Bialgebraic approach to rack cohomology

Abstract

We interpret the complexes defining rack cohomology in terms of a certain differential graded bialgebra. This yields elementary algebraic proofs of old and new structural results for this cohomology theory. For instance, we exhibit two explicit homotopies controlling structure defects on the cochain level: one for the commutativity defect of the cup product, and the other one for the "Zinbielity" defect of the dendriform structure. We also show that, for a quandle, the cup product on rack cohomology restricts to, and the Zinbiel product descends to quandle cohomology. Finally, for rack cohomology with suitable coefficients, we complete the cup product with a compatible coproduct.