CONICET | Buscador de Institutos y Recursos Humanos

This paper deals with left non-degenerate set-theoretic solutions to theYang-Baxter equation (=LND solutions), a vast class of algebraic structuresencompassing groups, racks, and cycle sets. To each such solution is associateda shelf (i.e., a self-distributive structure) which captures its majorproperties. We consider two (co)homology theories for LND solutions, one ofwhich was previously known, in a reduced form, for biracks only. An explicitisomorphism between these theories is described. For groups and racks werecover their classical (co)homology, whereas for cycle sets we get newconstructions. For a certain type of LND solutions, including quandles andnon-degenerate cycle sets, the (co)homologies split into the degenerate and thenormalized parts. We express 2-cocycles of our theories in terms of groupcohomology, and, in the case of cycle sets, establish connexions withextensions. This leads to a construction of cycle sets with interestingproperties.