Free object

Summary

A free object on a set $X$ is an object generated by $X$ with no relations other than those forced by the axioms. It is defined by a universal property: every map from $X$ into the underlying set of an object extends uniquely to a morphism. Free groups, free modules and free monoids are the main examples. When free objects exist on every set, they form a left adjoint of the forgetful functor.