A universal enveloping algebra for cocommutative rack bialgebras

We construct a bialgebra object in the category of linear maps {\mathcal{LM}} from a cocommutative rack bialgebra. The construction does extend to some non-cocommutative rack bialgberas, as is illustrated by a concrete example. As a separate result, we show that the Loday complex with adjoint coefficients embeds into the rack bialgebra deformation complex for the rack bialgebra defined by a Leibniz algebra.