An effective Proof of the Cartan Formula

The Cartan formula encodes the relationship between the cup product and the action of the Steenrod algebra in F_p-cohomology. In this work, we present an effective proof of the Cartan formula at the cochain level when the field is F_2. More explicitly, for an arbitrary pair of cocycles and any non-negative integer, we construct a natural coboundary that descends to the associated instance of the Cartan formula. Our construction of Cartan coboundaries works for general algebras over the Barratt-Eccles operad, in particular, for the singular cochains of spaces, a case that has been used in condensed mater physics and for which we have developed open source software.