Editing Forgetful functor (section)

== Left adjoints of forgetful functors ==
Forgetful functors tend to have [[left adjoint]]s, which are '[[free object|free]]' constructions. For example:
* [[free module]]: the forgetful functor from <math>\mathbf{Mod}(R)</math> (the category of <math>R</math>-[[module (mathematics)|modules]]) to <math>\mathbf{Set}</math> has left adjoint <math>\operatorname{Free}_R</math>, with <math>X\mapsto \operatorname{Free}_R(X)</math>, the free <math>R</math>-module with [[Basis (linear algebra)|basis]] <math>X</math>.
* [[free group]]
* [[free lattice]]
* [[tensor algebra]]
* [[free category]], adjoint to the forgetful functor from categories to [[quiver (mathematics)|quivers]]
* [[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 set ''X'' and an object (say, an ''R''-module) ''M'', maps ''of sets'' <math>X \to |M|</math> correspond to maps of modules <math>\operatorname{Free}_R(X) \to M</math>: 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:
:<math>\operatorname{Hom}_{\mathbf{Mod}_R}(\operatorname{Free}_R(X),M) = \operatorname{Hom}_{\mathbf{Set}}(X,\operatorname{Forget}(M)).</math>

The [[Free_object#Free_functor|unit of the free–forgetful adjunction]] is the "inclusion of a basis": <math>X \to \operatorname{Free}_R(X)</math>.

'''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.