Averaging Property of Wedge Product and Naturality in Discrete Exterior Calculus

In exterior calculus on smooth manifolds, the exterior derivative and wedge product are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial cochains play the role of discrete forms, the coboundary operator serves as the discrete exterior derivative, and the antisymmetrized cup product provides a discrete wedge product. In this paper we show that these discrete operations in DEC are natural with respect to abstract simplicial maps. A second contribution is a new combinatorial averaging interpretation of the discrete wedge product in DEC.