Averaging Property of Wedge Product and Naturality in Discrete Exterior Calculus

TL;DR

This paper demonstrates the naturality of discrete exterior calculus operations under simplicial maps and introduces an averaging interpretation of the discrete wedge product, linking it to Wilson's cochain product.

Contribution

It proves the naturality of DEC operations under simplicial maps and presents a new averaging perspective for the discrete wedge product, connecting it to existing cochain products.

Findings

Discrete exterior calculus operations are natural with respect to simplicial maps.

Introduces an averaging interpretation of the discrete wedge product.

Shows equivalence between the wedge product and Wilson's cochain product.

Abstract

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. We show that these discrete operations in DEC are natural with respect to abstract simplicial maps. A second contribution is a new averaging interpretation of the discrete wedge product in DEC. We also show that this wedge product is the same as Wilson's cochain product defined using Whitney and de Rham maps.