Differential 2-rigs

We study the notion of a "differential 2-rig", a category R with coproducts and a monoidal structure distributing over them, also equipped with an endofunctor D : R -> R that satisfies a categorified analogue of the Leibniz rule. This is intended as a tool to unify various applications of such categories to computer science, algebraic topology, and enumerative combinatorics. The theory of differential 2-rigs has a geometric flavour but boils down to a specialization of the theory of tensorial strengths on endofunctors; this builds a surprising connection between apparently disconnected fields. We build "free 2-rigs" on a signature, and we prove various initiality results: for example, a certain category of colored species is the free differential 2-rig on a single generator.