Forgetful Functor - Left Adjoint: Free

Left Adjoint: Free

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

• free module: the forgetful functor from (the category of -module) to has left adjoint, with, the free -module with basis .
• free group
• free lattice
• tensor algebra
• free category, adjoint to the forgetful functor from categories to quivers

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 set X and an object (say, an R-module) M, maps of sets correspond to maps of modules : 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:

The counit of the free-forget 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.