For a vector space $T$ over a field $K$, Bourbaki defines affine space as a homogeneous $T$-set $E$ on which $T$ acts faithfully. The definitions found both on Wikipedia and nlab say that the action should be free. Note that Bourbaki defines a faithful action as one which embeds $T$ into the symmetric group of $E$. Is this a typo? Does he mean free instead?
Edit: Even the french version of the book uses "fidèlement" instead of "librement".
If $T$ is an abelian group and $E$ is a homogeneous $T$-set, then $E$ is faithful iff it is free, so the two definitions are equivalent. Indeed, the kernel of the action is the intersection of the stabilizers of all the points. Since $E$ is homogeneous, the stabilizers of any two points are conjugate, which means they are equal since $T$ is abelian. So the action is faithful iff the stabilizers are trivial, i.e. iff the action is free.