DIFFERENTIAL AUTOMATA THEORY

In this talk I will study generalized automata (in the sense of Adámek-Trnková) in Joyal’s category of combinatorial species; as an important preliminary step, I will provide examples of coalgebras for the "derivative" endofunctor ∂ and for the ‘Euler homogeneity operator’ L∂ arising from the adjunction L⊣∂⊣R. The theory is connected with, and in fact provides nontrivial examples of, differential 2-rigs—a concept I recently introduced by treating combinatorial species in the same way that a generic (differential) semiring (R,d) relates to the (differential) semiring N[[X]] of power series with natural coefficients. Joyal himself has long regarded species as categorified formal power series. This perspective aligns with a fundamental category-theoretic insight: free objects in the category of rings naturally acquire a canonical differential structure. At the heart of this phenomenon lies the representability of the prestack of derivations by an object of Kähler differentials. These ideas categorify elegantly within the 2-category of differential 2-rigs, revealing that species possess a universal property as differential 2-rigs. The desire to study categories of ‘state machines’ valued in an ambient monoidal category (K,⊗) gives a pretext to further develop the abstract theory of differential 2-rigs, proving lifting theorems of a differential 2-rig structure from (R,∂) to the category of ∂-algebras on objects of R, and to categories of Mealy automata valued in (R,⊗), as well as various constructions inspired by differential algebra such as jet spaces and modules of differential operators. This talk covers the content of the paper Automata and Coalgebras in Categories of Species (Proceedings of CMCS24, Luxembourg), as well as parts of an ongoing project with Todd Trimble.