisfy the axioms automatically also respects the axioms. Forgetful functors that forget structures need not be full; some morphisms don't respect the structure. These functors are still faithful however because distinct morphisms that do respect the structure are still distinct when the structure is forgotten. Functors that forget the extra sets need not be faithful, since distinct morphisms respecting the structure of those extra sets may be indistinguishable on the underlying set.

In the language of formal logic, a functor of the first kind removes axioms, a functor of the second kind removes predicates, and a functor of the third kind remove types^{[clarification needed]}. An example of the first kind is the forgetful functorAb →Grp. One of the second kind is the forgetful functorAb →Set. A functor of the third kind is the functorMod →Ab, whereMod is thefibred category of all modules over arbitrary rings. To see this, just choose a ring homomorphism between the underlying rings that does not change the ring action. Under the forgetful functor, this morphism yields the identity. Note that an object inMod is a tuple, which includes a ring and an abelian group, so which to forget is a matter of taste.

Forgetful functors tend to haveleft adjoints, which are 'free' constructions. For example:

• free module: the forgetful functor from${\displaystyle \mathbf{M}\mathbf{o}\mathbf{d}(R)}$  (the category of${\displaystyle R}$ -modules) to${\displaystyle \mathbf{S}\mathbf{e}\mathbf{t}}$  has left adjoint${\displaystyle {\mathrm{Free}}_R}$ , with${\displaystyle X\mapsto {\mathrm{Free}}_R(X)}$ , the free${\displaystyle R}$ -module withbasis${\displaystyle X}$ .
• free group
• free lattice
• tensor algebra
• free category, adjoint to the forgetful functor from categories toquivers
• universal enveloping algebra

For a more extensive list, see (Mac Lane 1997).

As this is a fundamental example of adjoints, we spell it out:adjointness means that given a setX and an object (say, anR-module)M, mapsof sets${\displaystyle X\to |M|}$  correspond to maps of modules${\displaystyle {\mathrm{Free}}_R(X)\to M}$ : every map of sets yields a map of modules, and every map of modules comes from a map of sets.

In the case of vector spaces, this is summarized as:"A map between vector spaces is determined by where it sends a basis, and a basis can be mapped to anything."

Symbolically:

${\displaystyle {\mathrm{Hom}}_{{\mathbf{M}\mathbf{o}\mathbf{d}}_R}({\mathrm{Free}}_R(X),M)={\mathrm{Hom}}_{\mathbf{S}\mathbf{e}\mathbf{t}}(X,\mathrm{Forget}(M)).}$ 

Theunit of the free–forgetful adjunction is the "inclusion of a basis":${\displaystyle X\to {\mathrm{Free}}_R(X)}$ .

Fld, the category of fields, furnishes an example of a forgetful functor with no adjoint. There is no field satisfying a free universal property for a given set.

• Adjoint functors
• Functors
• Projection (set theory)

• Forgetful functor at thenLab