We define self-distributive structures in the categories of coalgebras and cocommutative coalgebras. We obtain examples from vector spaces whose bases are the elements of finite quandles, the direct sum of a Lie algebra with its ground field, and Hopf algebras. The self-distributive operations of these structures provide solutions of the Yang–Baxter equation, and, conversely, solutions of the Yang–Baxter equation can be used to construct self-distributive operations in certain categories. Moreover, we present a cohomology theory that encompasses both Lie algebra and quandle cohomologies, is analogous to Hochschild cohomology, and can be used to study deformations of these self-distributive structures. All of the work here is informed via diagrammatic computations.
© 2008, J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi and Masahico Saito.
Carter, J.S.; Crans, A.; Elhamdadi, M.; and Saito, M. “Cohomology of Categorical Self-Distributivity.” Journal of Homotopy and Related Structures, Vol. 3 (2008), No. 1: 13 – 63.