Homologies of Algebraic Structures via Braidings and Quantum Shuffles

In this paper we construct “structural” pre-braidings characterizing different algebraic structures: a rack, an associative algebra, a Leibniz algebra and their representations. Some of these pre-braidings seem original. On the other hand, we propose a general homology theory for pre-braided vector spaces and braided modules, based on the quantum co-shuffle comultiplication. Applied to the structural pre-braidings above, it gives a generalization and a unification of many known homology theories. All the constructions are categorified, resulting in particular in their superand co-versions. Loday’s hyper-boundaries, as well as certain homology operations are efficiently treated using the “shuffle” tools.