Example [efr-003B]

C[\partial \Delta ^1] = C[0] \times C[0], and the inclusion of the boundary in the whole induces a map C[\Delta ^2] = C[2] \to C[0] \times C[0] = C[\partial \Delta ^1], which is precisely the pairing of the two face maps.

C[\partial \Delta ^2] is the "pullback" of three copies of C[1] over three copies of C[0]---one could write it as C[1] \times _{C[0]} C[1] \times {C[0] \times C[0]} C[1], where the map C[1] \times _{C[0]} C[1] \to C[0] \times C[0] picks out the domain of the first 1-cell and the codomain of the second one, and the map C[1] \to C[0]\times C[0] is the above discussed pairing of the face maps.

Thus, as expected, an element of C[\partial \Delta ^2] consists of a triple of 1-cells with compatible endpoints to make up a triangle (but no 2-cell to fill it). The restriction map C[\Delta ^2] \to C[\partial \Delta ^2], as expected, takes this boundary of a 2-cell.