An affine extension of a map between the vertices of a Simplicial complex and $\mathbb{R}^m$ is the restriction of {the unique linear extension of the map to {a map between {the Vector spaces spanned by the vertices ($\mathbb{R}^n$ for $n$ vertices)} and $\mathbb{R}^m$}} to the simplex. A map that is the affine extension of some such map is called affine.

See here