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.
Read more about this topic: Forgetful Functor
Famous quotes containing the words left and/or free:
“He was a lucky fox that left his tail in the trap. The muskrat will gnaw his third leg off to be free. No wonder man has lost his elasticity.”
—Henry David Thoreau (1817–1862)
“Will women find themselves in the same position they have always been? Or do we see liberation as solving the conditions of women in our society?... If we continue to shy away from this problem we will not be able to solve it after independence. But if we can say that our first priority is the emancipation of women, we will become free as members of an oppressed community.”
—Ruth Mompati (b. 1925)