Algebras of Open Dynamical Systems on the Operad of Wiring Diagrams

It is widely believed that complex systems of interest in the sciences and engineering are both modular and hierarchical. Network theory uses the tools and visual language of graph theory to model such systems, and has proven to be both effective and flexible in describing their modular character. However, the field has put less of an emphasis on finding powerful and versatile language for describing the hierarchical aspects of complex systems. There is growing confidence that category theory can provide the necessary conceptual setting for this project. This is seen, for example, in Mikhail Gromov’s well-known claim, “the mathematical language developed by the end of the 20th century by far exceeds in its expressive power anything, even imaginable, say, before 1960. Any meaningful idea coming from science can be fully developed in this language.” [Gro13]

Joyal and Street’s work on string diagrams [JS91] for monoidal categories and (with Verity) on traced monoidal categories [JSV96] has been used for decades to visualize compositions and feedback in networked systems, for example in the theory of flow charts [AMMO10]. Precursors, such as Penrose diagrams and flow diagrams, have been used in physics and the theory of computation, respectively, since the 1970’s [Sco71, BS11].

Over the past several years, the second author and collaborators have been developing a novel approach to modular hierarchical systems based on the language of operads and symmetric monoidal categories [Spi13, SR13]. The main contribution to the theory of string diagrams of the present research program is

the inclusion of an outer box, which allows for holarchic [Koe67] combinations of these diagrams. That is, the parts can be assembled into a whole, which can itself be a part. The composition of such assemblies can now be viewed as morphism composition in an operad. In fact, there is a strong connection between traced monoidal categories and algebras on these operads, such as our operad $\backslash mathcal\{OW\}$ of wiring diagrams, though it will not be explained here (see [SSR15] for details).

More broadly, category theory can organize graphical languages found in a variety of applied contexts. For example, it is demonstrated in [BS11] and [Coe13] that the theory of monoidal categories unifies the diagrams coming from diverse fields such as physics, topology, logic, computation, and linguistics. More recently, as in [BB12], there has been growing interest in viewing more traditionally applied fields, such as ecology, biology, chemistry, electrical engineering, and control theory through such a lens. Specifically, category theory has been used to draw connections among visual languages such as planar knot diagrams, Feynman diagrams, circuit diagrams, signal flow graphs, Petri nets, entity relationship diagrams, social networks, and flow charts. This research is building toward what John Baez has called “a foundation of applied mathematics” [Bae13].

The goal of the present paper is to show that open continuous time dynamical systems form an algebra over a certain (colored) operad, which we call the operad of wiring diagrams. It is a variant of the operad that appeared in [SR13]. That is, wiring diagrams provide a straightforward, diagrammatic language to understand how dynamical systems that describe processes can be built up from the systems that describe its sub-processes.

More precisely, we will define a symmetric monoidal category $\backslash mathbf\{W\}$ of black boxes and wiring diagrams. Its underlying operad $\backslash mathcal\{OW\}$ is a graphical language for building larger black boxes out of an interconnected set of smaller ones. We then define two $\backslash mathbf\{W\}$-algebras, $\backslash mathcal\{G\}$ and $\backslash mathcal\{L\}$, which encode open dynamical systems, i.e., differential equations of the form

where $Q$ represents an internal state vector, $\backslash dot\{Q\}\; =\; \backslash frac\{dQ\}\{dt\}$ represents its time derivative, and input and output represent inputs to and outputs from the system. In $\backslash mathcal\{G\}$, the functions $f^\{\backslash text\{in\}\}$ and $f^\{\backslash text\{out\}\}$ are smooth, whereas in the subalgebra $\backslash mathcal\{L\}\; \backslash subseteq\; \backslash mathcal\{G\}$, they are moreover linear. The fact that $\backslash mathcal\{G\}$ and $\backslash mathcal\{L\}$ are $\backslash mathbf\{W\}$-algebras captures the fact that these systems are closed under wiring diagram interconnection.

Our notion of interconnection is a generalization of that in Deville and Lerman [DL10], [DL15], [DL14]. Their version of interconnection produces a closed system from open ones, and can be understood in the present context as a morphism whose codomain is the closed box (see Definition 3.8). Graph fibrations between wiring diagrams form an important part of their formalism, though we do not discuss that aspect here.

This paper is the third in a series, following [SR13] and [Spi13], on using wiring diagrams to model interactions. The algebra we present here, that of open systems, is distinct from the algebras of relations and of propagators studied in earlier works. Beyond the dichotomy of discrete vs. continuous, these algebras are markedly different in structure. For one thing, the internal wires in [SR13] themselves carry state, whereas here, a wire should be thought of as instantaneously transmitting its contents from an output site to an input site. Another difference between our algebra and those of previous works is that the algebras here involve open systems in which, as in (1), the instantaneous change of state is a function of the current state and the input, whereas the output depends only on the current state (see Definition 4.2). The differences between these algebras is also reflected in a mild difference between the operad we use here and the one used in previous work.

1.1. Motivating example. The motivating example for the algebras in this paper comes from classical differential equations pedagogy; namely, systems of tanks containing salt water concentrations, with pipes carrying fluid among them. The systems of ODEs produced by such applications constitute a subset of those our language can address; they are linear systems with a certain form (see Example 5.7). To ground the discussion, we consider a specific example.

Example 1.1. Figure 1 below reimagines a problem from Boyce and DiPrima's canonical text [BD65, Figure 7.1.6] as a dynamical system over a wiring diagram.

In this diagram, $X\_1$ and $X\_2$ are boxes that represent tanks consisting of salt water solution. The functions $Q\_1(t)$ and $Q\_2(t)$ represent the amount of salt (in ounces) found in 30 and 20 gallons of water, respectively. These tanks are interconnected with each other by pipes embedded within a total system $Y$. The prescription for how wires are attached among the boxes is formally encoded in the wiring diagram $\backslash Phi\; :\; X\_1,\; X\_2\; \backslash rightarrow\; Y$, as we will discuss in Definition 3.1.

Both tanks are being fed salt water concentrations at constant rates from the outside world. Specifically, $X\_1$ is fed a 1 ounce salt per gallon water solution at 1.5 gallons per minute and $X\_2$ is fed a 3 ounce salt per gallon water solution at 1 gallon per minute. The tanks also both feed each other their solutions, with $X\_1$ feeding $X\_2$ at 3 gallons per minute and $X\_2$ feeding $X\_1$ at 1.5 gallons per minute. Finally, $X\_2$ feeds the outside world its solution at 2.5 gallons per minute.

The dynamics of the salt water concentrations both within and leaving each tank $X\_i$ is encoded in a linear open system $f\_i$, consisting of a differential equation for $Q\_i$ and a readout map for each $X\_i$ output (see Definition 2.9). Our algebra $\backslash mathcal\{L\}$ allows one to assign a linear open system $f\_i$ to each tank $X\_i$, and by functoriality the morphism $\backslash Phi:\; X\_1,\; X\_2\; \backslash rightarrow\; Y$ produces a linear open system for the larger box $Y$. We will explore this construction in detail, in particular providing explicit formulas for it in the linear case, as well as for more general systems of ODEs.

Throughout this paper we use the language of monoidal categories and functors. Depending on the audience, appropriate background on basic category theory can be found in MacLane [ML98], Awodey [Awo10], or Spivak [Spi14]. Leinster [Lei04] is a good source for more specific information on monoidal categories and operads. We refer the reader to [KFA69] for an introduction to dynamical systems.

Notation. We denote the category of sets and functions by Set and the full subcategory spanned by finite sets as FinSet. We generally do not concern ourselves with cardinality issues. We follow Leinster [Lei04] and use $\backslash times$ for binary product and $\backslash Pi$ for arbitrary product, and dually $+$ for binary coproduct and $\backslash coprod$ for arbitrary coproduct in any category. By operad we always mean symmetric colored operad or, equivalently, symmetric multicategory.

2.1. Monoidal categories and operads. In Section 3, we will construct the symmetric monoidal category $(\backslash mathbf\{W\},\; \backslash oplus,\; 0)$ of boxes and wiring diagrams, which we often simply denote as W. We will sometimes consider the underlying operad $\backslash mathcal\{OW\}$, obtained by applying the fully faithful functor

to W. A brief description of this functor $\backslash mathcal\{O\}$ is given below in Definition 2.1.

Definition 2.1. Let SMC denote the category of symmetric monoidal categories and lax monoidal functors; and Opd be the category of operads and operad functors. Given a symmetric monoidal category $(\backslash mathcal\{C\},\; \backslash otimes,\; I\_\{\backslash mathcal\{C\}\})\; \backslash in\; \backslash mathbf\{Ob\}\; \backslash mathbf\{SMC\}$, we define the operad $\backslash mathcal\{OC\}$ as follows:

for any $n\; \backslash in\; \backslash mathbb\{N\}$ and objects $X\_1,\; \backslash dots,\; X\_n,\; Y\; \backslash in\; \backslash mathbf\{Ob\}\; \backslash mathcal\{C\}$.

Now suppose $F:\; (\backslash mathcal\{C\},\; \backslash otimes,\; I\_\{\backslash mathcal\{C\}\})\; \backslash rightarrow\; (\backslash mathcal\{D\},\; \backslash odot,\; I\_\{\backslash mathcal\{D\}\})$ is a lax monoidal functor in SMC. By definition such a functor is equipped with a morphism

natural in the $X\_i$, called the coherence map. With this map in hand, we define the operad functor $\backslash mathcal\{OF\}:\; \backslash mathcal\{OC\}\; \backslash rightarrow\; \backslash mathcal\{OD\}$ by stating how it acts on objects $X$ and morphisms $\backslash Phi:\; X\_1,\; \backslash dots,\; X\_n\; \backslash rightarrow\; Y$ in $\backslash mathcal\{OC\}$:

Example 2.2. Consider the symmetric monoidal category $(\backslash mathbf\{Set\},\; \backslash times,\; \backslash star)$, where $\backslash times$ is the cartesian product of sets and $\backslash star$ a one element set. Define Sets := $\backslash mathcal\{OSet\}$ as in Definition 2.1. Explicitly, Sets is the operad in which an object is a set and a morphism $f:\; X\_1,\; \backslash dots,\; X\_n\; \backslash rightarrow\; Y$ is a function $f:\; X\_1\; \backslash times\; \backslash dots\; \backslash times\; X\_n\; \backslash rightarrow\; Y$.

Definition 2.3. Let $\backslash mathcal\{C\}$ be a symmetric monoidal category and let $\backslash mathbf\{Set\}\; =\; (\backslash mathbf\{Set\},\; \backslash times,\; \backslash star)$ be as in Example 2.2. A $\backslash mathcal\{C\}$-algebra is a lax monoidal functor $\backslash mathcal\{C\}\; \backslash rightarrow\; \backslash mathbf\{Set\}$. Similarly, if $\backslash mathcal\{D\}$ is an operad, a $\backslash mathcal\{D\}$-algebra is defined as an operad functor $\backslash mathcal\{D\}\; \backslash rightarrow\; \backslash mathbf\{Sets\}$.

To avoid subscripts, we will generally use the formalism of SMCs in this paper. Definitions 2.1 and 2.3 can be applied throughout to recast everything we do in terms of operads. The primary reason operads may be preferable in applications is that they suggest more compelling pictures. Hence throughout this paper, depictions of wiring diagrams will often be operadic, i.e., have many input boxes wired together into one output box.

2.2. Typed sets. Each box in a wiring diagram will consist of finite sets of ports, each labelled by a type. To capture this idea precisely, we define the notion of typed finite sets. By a finite product category, we mean a category that is closed under taking finite products.

Definition 2.4. Let $\backslash mathcal\{C\}$ be a small finite product category. The category of $\backslash mathcal\{C\}$-typed finite sets, denoted $\backslash mathbf\{TFS\}\_\{\backslash mathcal\{C\}\}$, is defined as follows. An object in $\backslash mathbf\{TFS\}\_\{\backslash mathcal\{C\}\}$ is a map from a finite set to the objects of $\backslash mathcal\{C\}$:

Intuitively, one can think of a typed finite set as a finite unordered list of $\backslash mathcal\{C\}$-objects. For any element $a\; \backslash in\; A$, we call the object $\backslash tau(a)$ its type. If the typing function $\backslash tau$ is clear from context, we may denote $(A,\; \backslash tau)$ simply by $A$.

A morphism $q:\; (A,\; \backslash tau)\; \backslash rightarrow\; (A\text{'},\; \backslash tau\text{'})$ in $\backslash mathbf\{TFS\}\_\{\backslash mathcal\{C\}\}$ consists of a function $q:\; A\; \backslash rightarrow\; A\text{'}$ that makes the following diagram of finite sets commute:

Note that $\backslash mathbf\{TFS\}\_\{\backslash mathcal\{C\}\}$ is a cocartesian monoidal category.

We refer to the morphisms of $\backslash mathbf\{TFS\}\_\{\backslash mathcal\{C\}\}$ as $\backslash mathcal\{C\}$-typed functions. If a $\backslash mathcal\{C\}$-typed function $q$ is bijective, we call it a $\backslash mathcal\{C\}$-typed bijection.

In other words, $\backslash mathbf\{TFS\}\_\{\backslash mathcal\{C\}\}$ is the comma category for the diagram

where $i$ is the inclusion.

Definition 2.5. Let $\backslash mathcal\{C\}$ be a finite product category, and let $(A,\; \backslash tau)\; \backslash in\; \backslash mathbf\{Ob\}\; \backslash mathbf\{TFS\}\_\{\backslash mathcal\{C\}\}$ be a $\backslash mathcal\{C\}$-typed finite set. Its dependent product $\backslash overline\{(A,\; \backslash tau)\}\; \backslash in\; \backslash mathbf\{Ob\}\; \backslash mathcal\{C\}$ is defined as

Coordinate projections and diagonals are generalized as follows. Given a typed function $q:\; (A,\; \backslash tau)\; \backslash rightarrow\; (A\text{'},\; \backslash tau\text{'})$ in $\backslash mathbf\{TFS\}\_\{\backslash mathcal\{C\}\}$ we define

to be the unique morphism for which the following diagram commutes for all $a\; \backslash in\; A$:

By the universal property for products, this defines a functor,

Lemma 2.6. The dependent product functor $\backslash mathbf\{TFS\}\_\{\backslash mathcal\{C\}\}^\{\backslash text\{op\}\}\; \backslash rightarrow\; \backslash mathcal\{C\}$ is strong monoidal. In particular, for any finite set $I$ whose elements index typed finite sets $(A\_i,\; \backslash tau\_i)$, there is a canonical isomorphism in $\backslash mathcal\{C\}$,

Remark 2.7. The category of second-countable smooth manifolds and smooth maps is essentially small (by the embedding theorem) so we choose a small representative and denote it Man. Note that Man is a finite product category. Manifolds will be our default typing, in the sense that we generally take $\backslash mathcal\{C\}\; :=\; \backslash mathbf\{Man\}$ in Definition 2.4 and denote

We thus refer to the objects, morphisms, and isomorphisms in TFS simply as typed finite sets, typed functions, and typed bijections, respectively.

Remark 2.8. The ports of each box in a wiring diagram will be labeled by manifolds because they are the natural setting for geometrically interpreting differential equations (see [Spi65]). For simplicity, one may wish to restrict attention to the full subcategory Euc of Euclidean spaces $\backslash mathbb\{R\}^n$ for $n\; \backslash in\; \backslash mathbb\{N\}$, because they are the usual domains for ODEs found in the literature; or to the (non-full) subcategory Lin of Euclidean spaces and linear maps between them, because they characterize linear systems of ODEs. We will return to TFS_Lin in Section 5.

2.3. Open systems. As a final preliminary, we define our notion of open dynamical system. Recall that every manifold $M$ has a tangent bundle manifold, denoted $TM$, and a smooth projection map $p:\; TM\; \backslash rightarrow\; M$. For any point $m\; \backslash in\; M$, the preimage $T\_m\; M\; :=\; p^\{-1\}(m)$ has the structure of a vector space, called the tangent space of $M$ at $m$. If $M\; \backslash cong\; \backslash mathbb\{R\}^n$ is a Euclidean space then also $T\_m\; M\; \backslash cong\; \backslash mathbb\{R\}^n$ for every point $m\; \backslash in\; M$. A vector field on $M$ is a smooth map $g:\; M\; \backslash rightarrow\; TM$ such that $p\; \backslash circ\; g\; =\; \backslash text\{id\}\_M$. See [Spi65] or [War83] for more background.

For the purposes of this paper we make the following definition of open systems; this may not be completely standard.

Definition 2.9. Let $M,\; U^\{\backslash text\{in\}\},\; U^\{\backslash text\{out\}\}\; \backslash in\; \backslash text\{Ob\; \}\; \backslash mathbf\{Man\}$ be smooth manifolds and $TM$ be the tangent bundle of $M$. Let $f\; =\; (f^\{\backslash text\{in\}\},\; f^\{\backslash text\{out\}\})$ denote a pair of smooth maps

where, for all $(m,\; u)\; \backslash in\; M\; \backslash times\; U^\{\backslash text\{in\}\}$ we have $f^\{\backslash text\{in\}\}(m,\; u)\; \backslash in\; T\_m\; M$; that is, the following diagram commutes:

We sometimes use $f$ to denote the whole tuple,

which we refer to as an open dynamical system (or open system for short). We call $M$ the state space, $U^\{\backslash text\{in\}\}$ the input space, $U^\{\backslash text\{out\}\}$ the output space, $f^\{\backslash text\{in\}\}$ the differential equation, and $f^\{\backslash text\{out\}\}$ the readout map of the open system.

Note that the pair $f\; =\; (f^\{\backslash text\{in\}\},\; f^\{\backslash text\{out\}\})$ is determined by a single smooth map

which, by a minor abuse of notation, we also denote by $f$.

In the special case that $M,\; U^\{\backslash text\{in\}\},\; U^\{\backslash text\{out\}\}\; \backslash in\; \backslash text\{Ob\; \}\; \backslash mathbf\{Lin\}$ are Euclidean spaces and $f$ is a linear map (or equivalently $f^\{\backslash text\{in\}\}$ and $f^\{\backslash text\{out\}\}$ are linear), we call $f$ a linear open system.

Remark 2.10. Let $M$ be a smooth manifold, and let $U^\{\backslash text\{in\}\}\; =\; U^\{\backslash text\{out\}\}\; =\; \backslash mathbb\{R\}^0$ be trivial. Then an open system in the sense of Definition 2.9 is a smooth map $f:\; M\; \backslash rightarrow\; TM$ over $M$, in other words, a vector field on $M$. From the geometric point of view, vector fields are autonomous (i.e., closed!) dynamical systems; see [Tes12].

Remark 2.11. For an arbitrary manifold $U^\{\backslash text\{in\}\}$, a map $M\; \backslash times\; U^\{\backslash text\{in\}\}\; \backslash rightarrow\; TM$ can be considered as a function $U^\{\backslash text\{in\}\}\; \backslash rightarrow\; \backslash mathbf\{VF\}(M)$, where $\backslash mathbf\{VF\}(M)$ is the set of vector fields on $M$. Hence, $U^\{\backslash text\{in\}\}$ controls the behavior of the system in the usual sense.

Remark 2.12. Given an open system $f$ we can form a new open system by feeding the readout of $f$ into the inputs of $f$. For example suppose the open system is of the form

where $A,\; B,\; C$ and $M$ are manifolds. Define $F\text{'}:\; M\; \backslash times\; A\; \backslash rightarrow\; TM$ by

Then

is a new open system obtained by plugging a readout of $f$ into the space of inputs $B$. Compare with Figure 3.

This looks a little boring. It becomes more interesting when we start with several open systems, take their product and then plug (some of the) outputs into inputs. For example suppose we start with two open systems

and

Here, again, all capital letters denote manifolds. Take their product; we get

Now plug in the functions $g\_B$ and $g\_1$ into inputs. We get a new system

where

Compare with Figure 7. Making these kinds of operations on open systems precise for an arbitrary number of interacting systems is the point of our paper.

By defining the appropriate morphisms, we can consider open dynamical systems as being objects in a category. We are not aware of this notion being defined previously in the literature, but it is convenient for our purposes.

Definition 2.13. Suppose that $M\_i,\; U\_i^\{\backslash text\{in\}\},\; U\_i^\{\backslash text\{out\}\}\; \backslash in\; \backslash text\{Ob\; \}\; \backslash mathbf\{Man\}$ and $(M\_i,\; U\_i^\{\backslash text\{in\}\},\; U\_i^\{\backslash text\{out\}\},\; f\_i)$ is an open system for $i\; \backslash in\; \backslash \{1,\; 2\backslash \}$. A morphism of open systems

is a triple $(\backslash zeta\_M,\; \backslash zeta\_\{U^\{\backslash text\{in\}\}\},\; \backslash zeta\_\{U^\{\backslash text\{out\}\}\})$ of smooth maps $\backslash zeta\_M:\; M\_1\; \backslash rightarrow\; M\_2$, $\backslash zeta\_\{U^\{\backslash text\{in\}\}\}:\; U\_1^\{\backslash text\{in\}\}\; \backslash rightarrow\; U\_2^\{\backslash text\{in\}\}$, and $\backslash zeta\_\{U^\{\backslash text\{out\}\}\}:\; U\_1^\{\backslash text\{out\}\}\; \backslash rightarrow\; U\_2^\{\backslash text\{out\}\}$, such that the following diagram commutes:

This defines the category ODS of open dynamical systems. We define the subcategory ODS_Lin $\backslash subseteq$ ODS by restricting our objects to linear open systems, as in Definition 2.9, and imposing that the three maps in $\backslash zeta$ are linear.

As in Remark 2.12, we will often want to combine two or more interconnected open systems into one larger one. As we shall see in Section 4, this will involve taking a product of the smaller open systems. Before we define this formally, we first remind the reader that the tangent space functor $T$ is strong monoidal, i.e., it canonically preserves products,

Lemma 2.14. The category ODS of open systems has all finite products. That is, if $I$ is a finite set and $f\_i\; =\; (M\_i,\; U\_i^\{\backslash text\{in\}\},\; U\_i^\{\backslash text\{out\}\},\; f\_i)\; \backslash in\; \backslash text\{Ob\; \}\; \backslash mathbf\{ODS\}$ is an open system for each $i\; \backslash in\; I$, then their product is

with the obvious projection maps.

In this section, we define the symmetric monoidal category $(\backslash mathbf\{W\},\; \backslash oplus,\; 0)$ of wiring diagrams. We then use Definition 2.1 to define the wiring diagram operad $\backslash mathcal\{OW\}$, which situates our pictorial setting. We begin by formally defining the underlying category $\backslash mathbf\{W\}$ and continue with some concrete examples to explicate this definition.

Definition 3.1. The category $\backslash mathbf\{W\}$ has objects boxes and morphisms wiring diagrams. A box $X$ is an ordered pair of $\backslash mathbf\{Man\}$-typed finite sets (Definition 2.4),

Let $X^\{\backslash text\{in\}\}\; =\; (A,\; \backslash tau)$ and $X^\{\backslash text\{out\}\}\; =\; (A\text{'},\; \backslash tau\text{'})$. Then we refer to elements $a\; \backslash in\; A$ and $a\text{'}\; \backslash in\; A\text{'}$ as input ports and output ports, respectively. We call $\backslash tau(a)\; \backslash in\; \backslash text\{Ob\; \}\; \backslash mathbf\{Man\}$ the type of port $a$, and similarly for $\backslash tau\text{'}(a\text{'})$.

A wiring diagram $\backslash Phi:\; X\; \backslash rightarrow\; Y$ in $\backslash mathbf\{W\}$ is a triple $(X,\; Y,\; \backslash varphi)$, where $\backslash varphi$ is a typed bijection (see Definition 2.4)

satisfying the following condition:

no passing wires: $\backslash varphi(Y^\{\backslash text\{out\}\})\; \backslash cap\; Y^\{\backslash text\{in\}\}\; =\; \backslash emptyset$, or equivalently $\backslash varphi(Y^\{\backslash text\{out\}\})\; \backslash subseteq\; X^\{\backslash text\{out\}\}$.

This condition allows us to decompose $\backslash varphi$ into a pair $\backslash varphi\; =\; (\backslash varphi^\{\backslash text\{in\}\},\; \backslash varphi^\{\backslash text\{out\}\})$:

We often identify the wiring diagram $\backslash Phi\; =\; (X,\; Y,\; \backslash varphi)$ with the typed bijection $\backslash varphi$, or equivalently its corresponding pair $(\backslash varphi^\{\backslash text\{in\}\},\; \backslash varphi^\{\backslash text\{out\}\})$.

By a wire in $\backslash Phi$, we mean a pair $(a,\; b)$, where $a\; \backslash in\; X^\{\backslash text\{in\}\}\; +\; Y^\{\backslash text\{out\}\}$, $b\; \backslash in\; X^\{\backslash text\{out\}\}\; +\; Y^\{\backslash text\{in\}\}$, and $\backslash varphi(a)\; =\; b$. In other words a wire in $\backslash Phi$ is a pair of ports connected by $\backslash phi$.

The identity wiring diagram $\backslash iota:\; X\; \backslash rightarrow\; X$ is given by the identity morphism $X^\{\backslash text\{in\}\}\; +\; X^\{\backslash text\{out\}\}\; \backslash rightarrow\; X^\{\backslash text\{in\}\}\; +\; X^\{\backslash text\{out\}\}$ in $\backslash mathbf\{TFS\}$.

Now suppose $\backslash Phi\; =\; (X,\; Y,\; \backslash varphi)$ and $\backslash Psi\; =\; (Y,\; Z,\; \backslash psi)$ are wiring diagrams. We define their composition as $\backslash Psi\; \backslash circ\; \backslash Phi\; =\; (X,\; Z,\; \backslash omega)$, where $\backslash omega\; =\; (\backslash omega^\{\backslash text\{in\}\},\; \backslash omega^\{\backslash text\{out\}\})$ is given by the pair of dashed arrows making the following diagrams commute.

Here $\backslash nabla:\; X^\{\backslash text\{out\}\}\; +\; X^\{\backslash text\{out\}\}\; \backslash rightarrow\; X^\{\backslash text\{out\}\}$ is the codiagonal map in $\backslash mathbf\{TFS\}$.

Remark 3.2. For any finite product category $\backslash mathcal\{C\}$, we may define the category $\backslash mathbf\{W\}\_\{\backslash mathcal\{C\}\}$ by replacing $\backslash mathbf\{Man\}$ with $\backslash mathcal\{C\}$, and $\backslash mathbf\{TFS\}$ with $\backslash mathbf\{TFS\}\_\{\backslash mathcal\{C\}\}$, in Definition 3.1. In particular, as in Remark 2.8, we have the symmetric monoidal category $\backslash mathbf\{W\}\_\{\backslash text\{Lin\}\}$ of linearly typed wiring diagrams.

What we are calling a box is nothing more than an interface; at this stage it has no semantics, e.g., in terms of differential equations. Each box can be given a pictorial representation, as in Example 3.3 below.

Example 3.3. As a convention, we depict a box $X\; =\; (\backslash \{a,\; b\backslash \},\; \backslash \{c\backslash \})$ with input ports connecting on the left and output ports connecting on the right, as in Figure 2 below. When types are displayed, we label ports on the exterior of their box and their types adjacently on the interior of the box with a ‘:’ symbol in between to designate typing. Reading types off of this figure, we see that the type of input port $a$ is the manifold $\backslash mathbb\{R\}$, that of input port $b$ is the circle $S^1$, and that of output port $c$ is the torus $T^2$.

A morphism in $\backslash mathbf\{W\}$ is a wiring diagram $\backslash Phi\; =\; (X,\; Y,\; \backslash varphi)$, the idea being that a smaller box $X$ (the domain) is nested inside of a larger box $Y$ (the codomain). The ports of $X$ and $Y$ are then interconnected by wires, as specified by the typed bijection $\backslash varphi$. We will now see an example of a wiring diagram, accompanied by a picture.

Example 3.4. Reading off the wiring diagram $\backslash Phi\; =\; (X,\; Y,\; \backslash varphi)$ drawn below in Figure 3, we have the following data for boxes:

Table 1 makes $\backslash varphi$ explicit via a list of its wires, i.e., pairs $(\backslash gamma,\; \backslash varphi(\backslash gamma))$.

Remark 3.5. The condition that $\backslash varphi$ be typed, as in Definition 2.4, ensures that if two ports are connected by a wire then the associated types are the same. In particular, in Example 3.4 above, $(a,\; b,\; n)$ must be the same type tuple as $(m,\; d,\; c)$.

Now that we have made wiring diagrams concrete and visual, we can do the same for their composition.

Example 3.6. In Figure 4, we visualize the composition of two wiring diagrams $\backslash Phi\; =\; (X,\; Y,\; \backslash varphi)$ and $\backslash Psi\; =\; (Y,\; Z,\; \backslash psi)$ to form $\backslash Psi\; \backslash circ\; \backslash Phi\; =\; (X,\; Z,\; \backslash omega)$. Composition is depicted by drawing the wiring diagram for $\backslash Psi$ and then, inside of the $Y$ box, drawing in the wiring diagram for $\backslash Phi$. Finally, to depict the composition $\backslash Psi\; \backslash circ\; \backslash Phi$ as one single wiring diagram, one simply “erases” the $Y$ box, leaving the $X$ and $Z$ boxes interconnected among themselves. Figure 4 represents such a procedure by depicting the $Y$ box with a dashed arrow.

It’s important to note that the wires also connect, e.g. if a wire in $\backslash Psi$ connects a $Z$ port to some $Y$ port, and that $Y$ port attaches via a $\backslash Phi$ wire to some $X$ port, then these wires “link together” to a total wire in $\backslash Psi\; \backslash circ\; \backslash Phi$, connecting a $Z$ port with an $X$ port. Table 2 below traces the wires of $\backslash Psi\; \backslash circ\; \backslash Phi$ through the $\backslash omega^\{\backslash text\{in\}\}$ and $\backslash omega^\{\backslash text\{out\}\}$ composition diagrams in (5) on its left and right side, respectively. The left portion of the table starts with $\backslash gamma\; \backslash in\; X^\{\backslash text\{in\}\}$ and ends at $\backslash omega^\{\backslash text\{in\}\}(\backslash gamma)\; \backslash in\; X^\{\backslash text\{out\}\}\; +\; Z^\{\backslash text\{in\}\}$, with intermediary steps of the composition denoted with superscripts $\backslash gamma^n$. The right portion of the table starts with $\backslash gamma\; \backslash in\; Z^\{\backslash text\{out\}\}$ then goes through the intermediary of $\backslash gamma\text{'}\; \backslash in\; Y^\{\backslash text\{out\}\}$ and finally reaches $\backslash omega^\{\backslash text\{out\}\}(\backslash gamma)\; \backslash in\; Z^\{\backslash text\{out\}\}$. We skip lines on the right portion to match the spacing on the left.

Remark 3.7. The condition that $\backslash varphi$ be both injective and surjective prohibits exposed ports and split ports, respectively, as depicted in Figure 5a. The no passing wires condition on $\backslash varphi(Y^\{\backslash text\{out\}\})$ prohibits wires that go straight across the $Y$ box, as seen in the intermediate box of Figure 5b.

Now that we have formally defined and concretely explicated the category $\backslash mathbf\{W\}$, we will make it into a monoidal category by defining its tensor product.

Definition 3.8. Let $X\_1,\; X\_2,\; Y\_1,\; Y\_2\; \backslash in\; \backslash text\{Ob\; \}\; \backslash mathbf\{W\}$ be boxes and $\backslash Phi\_1:\; X\_1\; \backslash rightarrow\; Y\_2$ and $\backslash Phi\_2:\; X\_2\; \backslash rightarrow\; Y\_2$ be wiring diagrams. The monoidal product $\backslash oplus$ is given by

The closed box $0\; =\; \backslash \{\backslash emptyset,\; \backslash emptyset\backslash \}$ is the monoidal unit.

Remark 3.9. Once we add semantics in Section 4, closed boxes will correspond to autonomous systems, which do not interact with any outside environment (see Remark 2.10).

We now make this monoidal product explicit with an example.

Example 3.10. Consider boxes $X\; =\; (\backslash \{x\_1,\; x\_2\backslash \},\; \backslash \{x\_3,\; x\_4\backslash \})$ and $Y\; =\; (\backslash \{y\_1\backslash \},\; \backslash \{y\_2,\; y\_3\backslash \})$ depicted below.

We depict their tensor $X\; \backslash oplus\; Y\; =\; (\backslash \{x\_1,\; x\_2,\; y\_1\backslash \},\; \backslash \{x\_3,\; x\_4,\; y\_2,\; y\_3\backslash \})$ by stacking boxes.

Similarly, consider the following wiring diagrams (with ports left unlabelled).

We can depict their composition via stacking.

We now prove that the above data characterizing $(\backslash mathbf\{W\},\; \backslash oplus,\; 0)$ indeed constitutes a symmetric monoidal category, at which point we can, as advertised, invoke Definition 2.1 to define the operad $\backslash mathcal\{OW\}$.

Proposition 3.11. The category $\backslash mathbf\{W\}$ in Definition 3.1 and the monoidal product $\backslash oplus$ with unit 0 in Definition 3.8 form a symmetric monoidal category $(\backslash mathbf\{W\},\; \backslash oplus,\; 0)$.

Proof. We begin by establishing that $\backslash mathbf\{W\}$ is indeed a category. We first show that our class of wiring diagrams is closed under composition. Let $\backslash Phi\; =\; (X,\; Y,\; \backslash varphi)$, $\backslash Psi\; =\; (Y,\; Z,\; \backslash psi)$, and $\backslash Psi\; \backslash circ\; \backslash Phi\; =\; (X,\; Z,\; \backslash omega)$.

To show that $\backslash omega$ is a typed bijection, we replace the pair of maps $(\backslash varphi^\{\backslash text\{in\}\},\; \backslash varphi^\{\backslash text\{out\}\})$ with a pair of bijections $(\backslash widetilde\{\backslash varphi\}^\{\backslash text\{in\}\},\; \backslash widetilde\{\backslash varphi\}^\{\backslash text\{out\}\})$ as follows. Let $X\_\{\backslash varphi\}^\{\backslash text\{exp\}\}\; \backslash subseteq\; X^\{\backslash text\{out\}\}$ (for exports) denote the image of $\backslash varphi^\{\backslash text\{out\}\}$, and $X\_\{\backslash varphi\}^\{\backslash text\{loc\}\}$ (for local ports) be its complement. Then we can identify $\backslash varphi$ with the following pair of typed bijections

Similarly, identify $\backslash psi$ with $(\backslash widetilde\{\backslash psi\}^\{\backslash text\{in\}\},\; \backslash widetilde\{\backslash psi\}^\{\backslash text\{out\}\})$. We can then rewrite the diagram defining $\backslash omega$ in (5) as one single commutative diagram of typed finite sets.

As a composition of typed bijections, $\backslash omega$ is also a typed bijection.

The following computation proves that $\backslash omega$ has no passing wires:

Therefore $\backslash mathbf\{W\}$ is closed under wiring diagram composition. To show that $\backslash mathbf\{W\}$ is a category, it remains to prove that composition of wiring diagrams satisfies the unit and associativity axioms. The former is straightforward and will be omitted. We now establish the latter.

Consider the wiring diagrams $\backslash Theta\; =\; (V,\; X,\; \backslash theta)$, $\backslash Phi\; =\; (X,\; Y,\; \backslash varphi)$, $\backslash Psi\; =\; (Y,\; Z,\; \backslash psi)$; and let $(\backslash Psi\; \backslash circ\; \backslash Phi)\; \backslash circ\; \backslash Theta\; =\; (V,\; Z,\; \backslash kappa)$ and $\backslash Psi\; \backslash circ\; (\backslash Phi\; \backslash circ\; \backslash Theta)\; =\; (V,\; Z,\; \backslash lambda)$. We readily see that $\backslash kappa^\{\backslash text\{out\}\}\; =\; \backslash lambda^\{\backslash text\{out\}\}$ by the associativity of composition in TFS. Proving that $\backslash kappa^\{\backslash text\{in\}\}\; =\; \backslash lambda^\{\backslash text\{in\}\}$ is equivalent to establishing the commutativity of the following diagram:

(6)

This diagram commutes in any category with coproducts, as follows from the associativity and naturality of the codiagonal map. We present a formal argument of this fact below in the language of string diagrams (See [JS91]). As in [Sel11], we let squares with blackened corners denote generic morphisms. We let triangles denote codiagonal maps. See Figure 6 below.

The first step of the proof follows from the topological nature of string diagrams, which mirror the axioms of monoidal categories. The second step invokes the associativity of codiagonal maps. The third and final step follows from the naturality of codiagonal maps, i.e., the commutativity of the following diagram.

Now that we have shown that $\backslash mathbf\{W\}$ is a category, we show that $(\backslash oplus,\; 0)$ is a monoidal structure on $\backslash mathbf\{W\}$. Let $X,\; X\text{'},\; X\text{'}\text{'}\; \backslash in\; \backslash text\{Ob\; \}\; \backslash mathbf\{W\}$ be boxes. We readily observe the following canonical isomorphisms.

Hence the monoidal product $\backslash oplus$ is well behaved on objects. It is similarly easy, and hence will be omitted, to show that $\backslash oplus$ is functorial. This completes the proof that $(\backslash mathbf\{W\},\; \backslash oplus,\; 0)$ is a symmetric monoidal category. $\backslash square$

Having established that $(\backslash mathbf\{W\},\; \backslash oplus,\; 0)$ is an SMC, we can now speak about the operad $\backslash mathcal\{OW\}$ of wiring diagrams. In particular, we can draw operadic pictures, such as the one in our motivating example in Figure 1, to which we now return.

Example 3.12. Figure 7 depicts an $\backslash mathcal\{OW\}$ wiring diagram $\backslash Phi:\; X\_1,\; X\_2\; \backslash rightarrow\; Y$, which we may formally denote by the tuple $\backslash Phi\; =\; (X\_1,\; X\_2;\; Y;\; \backslash varphi)$. Reading directly from Figure 7, we have the boxes:

The wiring diagram $\backslash Phi$ is visualized by nesting the domain boxes $X\_1,\; X\_2$ within the codomain box $Y$, and drawing the wires prescribed by $\backslash varphi$, as recorded below in Table 3.

To reconceptualize $\backslash Phi:\; X\_1,\; X\_2\; \backslash rightarrow\; Y$ as a wiring diagram in $\backslash mathbf\{W\}$, we simply consider the tensor $\backslash Phi:\; X\_1\; \backslash oplus\; X\_2\; \backslash rightarrow\; Y$, as given in Figure 8 below. This demonstrates the fact that operadic pictures are easier to read and hence are more illuminating.

The following remark explains that our pictures of wiring diagrams are not completely ad hoc—they are depictions of 1-dimensional oriented manifolds with boundary. The boxes in our diagrams simply tie together the positively and negatively oriented components of an individual oriented 0-manifold.

Remark 3.13. For any set $S$, let $1\backslash text\{-Cob\}/S$ denote the symmetric monoidal category of oriented 0-manifolds over $S$ and the 1-dimensional cobordisms between them. We call its objects oriented $S$-typed 0-manifolds. Recall that $\backslash mathbf\{W\}\; =\; \backslash mathbf\{W\}\_\{\backslash mathbf\{Man\}\}$ is our category of Man-typed wiring diagrams; let $\backslash mathbf\{M\}\; :=\; \backslash text\{Ob\; \}\; \backslash mathbf\{Man\}$ denote the set of manifolds (see Remark 2.7). There is a faithful, essentially surjective, strong monoidal functor

sending a box $(X^\{\backslash text\{in\}\},\; X^\{\backslash text\{out\}\})$ to the oriented $\backslash mathbf\{M\}$-typed 0-manifold $X^\{\backslash text\{in\}\}\; +\; X^\{\backslash text\{out\}\}$ where $X^\{\backslash text\{in\}\}$ is oriented positively and $X^\{\backslash text\{out\}\}$ negatively. Under this functor, a wiring diagram $\backslash Phi\; =\; (X,\; Y,\; \backslash varphi)$ is sent to a 1-dimensional cobordism that has no closed loops. A connected component of such a cobordism can be identified with either its left or right endpoint, which correspond to the domain or codomain of the bijection $\backslash varphi:\; X^\{\backslash text\{in\}\}\; +\; Y^\{\backslash text\{out\}\}\; \backslash xrightarrow\{\backslash cong\}\; X^\{\backslash text\{out\}\}\; +\; Y^\{\backslash text\{in\}\}$. See [SSR15].

In fact, with the no passing wires condition on morphisms (cobordisms) $X\; \backslash rightarrow\; Y$ (see Definition 3.1), the subcategory $\backslash mathbf\{W\}\; \backslash subseteq\; 1\backslash text\{-Cob\}/\backslash mathbf\{M\}$ is the left class of an orthogonal factorization system. See [Aba15].

Let $\backslash Phi\; =\; (X,\; Y,\; \backslash varphi)$ be a wiring diagram. Applying the dependent product functor (see Definition 2.5) to $\backslash varphi$, we obtain a diffeomorphism of manifolds

Equivalently, if $\backslash varphi$ is represented by the pair $(\backslash varphi^\{\backslash text\{in\}\},\; \backslash varphi^\{\backslash text\{out\}\})$, as in Definition 3.1, we can express $\backslash overline\{\backslash varphi\}$ in terms of its pair of component maps:

It will also be useful to apply the dependent product functor to the commutative diagrams in (5), which define wiring diagram composition. Note that, by the contravariance of the dependent product, the codiagonal $\backslash nabla:\; X^\{\backslash text\{out\}\}\; +\; X^\{\backslash text\{out\}\}\; \backslash rightarrow\; X^\{\backslash text\{out\}\}$ gets sent to the diagonal map $\backslash Delta:\; \backslash overline\{X^\{\backslash text\{out\}\}\}\; \backslash rightarrow\; \backslash overline\{X^\{\backslash text\{out\}\}\}\; \backslash times\; \backslash overline\{X^\{\backslash text\{out\}\}\}$. Thus we have the following commutative diagrams:

In this section we define an algebra $\backslash mathcal\{G\}:\; (\backslash mathbf\{W\},\; \backslash oplus,\; 0)\; \backslash rightarrow\; (\backslash mathbf\{Set\},\; \backslash times,\; \backslash star)$ (see Definition 2.3) of general open dynamical systems. A $\backslash mathbf\{W\}$-algebra can be thought of as a choice of semantics for the syntax of $\backslash mathbf\{W\}$, i.e., a set of possible meanings for boxes and wiring diagrams. As in Definition 2.1, we may use this to construct the corresponding operad algebra $\backslash mathcal\{OG\}:\; \backslash mathcal\{OW\}\; \backslash rightarrow\; \backslash mathbf\{Sets\}$. Before we define $\backslash mathcal\{G\}$, we revisit Example 1.1 for inspiration.

Example 4.1. As the textbook exercise [BD65, Problem 7.21] prompts, let's begin by writing down the system of equations that governs the amount of salt $Q\_i$ within the tanks $X\_i$. This can be done by using dimensional analysis for each port of $X\_i$ to find the rate of salt being carried in ounces per minute, and then equating the rate $\backslash dot\{Q\}\_i$ to the sum across these rates for $X\_i^\{\backslash text\{in\}\}$ ports minus $X\_i^\{\backslash text\{out\}\}$ ports.

Dropping the physical units, we are left with the following system of ODEs:

The derivations for the equations in (9) involved a hidden step in which the connection pattern in Figure 1, or equivalently Figure 7, was used. Our wiring diagram approach explains this step and makes it explicit. Each box in a wiring diagram should only “know” about its own inputs and outputs, and not how they are connected to others. That is, we can only define a system on $X\_i$ by expressing $Q\_i$ just in terms of $Q\_i$ and $X\_i^\{\backslash text\{in\}\}$—this is precisely the data of an open system (see Definition 2.9). We now define our algebra $\backslash mathcal\{G\}$, which assigns a set of open systems to a box. Given a wiring diagram and an open system on its domain box, it also gives a functorial procedure for assigning an open system to the codomain box. We will then use this new machinery to further revisit Example 4.1 in Example 5.7.

Definition 4.2. We define $\backslash mathcal\{G\}\; :\; (\backslash mathbf\{W\},\; \backslash oplus,\; 0)\; \backslash rightarrow\; (\backslash mathbf\{Set\},\; \backslash times,\; \backslash star)$ as follows. Let $X\; \backslash in\; \backslash mathbf\{Ob\}\; \backslash mathbf\{W\}$. The set of open systems on $X$, denoted $\backslash mathcal\{G\}(X)$, is defined as

We call $S$ the set of state variables and its dependent product $\backslash overline\{S\}$ the state space.

Let $\backslash Phi\; =\; (X,\; Y,\; \backslash varphi)$ be a wiring diagram. Then $\backslash mathcal\{G\}(\backslash Phi)\; :\; \backslash mathcal\{G\}(X)\; \backslash rightarrow\; \backslash mathcal\{G\}(Y)$ is given by $(S,\; f)\; \backslash mapsto\; (\backslash mathcal\{G\}(\backslash Phi)S,\; \backslash mathcal\{G\}(\backslash Phi)f)$, where $\backslash mathcal\{G\}(\backslash Phi)S\; =\; S$ and $g\; =\; \backslash mathcal\{G\}(\backslash Phi)f\; :\; \backslash overline\{S\}\; \backslash times\; \backslash overline\{Y\}^\{\backslash text\{in\}\}\; \backslash rightarrow\; \backslash overline\{T\backslash overline\{S\}\}\; \backslash times\; \backslash overline\{Y\}^\{\backslash text\{out\}\}$ is defined by the dashed arrows ($g^\{\backslash text\{in\}\},\; g^\{\backslash text\{out\}\}$) (see Definition 2.9) that make the diagrams below commute:

One may note strong resemblance between the diagrams in (10) and those in (5).

We give $\backslash mathcal\{G\}$ a lax monoidal structure: for any pair $X,\; X\text{'}\; \backslash in\; \backslash mathbf\{W\}$ we have a coherence map $\backslash mu\_\{X,\; X\text{'}\}\; :\; \backslash mathcal\{G\}(X)\; \backslash times\; \backslash mathcal\{G\}(X\text{'})\; \backslash rightarrow\; \backslash mathcal\{G\}(X\; \backslash oplus\; X\text{'})$ given by

where $f\; \backslash times\; f\text{'}$ is as in Lemma 2.14.

Remark 4.3. Recall from Remark 2.7 that $\backslash mathbf\{Man\}$ is small, so the collection $\backslash mathcal\{G\}(X)$ of open systems on $X$ is indeed a set.

Remark 4.4. One may also encode an initial condition in $\backslash mathcal\{G\}$ by using $\backslash mathbf\{Man\}\_*$ instead of $\backslash mathbf\{Man\}$ in Remark 2.7 as the default choice of finite product category, where $\backslash mathbf\{Man\}\_*$ is the category of pointed smooth manifolds and base point preserving smooth maps. The base point represents the initialization of the state variables.

We now establish that $\backslash mathcal\{G\}$ is indeed an algebra.

Proposition 4.5. The pair $(\backslash mathcal\{G\},\; \backslash mu)$ of Definition 4.2 is a lax monoidal functor, i.e., $\backslash mathcal\{G\}$ is a $\backslash mathbf\{W\}$-algebra.

Proof. Let $\backslash Phi\; =\; (X,\; Y,\; \backslash varphi)$ and $\backslash Psi\; =\; (Y,\; Z,\; \backslash psi)$ be wiring diagrams in $\backslash mathbf\{W\}$. To show that $\backslash mathcal\{G\}$ is a functor, we must have that $\backslash mathcal\{G\}(\backslash Psi\; \backslash circ\; \backslash Phi)\; =\; \backslash mathcal\{G\}(\backslash Psi)\; \backslash circ\; \backslash mathcal\{G\}(\backslash Phi)$. Immediately we have $\backslash mathcal\{G\}(\backslash Psi\; \backslash circ\; \backslash Phi)S\; =\; S\; =\; \backslash mathcal\{G\}(\backslash Psi)(\backslash mathcal\{G\}(\backslash Phi)S)$.

Now let $h\; :=\; \backslash mathcal\{G\}(\backslash Psi\; \backslash circ\; \backslash Phi)f$ and $k\; :=\; \backslash mathcal\{G\}(\backslash Psi)(\backslash mathcal\{G\}(\backslash Phi)f)$. It suffices to show $h\; =\; k$, or equivalently $(h^\{\backslash text\{in\}\},\; h^\{\backslash text\{out\}\})\; =\; (k^\{\backslash text\{in\}\},\; k^\{\backslash text\{out\}\})$. One readily sees that $h^\{\backslash text\{out\}\}\; =\; k^\{\backslash text\{out\}\}$. We use (8) and (10) to produce the following diagram; showing it commutes is equivalent to proving that $h^\{\backslash text\{in\}\}\; =\; k^\{\backslash text\{in\}\}$.

The commutativity of this diagram, which is dual to the one for associativity in (6), holds in an arbitrary category with products. Although the middle square fails to commute by itself, the composite of the first two maps equalizes it; that is, the two composite morphisms $\backslash overline\{S\}\; \backslash times\; \backslash overline\{Z^\{\backslash text\{in\}\}\}\; \backslash rightarrow\; \backslash overline\{S\}\; \backslash times\; \backslash overline\{X^\{\backslash text\{out\}\}\}\; \backslash times\; \backslash overline\{Y^\{\backslash text\{in\}\}\}$ agree.

Since we proved the analogous result via string diagrams in the proof of Proposition 3.11, we show it concretely using elements this time. Let $(s,\; z)\; \backslash in\; \backslash overline\{S\}\; \backslash times\; \backslash overline\{Z^\{\backslash text\{in\}\}\}$ be an arbitrary element. Composing six morphisms $\backslash overline\{S\}\; \backslash times\; \backslash overline\{Z^\{\backslash text\{in\}\}\}\; \backslash rightarrow\; \backslash overline\{S\}\; \backslash times\; \backslash overline\{X^\{\backslash text\{out\}\}\}\; \backslash times\; \backslash overline\{Y^\{\backslash text\{in\}\}\}$ through the left of the diagram gives the same answer as composing through the right; namely,

Since the diagram commutes, we have shown that $\backslash mathcal\{G\}$ is a functor. To prove that the pair $(\backslash mathcal\{G\},\; \backslash mu)$ constitutes a lax monoidal functor $\backslash mathbf\{W\}\; \backslash rightarrow\; \backslash mathbf\{Set\}$, i.e., a $\backslash mathbf\{W\}$-algebra, we must establish coherence. Since $\backslash mu$ simply consists of a coproduct and a product, this is straightforward and will be omitted. $\backslash square$

As established in Definition 2.1, the coherence map $\backslash mu$ allows us to define the operad algebra $\backslash mathcal\{OG\}$ from $\backslash mathcal\{G\}$. This finally provides the formal setting to consider open dynamical systems over operadic wiring diagrams, such as our motivating one in Figure 1. We note that, in contrast to the trivial equality $\backslash mathcal\{G\}(\backslash Phi)S\; =\; S$ found in Definition 4.2, in the operadic setting we have

This simply means that the set of state variables of the larger box $Y$ is the disjoint union of the state variables of its constituent boxes $X\_i$. Now that we have the tools to revisit Example 4.1, we do so in the following section, but first we will define the subalgebra $\backslash mathcal\{L\}$ to which it belongs—that of linear open systems.

In this section, we define the algebra $\backslash mathcal\{L\}:\; \backslash mathbf\{W\}\_\{\backslash mathbf\{Lin\}\}\; \backslash rightarrow\; \backslash mathbf\{Set\}$, which encodes linear open systems. Here $\backslash mathbf\{W\}\_\{\backslash mathbf\{Lin\}\}$ is the category of $\backslash mathbf\{Lin\}$-typed wiring diagrams, as in Remark 3.2. Of course, one can use Definition 2.1 to construct an operad algebra $\backslash mathcal\{OL\}:\; \backslash mathcal\{OW\}\_\{\backslash mathbf\{Lin\}\}\; \backslash rightarrow\; \backslash mathbf\{Sets\}$.

Before we give a formal definition for $\backslash mathcal\{L\}$, we first provide an alternative description for linear open systems and wiring diagrams in $\backslash mathbf\{W\}\_\{\backslash mathbf\{Lin\}\}$. The category $\backslash mathbf\{Lin\}$ enjoys special properties—in particular it is an additive category, as seen by the fact that there is an equivalence of categories $\backslash mathbf\{Lin\}\; \backslash cong\; \backslash mathbf\{Vect\}\_\{\backslash mathbb\{R\}\}$. Specifically, finite products and finite coproducts are isomorphic. Hence a morphism $f:\; A\_1\; \backslash times\; A\_2\; \backslash rightarrow\; B\_1\; \backslash times\; B\_2$ in $\backslash mathbf\{Lin\}$ canonically decomposes into a matrix equation

This matrix is naturally equivalent to the whole map $f$ by universal properties. We use these to rewrite our relevant $\backslash mathbf\{Lin\}$ maps in Definitions 5.1 and 5.2 below.

Definition 5.1. Suppose that $(M,\; U^\{\backslash text\{in\}\},\; U^\{\backslash text\{out\}\},\; f)$ is a linear open system and hence $f:\; M\; \backslash times\; U^\{\backslash text\{in\}\}\; \backslash rightarrow\; TM\; \backslash times\; U^\{\backslash text\{out\}\}$. Then $f$ decomposes into the four linear maps:

By Definition 2.9, we know $f^\{U,U\}\; =\; 0$. If we let $(m,\; u^\{\backslash text\{in\}\},\; u^\{\backslash text\{out\}\})\; \backslash in\; M\; \backslash times\; U^\{\backslash text\{in\}\}\; \backslash times\; U^\{\backslash text\{out\}\}$, these equations can be organized into a single matrix equation

We will exploit this form in Definition 5.4 to define how $\backslash mathcal\{L\}$ acts on wiring diagrams in terms of one single matrix equation, in place of the seemingly complicated commutative diagrams in (10). To do so, we also recast wiring diagrams in matrix format in Definition 5.2 below.

Definition 5.2. Suppose $\backslash Phi\; =\; (X,\; Y,\; \backslash varphi)$ is a wiring diagram in $\backslash mathbf\{W\}\_\{\backslash mathbf\{Lin\}\}$. Recalling (7), we apply the dependent product functor to $\backslash varphi$:

Since this is a morphism in $\backslash mathbf\{Lin\}$, it can be decomposed into four linear maps

By virtue of the no passing wires condition in Definition 3.1, we must have $\backslash overline\{\backslash varphi\}^\{Y,Y\}\; =\; 0$. We can then, as in (12), organize this information in one single matrix:

Remark 5.3. The bijectivity condition in Definition 3.1 implies that $\backslash overline\{\backslash varphi\}$ is a permutation matrix.

We now employ these matrix characterizations to define the algebra $\backslash mathcal\{L\}$ of linear open systems.

Definition 5.4. We define the algebra $\backslash mathcal\{L\}\; :\; (\backslash mathbf\{W\}\_\{\backslash text\{Lin\}\},\; \backslash oplus,\; 0)\; \backslash rightarrow\; (\backslash mathbf\{Set\},\; \backslash times,\; \backslash star)$ as follows. Let $X\; \backslash in\; \backslash text\{Ob\; \}\; \backslash mathbf\{W\}\_\{\backslash text\{Lin\}\}$. Then the set of linear open systems $\backslash mathcal\{L\}(X)$ on $X$ is defined as

Let $\backslash Phi\; =\; (X,\; Y,\; \backslash varphi)$ be a wiring diagram. Then, as in Definition 4.2, we define $\backslash mathcal\{L\}(\backslash Phi)(S,\; f)\; :=\; (S,\; g)$. We use the format of Definitions 5.1 and 5.2 to define $g$:

This is really just a linear version of the commutative diagrams in (10). For example, the equation $g^\{S,S\}\; =\; f^\{S,X\}\; \backslash overline\{\backslash varphi\}^\{X,X\}\; f^\{X,S\}\; +\; f^\{S,S\}$ can be read off the diagram for $g^\{\backslash text\{in\}\}$ in (10), using the additivity of $\backslash mathbf\{Lin\}$.

Finally, The coherence map $\backslash mu\_\{\backslash text\{Lin\}\_\{X,X\text{'}\}\}\; :\; \backslash mathcal\{L\}(X)\; \backslash times\; \backslash mathcal\{L\}(X\text{'})\; \backslash rightarrow\; \backslash mathcal\{L\}(X\; \backslash oplus\; X\text{'})$ is given, as in Definition 4.2, by $((S,\; f),\; (S\text{'},\; f\text{'}))\; \backslash mapsto\; (S\; +\; S\text{'},\; f\; \backslash times\; f\text{'})$.

We now establish that this constitutes an algebra.

Proposition 5.5. The pair $(\backslash mathcal\{L\},\; \backslash mu\_\{\backslash text\{Lin\}\})$ of Definition 5.4 is a lax monoidal functor, i.e. a $\backslash mathbf\{W\}\_\{\backslash text\{Lin\}\}$-algebra.

Proof. Since coherence is identical to that in Proposition 4.5, it will suffice to show functoriality. Let $\backslash Phi\; =\; (X,\; Y,\; \backslash varphi)$ and $\backslash Psi\; =\; (Y,\; Z,\; \backslash psi)$ be wiring diagrams with composition $\backslash Psi\; \backslash circ\; \backslash Phi\; =\; (X,\; Z,\; \backslash omega)$. We now rewrite $\backslash omega$ using a matrix equation in terms of $\backslash overline\{\backslash varphi\}$ and $\backslash overline\{\backslash psi\}$ by recasting (5) in matrix form below.

We now prove that $\backslash mathcal\{L\}(\backslash Psi\; \backslash circ\; \backslash Phi)\; =\; \backslash mathcal\{L\}(\backslash Psi)\; \backslash circ\; \backslash mathcal\{L\}(\backslash Phi)$. We immediately have $\backslash mathcal\{L\}(\backslash Psi\; \backslash circ\; \backslash Phi)S\; =\; S\; =\; \backslash mathcal\{L\}(\backslash Psi)(\backslash mathcal\{L\}(\backslash Phi)S)$. Let $h\; :=\; \backslash mathcal\{L\}(\backslash Psi\; \backslash circ\; \backslash Phi)f$ and $k\; :=\; \backslash mathcal\{L\}(\backslash Psi)(\backslash mathcal\{L\}(\backslash Phi)f)$. We must show $h\; =\; k$. Let $g\; =\; \backslash mathcal\{L\}(\backslash Psi)f$ and $\backslash Psi\; \backslash circ\; \backslash Phi\; =\; (X,\; Z,\; \backslash omega)$. It is then straightforward matrix arithmetic to see that

(15)

Therefore, the pair $(\backslash mathcal\{L\},\; \backslash mu\_\{\backslash text\{Lin\}\})$ constitutes a lax monoidal functor $\backslash mathbf\{W\}\_\{\backslash text\{Lin\}\}\; \backslash rightarrow\; \backslash mathbf\{Set\}$, i.e., a $\backslash mathbf\{W\}\_\{\backslash text\{Lin\}\}$-algebra. $\backslash square$

Remark 5.6. Although we've been referring to $\backslash mathcal\{L\}$ as a subalgebra of $\backslash mathcal\{G\}$, this is technically not the case since they have different source categories. The following diagram illustrates precisely the relationship between the $\backslash mathbf\{W\}\_\{\backslash text\{Lin\}\}$-algebra $\backslash mathcal\{L\}$, defined above, and the $\backslash mathbf\{W\}$-algebra $\backslash mathcal\{G\}$, defined in Section 4.

Here, the natural inclusion $\backslash mathbf\{W\}\_i:\; \backslash mathbf\{W\}\_\{\backslash text\{Lin\}\}\; \backslash hookrightarrow\; \backslash mathbf\{W\}$ corresponds to $i:\; \backslash mathbf\{Lin\}\; \backslash hookrightarrow\; \backslash mathbf\{Man\}$, and we have a natural transformation $\backslash epsilon:\; \backslash mathcal\{L\}\; \backslash rightarrow\; \backslash mathcal\{G\}\; \backslash circ\; i$. Hence for each $X\; \backslash in\; \backslash text\{Ob\; \}\; \backslash mathbf\{W\}\_\{\backslash text\{Lin\}\}$, we have a function $\backslash epsilon\_X:\; \backslash mathcal\{L\}(X)\; \backslash rightarrow\; \backslash mathcal\{G\}(i(X))\; =\; \backslash mathcal\{G\}(X)$ that sends the linear open system $(S,\; f)\; \backslash in\; \backslash mathcal\{L\}(X)$ to the open system $(\backslash mathbf\{TFS\}\_i(S),\; i(f))\; =\; (S,\; f)\; \backslash in\; \backslash mathcal\{G\}(X)$.

As promised, we now reformulate Example 1.1 in terms of our language.

Example 5.7. For the reader's convenience, we reproduce Figure 1 and Table 3.

We can invoke the yoga of Definition 5.2 to write $\backslash overline\{\backslash varphi\}$ as a matrix below:

One can think of $\backslash overline\{\backslash varphi\}$ as a block permutation matrix consisting of identity and zero matrix blocks. An identity matrix in block entry $(i,\; j)$ represents the fact that the port whose state space corresponds to row $i$ and the one whose state space corresponds to column $j$ get linked by $\backslash Phi$. In general, the dimension of each $I$ is equal to the dimension of the corresponding state space and hence the formula in (17) is true, independent of the typing. In the specific example of this system, however, all of these ports are typed in $\backslash mathbb\{R\}$, and so we have $I\; =\; 1$ in (17).

As promised in Example 4.1, we now write the open systems for the $X\_i$ in Figure 1 as elements of $\backslash mathcal\{L\}(X\_i)$. The linear open systems below in (18) represent $f\_1$ and $f\_2$, respectively.

Note the proportion of zeros and ones in the $f$-matrices of (18)—this is perhaps why the making explicit of these details was an afterthought in (9). Because we may have arbitrary nonconstant coefficients, our formalism can capture more intricate systems.

We then use (17) to establish that $X\_\{1b\}^\{\backslash text\{in\}\}\; =\; X\_\{2b\}^\{\backslash text\{out\}\}$ and $X\_\{2b\}^\{\backslash text\{in\}\}\; =\; X\_\{1a\}^\{\backslash text\{out\}\}$. This allows us to recover the equations in (9):

The coherence map in Definition 5.4 gives us the combined tank system:

This system can then be written out as a matrix below

Finally, we can apply formula (13) to (19) above to express as a matrix the open system $(Q,\; g)\; =\; (\backslash Phi)f\; \backslash in\; \backslash mathcal\{L\}(Y)$ for the outer box $Y$.