A simplicial map is a family of functions that commutes with and . If each is a subset of such that the inclusions is a simplicial map, then is said to be a simplicial subset of .
The -simplex is defined as follows:
The face is defined by , i.e. deleting . The degeneracy is given by , i.e. repeating . Let . Any element in can be written as iterated compositions of faces and degeneracies of .