In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output.
If the signature is left as an empty list, the functor is simply to take the underlying 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 the category of commutative rings.
A (unital) ring, described in the language of universal algebra, is an ordered tuple
and 1 yields a functor to the category of abelian groups, which assigns to each ring
Deleting all the operations gives the functor to the underlying set
There is a functor from the category CRing to Ring 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, a commonly considered forgetful functor is as follows.
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 functor Ab → Grp.
One of the second kind is the forgetful functor Ab → Set.
Under the forgetful functor, this morphism yields the identity.
Note that an object in Mod is a tuple, which includes a ring and an abelian group, so which to forget is a matter of taste.
Forgetful functors tend to have left adjoints, which are 'free' constructions.
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: The unit of the free–forgetful adjunction is the "inclusion of a basis":
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.