LAUSR

We factor the classical functors $${ As}\mathop {\longrightarrow }\limits ^{-} { Lie}$$As⟶-Lie and $${ Dias}\mathop {\longrightarrow }\limits ^{-}{ Leib}$$Dias⟶-Leib through the categories $${ Pre}\hbox {-}{} { Lie}$$Pre-Lie and $${ Pre}\hbox {-}{} { Leib}$$Pre-Leib of two new types of algebras. Thanks to Koszul duality for binary quadratic operads, we deduce two more categories of algebras $${ Perm}$$Perm and $${ Ricod}$$Ricod giving rise to other factorizations. This yields a triangulation of Loday’s commutative diagram of functors on Leibniz algebras and associated operads. As an application, we define a notion of extended Leibniz algebras.