# Bialgebraic approach to rack cohomology

## TL;DR

This paper introduces a bialgebraic framework for rack cohomology, providing new algebraic proofs and explicit homotopies that clarify the structure and relationships within the cohomology theory.

## Contribution

It develops a bialgebraic approach to rack cohomology, offering elementary proofs and explicit homotopies for structural properties, and explores the interplay between cup and Zinbiel products.

## Key findings

- Explicit homotopies for structure defects in cohomology
- Cup product restricts to quandle cohomology
- Compatible coproduct completes the cohomology structure

## 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.