Home > Forgetful functor

For example, the forgetful functor from the category of rings to the category of abelian groups assigns to each ring the underlying additive abelian group of . 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:

in the underlying set of

in the morphism, , as a map of sets.

The functor is then the forgetful functor from to , the category of sets.

Forgetful functors are always faithful. Concrete categoriesIn mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of function 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 ' freeThe idea of a free object in mathematics is one of the basics of abstract algebra. It is part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations); but on the other hand it has a clean formulati' constructions. For example, the forgetful functor from (the category of - moduleAlgebra In abstract algebra, a module is a generalization of a vector space. In a vector space the set of scalars forms a field whereas in a module the scalars just form a ring. Much of the theory of modules consists of recovering desirable properties of) to has left adjoint , with , the free -module with basisIn mathematics, a basis or set of generators is a collection of objects that can be systematically combined to produce a larger collection of objects. See: Basis (linear algebra) Basis (topology) Greedoid (basis as a maximal feasible set) In economics or . For a more extensive list, see [Mac Lane].

References

• [Mac Lane] Categories for the Working Mathematician, Saunders Mac LaneSaunders Mac Lane (born 4 August 1909) is a US mathematician. He is one of the founders of category theory. Mac Lane was born in Taftville, Connecticut. He studied at the Yale University from 1926 to 1930 before going to the University of Chicago on a one, Springer Graduate Texts in Mathematics 5, 1997.