Higher algebra over the Leibniz operad

Abstract

We study oidication of Leibniz algebras and introduce two subclasses of classical Leibniz algebroids, Loday algebroids and symmetric Leibniz algebroids. The algebroids of the rst subclass have true dierential geometric brackets and the ones of the second are the main ingredients of generalized Courant algebroids, a broader category that we dene and investigate, proving in particular that it admits free objects. Regarding homotopycation of Leibniz algebras, we review and introduce ve concepts of homotopy between Leibniz innity morphisms in particular an explicit notion of operadic homotopy and show that they are all equivalent. Further, we prove that the category of Leibniz innity algebras carries an ∞-category structure. The latter projects onto the strict 2-category structure obtained on 2-term Leibniz innity algebras via transport of the canonical 2-category structure on categoried Leibniz algebras.