Free Functor

A free functor is a functor, typically with the category of sets as its domain, which sends sets to freely-constructed mathematical objects. They are generally [[Adjoint Functor|adjoint functors]] with forgetful functors.

Examples of Free Functors

One can construct a free functor for the majority of mathematical objects for which there is an underlying set:

1. There exists a free functor from \(\text{Set}\) to \(\text{Grp}\) which takes a set \(X\) to the [[Free Group|free group]] on \(X\).
2. There are two free functors from \(\text{Set}\) to \(\text{Top}\): one which sets a set to the discrete topology over that set, and the other which sets a set to the indiscrete topology over that set.
3. Notably, there is not a free functor from \(\text{Set}\) to the category of fields, because there are no free fields over non-empty sets.