Automata and coalgebras in categories of species

We study generalized automata (in the sense of Ad\'amek-Trnkov\'a) in Joyal's category of (set-valued) combinatorial species, and as an important preliminary step, we study coalgebras for its derivative endofunctor $\partial$ and for the "Euler homogeneity operator" $L\circ\partial$ arising from the adjunction $L\dashv\partial\dashv R$. The theory is connected with, and in fact provides relatively nontrivial examples of, "differential 2-rigs", a notion recently introduced by the author putting combinatorial species on the same relation a generic (differential) semiring $(R,d)$ has with the (differential) semiring $\mathbb N[\![ X]\!]$ of power series with natural coefficients. The desire to study categories of "state machines" valued in an ambient monoidal category $(\mathcal K,\otimes)$ gives a pretext to further develop the abstract theory of differential 2-rigs, proving lifting theorems of a differential 2-rig structure from $(\mathcal R,\partial)$ to the category of $\partial$-algebras on objects of $\mathcal R$, and to categories of Mealy automata valued in $(\mathcal R,\otimes)$, as well as various constructions inspired by differential algebra such as jet spaces and modules of differential operators. These theorems adapt to various "species-like" categories such as coloured species, $k$-vector species (both used in operad theory), linear species (introduced by Leroux to study combinatorial differential equations), M\"obius species, and others.