nLab
free functor

Idea

Informally, a free functor is a left adjoint to a forgetful functor. (This is informal because the concept of forgetful functor is informal; any functor might be viewed as forgetful, so any left adjoint might be viewed as free, while in practice only some are.)

Examples

• the free monoid functor

• the free module functor

One formal sort of free functor is the left adjoint $C\to C^T$, where $T$ is a monad on the category $C$ and $C^T$ is its Eilenberg-Moore category (the category of $T$-algebras). This includes:

• The “free group” functor $\mathrm{Set}\to \mathrm{Grp}$

• The “free abelian group” functor $\mathrm{Set}\to \mathrm{Ab}$

• The “free category” functor $\mathrm{DiGraph}\to \mathrm{Cat}$

and many others.

In general, if $U:C\to D$ is thought of as a forgetful functor and $F:D\to C$ is its left adjoint, then $F(x)$ is the free $C$-object on an object $x$ of $D$.