Higher algebra over the Leibniz operad
We study oidification of Leibniz algebras and introduce two subclasses of classical Leibniz algebroids, Loday algebroids and symmetric Leibniz algebroids. The algebroids of the first subclass have true differential geometric brackets and the ones of the second are the main ingredients of generalized Courant algebroids, a broader category that we define and investigate, proving in particular that it admits free objects. Regarding homotopyfication of Leibniz algebras, we review and introduce five concepts of homotopy between Leibniz infinity morphisms—in particular an explicit notion of operadic homotopy—and show that they are all equivalent. Further, we prove that the category of Leibniz infinity algebras carries an $\infty$-category structure. The latter projects onto the strict 2-category structure obtained on 2-term Leibniz infinity algebras via transport of the canonical 2-category structure on categorified Leibniz algebras.