1. Introduction

Combinatorial species arose in the work of André Joyal (Reference Joyal1985, Reference Joyal1981) as categorification of the theory of generating functions (Wilf Reference Wilf1990). The idea is as simple as it is fruitful: a convenient way to study a sequence of positive integers $\boldsymbol{b}=(b_0,b_1,b_2,\dots )$ equipped with some combinatorial meaning is to consider them as the coefficients of a formal power series $F_{\boldsymbol{b}}(X)\in \mathbb{Q}[\![ X]\!]$ (the generating series or generating function of $\boldsymbol{b}$ ), most often in exponential form, that is, $F_{\boldsymbol{b}}(X) := \sum _{n\ge 0}\frac {b_n}{n!}X^n$ . The properties of $F_{\boldsymbol{b}}(X)$ as an algebraic object reflect onto the combinatorial properties of $\boldsymbol{b}$ (species can be added, multiplied, functionally composed; each of these operations performed on $F_{\boldsymbol{a}},F_{\boldsymbol{b}}$ has meaning in terms of the combinatorial object described by $\boldsymbol{a},\boldsymbol{b}$ ), and vice versa, the combinatorial properties of $\boldsymbol{b}$ (e.g., the fact that its elements satisfy a certain recurrence relation) reflect on the algebraic properties of its generating series. ‘Generatingfunctionology’ is, among other things, the study of combinatorics inspired by the manipulations of formal power series whose coefficients have combinatorial meaning.

Indeed, crafting a bijective proof to grok numerical identities in terms of bijections between finite sets is acknowledged as the fundamental problem in enumerative combinatorics (cf. e.g., the introduction of Méndez and Yang Reference Méndez and Yang1991), and generating functionology is of great help in this respect: for Joyal, a ‘species of structure’ arises as a categorification of the notion of generating series.

A species of structure consists of a functor $F : {\textsf{P}} \to \textsf{Set}$ having domain the category of finite sets and bijections: instead of a countable sequence of numbers $\{b_n\mid n\ge 0\}$ , a countable sequence of sets $\{F[n]\mid n\ge 0\}$ ; properties of the category of all such functors can now be given combinatorial meaning, combinatorial identities acquire meaning as bijective proofs (=isomorphisms of functors), and operations performed on functors express a possibly complicated object as (monoidal) product of simpler bits (we mention some of these isomorphisms in Remark 2.11). Among Joyal’s first applications for the language of species, there was a particularly insightful proof of Cayley’s counting theorem for trees (Cayley Reference Cayley1889), a result which paved the way to a booming development of techniques (propelled by the support of an insider of enumerative combinatorics, and genius, as C.G. Rota) in domains such as representation theory of groups (Chen Reference Chen1993; Leroux and Viennot Reference Leroux and Viennot1988; Rajan Reference Rajan1993; Yeh Reference Yeh, Labellea and Leroux1986), the study of set partitions (Bonetti et al. Reference Bonetti, Rota, Senato and Venezia1992; Joni and Rota Reference Joni and Rota1979; Méndez and Nava Reference Méndez and Nava1993), Möbius functions (Méndez and Yang Reference Méndez and Yang1991; Rota Reference Rota1964; Senato et al. Reference Senato, Venezia and Yang1997) and graph theory (Méndez Reference Dixon and Mortimer1996), up to the exciting field of combinatorial differential equations (Bergeron and Reutenauer Reference Bergeron and Reutenauer1990; Leroux and Viennot Reference Leroux, Viennot, Labellea and Leroux1986, 1988; Menni Reference Menni2008).

This wealth of applications is by no means limited to the field of enumerative combinatorics; the operation of plethystic substitution (Bergeron Reference Bergeron1987; Nava Reference Nava1987; Nava and Rota Reference Nava and Rota1985) is recognised as the fundamental building block in the theory of operads envisioned by May (1972, 1997) and natural instances of operadic composition arise in algebraic topology and algebraic geometry (Fresse 1967; Getzler and Kapranov Reference Getzler and Kapranov1998; Loday and Vallette Reference Loday and Vallette2012; Obradović Reference Obradović2017), logic and computer science (Gambino and Joyal Reference Gambino and Joyal2017; Gambino et al. Reference Gambino, Garner and Vasilakopoulou2022; Yorgey Reference Yorgey2014) (especially due to their link with multicategory theory; Lambek Reference Lambek1969; Lambek et al. Reference Lambek1989), theoretical physics (Getzler Reference Getzler2009; Getzler and Jones Reference Getzler and Jones1994) and more.

At about the same time, another application of category theory gained momentum: the idea of interpreting abstract state machines inside general categories. The line of research initiated by Arbib–Manes (Arbib and Manes Reference Arbib and Manes1975; Pohl and Arbib Reference Pohl and Arbib1970), Goguen (Goguen Reference Goguen1975, 1973; Goguen et al. Reference Goguen1975), Naudé (Reference Naudé1977, 1979) and developed in recent times by van Glabbeek (Reference van Glabbeek1999), Hermida and Mateus (Reference Hermida and Mateus2004), Jacobs (Reference Jacobs2006) and Venema (Reference Venema2006) culminated into Ehrig’s monograph (Ehrig et al. Reference Ehrig, Kiermeier, Kreowski and Kühnel1974) on automata ‘valued’ in an abstract monoidal category $\mathcal{K}$ .

The intuition the reader should have is that an automaton is a span

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whose legs represent, respectively, a dynamical system (yielding a representation of $A$ over a state space $X$ ), and a function $s$ whose role is to give a final state (or output, or answer) to the computation performed by $d$ .

The treatment made by Ehrig et al. (Reference Ehrig, Kiermeier, Kreowski and Kühnel1974) provides a systematic, category-theoretic insight into the transition from determinism to non-determinism that can be seen as the passage from automata in a monoidal category (Meseguer and Sols Reference Meseguer and Sols1975), to automata in the Kleisli category of an opmonoidal monad (Guitart Reference Guitart1980; Jacobs Reference Jacobs2016) (e.g., the probability distribution monads for convex spaces, Doberkat Reference Doberkat2007; Fritz Reference Fritz2015; Jacobs Reference Jacobs2018, 2010; Mateus et al. Reference Mateus, Sernadas and Sernadas2000) or one of its companions – the subdistribution or unnormalised distribution monad).

The category-theoretic content of such an approach to ‘machines’ goes a long way: a tentative chronology follows, but it can only scratch the surface of an immense, often submerged, body of research.

• • Adámek (Reference Adámek1974) and Adámek and Trnková (Reference Adámek and Trnková1990) introduced the notion of an $F$ -automaton in order to abstract even further from the monoidal case the ‘dynamics’ igniting the behaviour of an abstract machine; the progression in abstraction is as follows: from Cartesian machines in a category $\mathcal{K}$ with finite products, that is, spans $E \leftarrow A\times E \to B$ , one goes to monoidal ones, that is, spans $E \leftarrow A\otimes E \to B$ valued in a monoidal category $(\mathcal{K},\otimes )$ ; these are the objects of categories ${\textsf{Mly}}_{\mathcal{K}}{(A, B)}$ . Subsequently, one abstracts the action of $A\,\otimes \,\_\,$ on $E$ even further, using a generic endofunctor $F : \mathcal{K}\to \mathcal{K}$ , this is, the category ${\textsf{Mly}}_{\mathcal{K}}{(F, B)}$ .

• • Only few years prior, extensive work of Betti and Kasangian (Reference Betti and Kasangian1981, Reference Betti and Kasangian1982) and Kasangian and Rosebrugh (Reference Kasangian and Rosebrugh1990) pushed for the adoption of ‘profunctorial’ models for automata, capable to pinpoint their behaviour, and their minimisation, as a universal property (Goguen Reference Goguen1972; Goguen et al. Reference Goguen1975).

• • An insightful idea of Katis et al. (Reference Katis, Sabadini and Walters1997, Reference Katis, Sabadini and Walters2010) recognised that categories ${\textsf{Mly}}_{\mathcal{K}}{(A, B)}$ in a Cartesian category give rise to a composition operation

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  and thus organise as the hom-categories of a bicategory $\textsf{KSW}(\mathcal{K})$ .Footnote ¹ The bicategory so obtained can be concisely described as the bicategory of pseudofunctors, lax natural transformations and modifications $\textbf {B}\mathbb{N} \to \mathcal{K}$ , where $\textbf {B}\mathbb{N}$ is the monoid of natural numbers, regarded as a bicategory with a single object together with $\mathcal{K}$ . This definition extends to monoidal automata in a straightforward way, but there one loses the description as spans, given that the monoidal product isn’t universal.

• • In Guitart (1974, 1980), René Guitart introduces the bicategory $\textsf{Mac}$ as a refinement of a bicategory of spans.Footnote ² In Guitart and Van den Bril (Reference Guitart and Van den Bril1977), Guitart proves that $\textsf{Mac}$ is simply the Kleisli bicategory (Fiore et al. Reference Fiore, Gambino, Hyland and Winskel2016, Section 4; Gambino et al. Reference Gambino, Garner and Vasilakopoulou2022) of the 2-monad of cocompletion under lax colimits. This theme is reprised in Guitart (Reference Guitart1978) where Guitart introduces the notion of lax coend (Hirata Reference Hirata2022; Loregian Reference Loregian2021) as a technical preliminary to expand on the theme of Guitart and Van den Bril (Reference Guitart and Van den Bril1977).

• • Building on Ehrig et al. (Reference Ehrig, Kiermeier, Kreowski and Kühnel1974), but apparently unaware of Guitart (Reference Guitart1974), Paré proposed in (Reference Paré2010) the notion of a Mealy morphism as a proxy between strong functors and profunctors in any $\mathcal{V}$ -enriched category $\mathcal{C}$ . The paper culminates in the impressively general and elegantFootnote ³ result that the bicategory of $\mathcal{V}$ -Mealy maps is simply the Kleisli bicategory of the lax idempotent 2-monad of $\mathcal{V}$ -copower completion.Footnote ⁴

• • In a joint work (Boccali et al. Reference Boccali, Femić, Laretto, Loregian and Luneia2023), the present author explores how KSW’s ‘circuits’ and Guitart’s $\textsf{Mac}$ connect via a local adjunction (Jay Reference Jay1988; Kasangian et al. Reference Kasangian, Kelly and Rossi1983) and can be used to enhance categorical automata into widgets ‘typed’ over a bicategory with possibly more than one object; in short, it allows the passage from a bicategory of automata to automata in a bicategory, drawing some ideas from Bainbridge’s (Reference Bainbridge and Manes1975, Reference Bainbridge1972). Despite its relative obscurity, likely due to its cutting-edge nature, Bainbridge recognised and made clear the importance of bicategory theory as a foundational language for the theory of abstract automata and, in particular, proposed the idea of left/right Kan extensions along an ‘input scheme’ to analyse behaviour and minimisation.

Pushing further these ideas, building on all this work, intersects the most prolific branches of modern category theory. To sum up, we find ourselves in the following situation today: a forgotten school of category theorists hid an exciting claim behind a curtain of two-dimensional algebra:

This intriguing hypothesis is scattered across various sources, often unaware of each other; it has been hinted at multiple times and continues to leave traces of its presence for those willing to follow it. We are left with a conjecture and a clear work plan: can this fundamental guiding principle be taken seriously and formalised? Whoever is willing to take up the challenge of verifying this claim is now tasked with lifting the curtain and exploring a rich fauna of categorical widgets.

The present work grafts on top of the wide branches of this overarching project, studying categorical automata theory specialised to the differential 2-rig (a notion introduced by the author in Loregian and Trimble (Reference Loregian and Trimble2023) of Joyal’s combinatorial species. In order to do so, it develops further the basic theory of differential 2-rigs, expounded in Loregian and Trimble (Reference Boccali, Femić, Laretto, Loregian and Luneia2023): loosely speaking, a differential 2-rig (‘for the doctrine of coproducts’) consists of a monoidal category $(\mathcal{R},\otimes ,\partial )$ satisfying the following assumptions:

• • each functor $A\,\otimes \,\_\,$ and $\,\_\,\otimes B$ commutes with finite coproducts, to the effect that $\mathcal{R}$ embodies the structure of a categorified semiring, realised qua category equipped with a ‘bilinear’ tensor product;

• • there exists a functor $\partial : \mathcal{R}\to \mathcal{R}$ which commutes with coproducts, that is, there is a canonical natural isomorphism

  \begin{align*} \partial A + \partial B\cong \partial (A+B) \end{align*}
  and ‘satisfies the Leibniz rule’, that is, there is a natural isomorphism
  \begin{align*} \partial A\otimes B+A\otimes \partial B \cong \partial (A\otimes B). \end{align*}

Starting from this definition, one develops a categorification of differential algebra (intended, e.g., in the sense of Kaplansky Reference Kaplansky1976, Kolchin Reference Kolchin1973 and Marker Reference Marker, Haskell, Pillay and Steinhorn2000), recognising to the category of Joyal species the same role that in commutative algebra is covered by the ring of polynomials $k[X]$ .

On one side, the desire to better understand the theory of differential 2-rigs forces to find examples that motivate general definitions; species constitute such an example: the category ${\textsf{Spc}}$ of species is a presheaf topos equipped with a plethora of tightly knit monoidal structures interacting with a differential structure; this richness implies that when used as an ambient category for monoidal/functorial automata, it gives rise to an interesting theory that, when stated at the correct level of abstraction, is ‘stable under small perturbations’, which means that similar results to the ones presented here export without much effort to presheaf categories equipped with a plethystic substitution operation, such as coloured species (Méndez and Nava 1993), linear species (both in the sense of Leroux and Viennot 1986 and in the sense of $k\text{-}\textsf{Mod}$ -enriched, Aguiar and Mahajan Reference Aguiar and Mahajan2010; Getzler and Kapranov Reference Getzler and Kapranov1998), Möbius species (Méndez and Yang Reference Méndez and Yang1991) and nominal sets (Pitts Reference Pitts2013), and it allows to predict what happens when abstract automata are interpreted in a differential 2-rig other than ${\textsf{Spc}}$ , generalising Theorem 4.1.

On the other hand, the desire to understand better the theory of combinatorial species forces one to regard the category ${\textsf{Spc}}$ as an object of a larger universe of ‘2-rigs’ a fundamental feature of ${\textsf{Spc}}$ is that its derivative endofunctor $\partial$ admits both a left and a right adjoint denoted $L$ and $R$ . The categories of automata

(3) \begin{align} {\textsf{Mly}}_{\textsf{Spc}}(\partial ,B),{\textsf{Mly}}_{\textsf{Spc}}(L,B),{\textsf{Mly}}_{\textsf{Spc}}(L\partial ,B),{\textsf{Mly}}_{\textsf{Spc}}(\partial L,B) \end{align}

all constitute interesting examples of categories of automata, yielding a ‘dynamical system’ interpretation for diagrams of the form $X\leftarrow \partial X\to B$ , $X\leftarrow L\partial X\to B$ , etc., at the same time motivating a study of the categories of the endofunctor algebras for such functors (there is to date no reference for such a study).

The present work draws from both the desire to understand ${\textsf{Spc}}$ and ${\mathcal{M\!l\!y}}^\otimes _{\textsf{Spc}}[{{\textsf{Spc}}}]$ as (differential) 2-rigs and the desire to understand (differential) 2-rigs through specific examples; let’s now outline more precisely the structure of the paper.