  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.
