Networks of open systems

TL;DR

This paper develops an algebraic framework for networks of open systems, enabling construction of complex systems from smaller parts and tracking maps between them, with applications to synchronization phenomena.

Contribution

It unifies algebraic and categorical approaches to systems of systems, allowing for composition and morphisms of open systems within a single structure.

Findings

Provides a new algebraic structure for open systems

Enables construction of large systems from smaller subsystems

Generalizes existing synchronization results

Abstract

Many systems of interest in science and engineering are made up of interacting subsystems. These subsystems, in turn, could be made up of collections of smaller interacting subsystems and so on. In a series of papers David Spivak with collaborators formalized these kinds of structures (systems of systems) as algebras over presentable colored operads. It is also very useful to consider maps between dynamical systems, which in effect amounts to viewing dynamical systems as objects in an appropriate category. This is the point of view taken by DeVille and Lerman in the study of dynamics on networks. The goal of this paper is to describe an algebraic structure that encompasses both approaches to systems of systems. This allows us, on one hand, build new large open systems out of collections of smaller open subsystems and on the other keep track of maps between open systems. Consequently we obtain synchrony results for open systems which generalize the synchrony results of Golubitsky, Stewart and their collaborators for groupoid invariant vector fields on coupled cell networks.

Equations

$$a\mapsto \mathsf{Crl}(a)$$

$$\mathsf{Crl}:({\mathsf{SSub}}^{\mathsf{int}}{)}^{\mathsf{op}}\to \mathsf{Vect},$$

$$R:U\to \mathcal{X}(M),$$

$$\hat{R}:U\times M\to TM$$

$$\hat{R}(u,m)\in T_{m}M\text{ for all }(u,m)\in U\times M.$$

$$M\times U$$

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$$\mathsf{Crl}(Q\stackrel{p}{\to }M):=\{F:Q\to TM\mid \pi \circ F=p\},$$

$$\frac{d}{dt}(p\circ \gamma )(t)=F(\gamma (t)).$$

$$\gamma (t)=(x(t),u(t))$$

$$\frac{d}{dt}x(t)=F(x(t),u(t)).$$

$$Tf\circ X=Y\circ f.$$

$$M$$

$$Q$$

$$a=(a_{\mathsf{tot}}\stackrel{p_a}{\to }{a}_{\mathsf{st}})$$

$$Q$$

$$p_1\times p_2:Q_1\times Q_2\to M_1\times M_2,(p_1\times p_2)(q_1,q_2)=(p_1(q_1),p_2(q_2)).$$

$$(Q_1\times Q_2\stackrel{p_1\times p_2}{\to }{M}_1\times M_2,F_1\times F_2:Q_1\times Q_2\to T{M}_1\times T{M}_2\simeq T(M_1\times M_2)),$$

$$(F_1\times F_2)(q_1,q_2)=(F_1(q_1),F_2(q_2))$$

$$u(a,F)=a.$$

$$\mathsf{Crl}(f):=\{(G,F)\in \mathsf{Crl}(b)\times \mathsf{Crl}(a)\mid F,G\text{ are }f\text{-related}\}.$$

$$\mathsf{Crl}(g)\circ \mathsf{Crl}(f)\subset \mathsf{Crl}(g\circ f).$$

$$S\circ R:=\{(z,x)\in Z\times X\mid \text{ there exist }(z,y)\in S\text{ and }(y^{\prime },x)\in R\text{ with }y=y^{\prime }\}.$$

$$S\circ R\subset S^{\prime }\circ R^{\prime }.$$

$$\mathsf{graph}(W\stackrel{T}{\leftarrow }V):=\{(w,v)\in W\times V\mid w=T(v)\}.$$

$$\mathsf{Crl}(g)\circ \mathsf{Crl}(f)\subset \mathsf{Crl}(g\circ f).$$

$$f:\mathbb{R}\hookrightarrow {\mathbb{R}}^2,f(x)=(x,0)$$

$$g:{\mathbb{R}}^2\hookrightarrow {\mathbb{R}}^3,g(x,y)=(x,y,0).$$

$$\mathsf{Crl}(g)\circ \mathsf{Crl}(f)⊊\mathsf{Crl}(g\circ f).$$

$$M\times U\times V\to M$$

Full text

It is also very useful to consider maps between dynamical systems. This is the point of view taken by DeVille and Lerman in the study of dynamics on networks [DL1], [DL2], [DL3]. The work of DeVille and Lerman was inspired by the coupled cell networks of Golubitsky, Stewart and their collaborators [SGP, GST, GS].

The goal of this paper is to describe an algebraic structure that encompasses both approaches to systems of systems. More specifically we define a double category of open systems and construct a functor from this double category to the double category of vector spaces, linear maps and linear relations. This allows us, on one hand, to build new open systems out of collections of smaller open subsystems and on the other to keep track of maps between open systems. Consequently we obtain synchrony results for open systems which generalize the synchrony results of Golubitsky, Stewart and their collaborators for groupoid invariant vector fields on coupled cell networks.

1. 1 Introduction
2. 2 Open systems
3. 3 Interconnections and networks
4. 4 Networks as morphisms in a colored operad
5. 5 Categories of lists
6. 6 Wiring diagrams
7. 7 Extension of the functor $\mathsf{Crl}$ to the category $(\mathsf{FinSet}/\mathsf{SSub})^{\Leftarrow}$ of lists of submersions
8. 8 Double categories
9. 9 Maps between networks of open systems
10. 10 Networks of manifolds

1. Introduction

Many systems of interest in science and engineering are made up of interacting subsystems. These subsystems, in turn, could be made up of collections of smaller interacting subsystems and so on. These kind of structures have been formalized by David Spivak with collaborators as algebras over presentable colored operads [Sp, RS, VSL]. There are several variants of these operads; they depend on the kinds of systems one is interested in. Since the subsystems are supposed to receive input from other subsystems they are conveniently modeled as open (a.k.a. control) systems; we review open systems in Section 2. Informally an open system is a dynamical system that receive inputs from other systems. There are several formal models of open systems starting with collections of vector fields that depend on parameters. In [VSL] the input-state-output model is used.

One of the fundamental problems in the theory of (closed) dynamical systems is finding or, failing that, proving the existence of equilibria, periodic orbits and, more generally, other invariant submanifolds. This amounts to finding/proving the existence of maps between dynamical systems. For example, a map from a point to our favorite closed system is an equilibrium, maps from circles are periodic orbits, and so on. Thus it is highly desirable to have a systematic way of constructing maps between dynamical systems.

One can view a network as a pattern of interconnection of open system. In [VSL] a network is formalized as a morphism in the colored operad of wiring diagram — the morphism encodes the pattern. In the work of DeVille and Lerman [DL1, DL2, DL3], which was inspired by the coupled cell networks of Golubitsky, Stewart and their collaborators [SGP, GST, GS], networks are encoded by directed graphs. In contrast to [VSL] networks in [DL1, DL2, DL3] are viewed as objects in a category, and the main result is a good notions of a map between networks. The notion leads to a combinatorial recipe for a construction of maps of closed dynamical systems out of appropriate maps of graphs. We will show in this paper that the networks of [DL1, DL2, DL3] can be viewed as particular kinds of morphisms in a colored operad (Proposition 10.3). The morphisms of networks of [DL1, DL2, DL3], on the other hand, have no obvious interpretation in the operadic language.

In this paper we generalize both approaches (directed graphs and operadic) to networks of open systems. This allows us, on one hand, to build new large open systems out of collections of smaller open subsystems and on the other hand keep track of maps between open systems. Consequently we obtain synchrony results for open systems which generalize the synchrony results of Golubitsky, Stewart and their collaborators for groupoid invariant vector fields on coupled cell networks (see for example [SGP, GST, GS]).

Networks of open systems as such are not new. For example, networks of open systems are implicit in the work of Field [F]. They are also implicit in the work of Golubitsky, Pivato, Torok and Stewart [SGP, GST] and their collaborators. Special cases of networks of open systems present in the coupled cell network formalism were made explicit in [DL1, DL2, DL3]. Maps between open systems are not new either. For example the category of open systems has been explicitly introduced by Tabuada and Pappas [TP].

What is new in this paper is a general notion of maps between networks (Definition 9.1) and a dynamical/control system interpretation of these maps (Theorem 9.5). We frame this notion in terms of double categories. In particular the results of this paper subsume and extend the results of [DL1, DL2, DL3], as we explain in Section 10.

The paper assumes that the reader is comfortable with viewing continuous time dynamical systems as vector fields on manifolds. By necessity the paper also uses a certain amount of category theory, which we try to keep down to a minimum. We expect the reader to be comfortable with the universal properties of products and coproducts and have a nodding acquaintance with 2 categories, but not much more than that. Some of the results of the paper are expressed in the language of symmetric monoidal categories and the corresponding colored operads. A reader who may be unfamiliar with monoidal categories may safely skip the corresponding parts of the paper. We also use the language of (strict) double categories. Since double categories are somewhat less common, we do not expect any familiarity with them on the part of the reader. Strict double categories are reviewed in Section 8.

Organization of the paper

We start by recalling the definition of an open system (Definition 2.3) and reviewing the category of open systems of Tabuada and Pappas [TP]. We then constructing a symmetric monoidal category $(\mathsf{SSub}^{\mathsf{int}})^{\mathsf{op}}$ whose objects are surjective submersions. In a coordinate-free approach to control theory a surjective submersion $a$ gives rise to a vector space of $\mathsf{Crl}(a)$ of control (i.e., open) systems. We extend the assignment

[TABLE]

to a morphism of symmetric monoidal categories

[TABLE]

where $\mathsf{Vect}$ is the category of real vector spaces and linear maps with the monoidal product being given by direct sum $\oplus$.

Recall that a symmetric monoidal category $\mathsf{A}$ defines a colored operad ${\mathcal{O}}(\mathsf{A})$. We interpret a morphism in the operad ${\mathcal{O}}((\mathsf{SSub}^{\mathsf{int}})^{\mathsf{op}})$ as a pattern of interconnection of open systems and think of it as a network of open systems. The monoidal functor $\mathsf{Crl}$ turns the colored operad ${\mathcal{O}}(\mathsf{Vect},\oplus)$ into an algebra over the operad ${\mathcal{O}}((\mathsf{SSub}^{\mathsf{int}})^{\mathsf{op}})$ (Section 4).

In Section 5 we review the category of lists $\mathsf{FinSet}/\mathscr{C}$ in a category $\mathscr{C}$. The objects of $\mathsf{FinSet}/\mathscr{C}$ are finite unordered lists of objects of $\mathscr{C}$. This is done to facilitate the comparison of the operad ${\mathcal{O}}((\mathsf{SSub}^{\mathsf{int}})^{\mathsf{op}})$ with the operad of wiring diagrams of [VSL]. There are also other reasons for introducing the categories of lists that will become apparent later. We then revisit the algebra ${\mathcal{O}}\mathsf{Crl}:{\mathcal{O}}((\mathsf{SSub}^{\mathsf{int}})^{\mathsf{op}})\to{\mathcal{O}}\mathsf{Vect}$ introduced earlier in Subsection 5.2.

We carry out the comparison of the operad ${\mathcal{O}}((\mathsf{SSub}^{\mathsf{int}})^{\mathsf{op}})$ with the operad of wiring diagrams in Section 6. The main difference between the two operads and their respective algebras is philosophical. Namely, the approach of [VSL] is to treat an open system as a black box — the space of internal states is completely unknown while the algebra supplies all possible choices of internal state spaces. By contrast in this paper we treat the space of internal states (and the total space) as known and have the algebra supply the possible choices of dynamics on a given total space.

The next two sections are technical. The main result of Section 7 is Lemma 7.1. This lemma, in effect, is half of the proof of the main theorem of the paper, Theorem 9.5. The results of Section 7 are used to motivate the introduction of double categories, which is carried out in Section 8. The two main results of Section 8 are Lemma 8.8 (which is a reformulation of Lemma 7.1 in terms of double categories) and Lemma 8.12.

Finally in Section 9 we introduce our notion of maps between networks (Definition 9.1) and interpret it in terms of maps of open systems (Theorem 9.5). In Section 10 we show that the networks of [DL1] (hence the coupled cell networks of Golubitsky, Stewart et al.) are a special case of the networks in the sense of Definition 3.10. We then show that Theorem 3 of [DL1] (which is the main result of that paper) is an easy consequence of Theorem 9.5.

Acknowledgments

I thank Tobias Fritz, Joachim Kock and John Baez for a number of helpful comments.

This paper started out as a joint project with David Spivak. An earlier version of the paper is [LS].

2. Open systems

In this section we define open/control systems and maps between them. We then construct the functor $\mathsf{Crl}$ which assigns to a surjective submersion the vector space of all control systems supported by the submersion. The tricky part is figuring out the target category of $\mathsf{Crl}$.

Informally an open (a.k.a. a control) system is a dynamical system that receive inputs from other systems. There are several formal models of open systems. The simplest has the following form. Fix a manifold $M$ of internal states of the system and another manifold $U$ (the space of parameters). An open system is a map (of an appropriate regularity)

[TABLE]

where $\mathscr{X}(M)$ is a space of vector fields on $M$. Any map $R:U\to\mathscr{X}(M)$ corresponds to a map

[TABLE]

with the property that

[TABLE]

Equation (2.1) is equivalent to the commutativity of the diagram

[TABLE]

where $p:M\times U\to M$ is the projection on the first factor and $\pi:TM\to M$ is the canonical projection from the tangent bundle of $M$ to its base. We think of the manifold $M\times U$ as the total space of the open system with the factor $U$ representing the space of inputs or of control variables (we use the words “inputs” and “controls” interchangeably). However in many control systems of interests the factorization of the total space into internal states and inputs is not natural. For this reason we adopt a somewhat more general definition of a open/control system. Note that the map $p$ in (2.2) is a surjective submersion. Following Brockett [Bro] and Tabuada and Pappas [TP] we define a (continuous-time) open system as follows:

Definition 2.3** (open system).**

A continuous time open system on a surjective submersion $p:Q\to M$ is a smooth map $F:Q\to TM$ so that $F(q)\in T_{p(q)}M$ for all $q\in Q$. That is, the following diagram commutes:

[TABLE]

Thus an open system (or a control system) is a pair $(Q\xrightarrow{p}M,F)$ where $p$ is a surjective submersion and $F:Q\to TM$ is a smooth map satisfying (2.4). We refer to the manifold $Q$ as the total space and of the manifold $M$ as the state space.

Remark 2.5**.**

In the case when $Q=M\times U$ for some manifold $U$ and the surjective submersion $p:M\times U\to M$ is the projection on the first factor, we think of $U$ as the space of input variables and say that the open system $F:M\times U\to TM$ is an open system with a choice of factorization of the total space into inputs and states.

Note that in general even if a surjective submersion $p:Q\to M$ is a trivial fiber bundle with a typical fiber $U$ there may not be a preferred choice of a factorization $Q\xrightarrow{\simeq}M\times U$. The lack of natural factorization of variables of open systems into inputs and states has been emphasized by Willems [W].

Remark 2.6**.**

Fix a surjective submersion $p:Q\to M$. The set

[TABLE]

of all control systems for the given submersion $p$ has the structure of an infinite dimensional real vector space.

Remark 2.8**.**

If the surjective submersion $Q\to M$ is the identity map $\mathsf{id}:M\to M$ then the space of open systems $\mathsf{Crl}(M\xrightarrow{\mathsf{id}}M)$ is the space $\mathscr{X}(M)$ of vector fields on the manifold $M$. Thus closed systems (i.e., vector fields) can be thought of as open systems whose space of inputs is a point (provided we suppress the diffeomorphism $M\times\{*\}\xrightarrow{\,\simeq\,}M$.)

Just as vector fields have trajectories, so do open systems.

Definition 2.9**.**

A trajectory of an open system $(Q\xrightarrow{p}M,F:Q\to TM)$ is a curve $\gamma:(a,b)\to Q$ so that

[TABLE]

for all $t$ in the open interval $(a,b)$.

Remark 2.10**.**

• •

If $Q=M$ and $p$ is the identity map, the definition above reduces to the definition of an integral curve of a vector field.

• •

if $Q=M\times U$ and $p:M\times U\to M$ is the projection on the first factor then a trajectory $\gamma$ of an open system $F:M\times U\to TM$ is of the form

[TABLE]

with

[TABLE]

Such a definition of a trajectory is very common in the control theory literature.

Recall that a vector field $X$ on a manifold $M$ is $f$-related to a vector field $Y$ on a manifold $M$ (where $f:M\to N$ is a smooth map of manifolds) if

[TABLE]

Here and elsewhere in the paper $Tf:TM\to TN$ denotes the differential of $f$. Equivalently the diagram

[TABLE]

commutes. Recall also that if $X\in\mathscr{X}(M)$ is $M\xrightarrow{f}N$-related to $Y\in\mathscr{X}(N)$ and $\gamma$ is a trajectory (i.e., an integral curve) of $X$ then $f\circ\gamma$ is an integral curve of $Y$.

The analogues results holds for open systems. To state it we need to first recall the notion of a map of submersions and then the notion of morphism of open systems, where we follow [TP].

Definition 2.13**.**

A morphism $f$ from a submersion $p:Q\to M$ to a submersion $p^{\prime}:Q^{\prime}\to M^{\prime}$ is a pair of maps $f_{\mathsf{tot}}:Q\to Q^{\prime}$, $f_{\mathsf{st}}:M\to M^{\prime}$ so that the following square commutes:

[TABLE]

Definition 2.14** (The category $\mathsf{SSub}$ of surjective submersions).**

Definition 2.13 allows us to turns the collection of surjective submersions into a category. We denote it by $\mathsf{SSub}$. Explicitly an object $a$ of the category $\mathsf{SSub}$ is surjective submersion

[TABLE]

(Here and elsewhere in the paper $a_{\mathsf{tot}}$ stands for the total space of $a$, $a_{\mathsf{st}}$ for the state space and $p_{a}$ is the submersions). A morphism $a\xrightarrow{f}b$ in the category $\mathsf{SSub}$ is a map of submersions. That is, it is a pair of smooth maps $(f_{\mathsf{tot}},f_{\mathsf{st}})$ so that $p_{b}\circ f_{\mathsf{tot}}=f_{\mathsf{st}}\circ p_{a}$.

Notation 2.15**.**

We will use two types of notation for surjective submersions: $a=(a_{\mathsf{tot}}\xrightarrow{p_{a}}a_{\mathsf{st}})$ and $Q\xrightarrow{p}M$. The usage will depend on convenience.

Remark 2.16**.**

Surjective submersions are also known as fibered manifolds. The term “fibered manifold” goes back to Seifert and Whitney and has been in use since the early 1930’s.

Definition 2.17** (cf. [TP, Definition 4.1]).**

A morphism from an open system $(Q\to M,F:Q\to TM)$ to an open system $(Q^{\prime}\to M^{\prime},F^{\prime}:Q^{\prime}\to TM^{\prime})$ is a morphism of submersions $f=(f_{\mathsf{tot}},f_{\mathsf{st}}):(Q\to M)\to(Q^{\prime}\to M^{\prime})$ for which the following diagram commutes:

[TABLE]

In this case, we say that the open systems $(Q\to M,F)$ and $(Q^{\prime}\to M^{\prime},F^{\prime})$ are $f$-related.

Remark 2.18**.**

It is easy to see that if $f:(Q\to M,F)\to(Q^{\prime}\to M^{\prime})$ is a morphism of an open systems and $\gamma$ is a trajectory of $F$ then $f\circ\gamma$ is a trajectory of $F^{\prime}$.

We can now in position to recall the definition of the category $\mathsf{OS}$ of open systems (it is called $\mathsf{Con}$ for control in [TP]).

Definition 2.19** (The category $\mathsf{OS}$).**

The objects of the category $\mathsf{OS}$ of open systems are open systems $(Q\to M,F)$ as in Definition 2.3. Morphisms of $\mathsf{OS}$ are morphisms of open systems as in Definition 2.17.

Remark 2.20**.**

The categories $\mathsf{SSub}$ of surjective submersions and the category $\mathsf{OS}$ of open systems have finite products. The product of two surjective submersions $Q_{i}\xrightarrow{p_{i}}M_{i}$, $i=1,2$, is the submersion

[TABLE]

The product of two open systems $(Q_{i}\xrightarrow{p_{i}}M_{i},F_{i}:Q_{i}\to TM_{i})$, $i=1,2$, is the open system

[TABLE]

where

[TABLE]

for all $(q_{1},q_{2})\in Q_{1}\times Q_{2}$.

Remark 2.21**.**

We have the evident forgetful functor $u:\mathsf{OS}\to\mathsf{SSub}$ from open systems to submersions that forgets the dynamics. That is, on objects the functor $u$ is given by

[TABLE]

Note that the functor $u$ preserves finite products.

For a surjective submersion $a$ the fiber $u^{-1}(a)$ of the functor $u:\mathsf{OS}\to\mathsf{SSub}$ is (isomorphic to) the space $\mathsf{Crl}(a)$. This suggest that the assignment which sends a surjective submersion $a$ to the space of open systems $\mathsf{Crl}(a)$ should extend to a functor from the category $\mathsf{SSub}$ to some category. The objects of the target category should be real vector spaces since $\mathsf{Crl}(a)$ is a real vector space. But what are the morphisms? A map $f:a\to b$ between two submersions does not in general give rise to a linear map from $\mathsf{Crl}(a)$ to $\mathsf{Crl}(b)$. However a morphism of submersions $f:a\to b$ defines a linear relation

[TABLE]

This suggests that the assignment $a\mapsto\mathsf{Crl}(a)$ extends to a functor from the category $\mathsf{SSub}$ of submersions to the category whose objects are vector spaces and morphisms are linear relations. This is not quite correct. If $f:a\to b$ and $g:b\to c$ are two morphisms of submersions then it easy to see that

[TABLE]

However in general there is no reason for the inclusion to be an equality of linear subspaces. In fact the inclusion can be strict (see Example 2.25 below). Thus our best hope is to make $\mathsf{Crl}$ into a lax 2-functor with the target 2-category $\mathsf{LinRel}$ of vector spaces, linear relations and inclusions. We now proceed to formally define the 2-category $\mathsf{LinRel}$.

Definition 2.23** **(The 2-category $\mathsf{LinRel}$ of real vector spaces,

linear relations and inclusions).

The objects of the category $\mathsf{LinRel}$ are (real) vector spaces. A 1-morphism from a vector space $V$ to a vector space $W$ is a subspace $R\subset W\times V$. We write R:V\leavevmode\hbox to18pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 8.99997pt\lower-3.09982pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.99997pt}{-2.99998pt}\pgfsys@invoke{ }\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip 3.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 3.0pt\hfil&\hfil\hskip 14.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 3.0pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-5.79997pt}{-0.49998pt}\pgfsys@lineto{5.40005pt}{-0.49998pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.60004pt}{-0.49998pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{1.22223pt}\pgfsys@lineto{0.0pt}{-2.2222pt}\pgfsys@stroke\pgfsys@invoke{ } }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}W and say that $R$ is a linear relation from $V$ to $W$.

Given two linear relations $S\subset Z\times Y$ and $R\subset Y\times X$ we define their composite $S\circ R$ to be the linear relation

[TABLE]

It is easy to see that the composition of relations is associative; hence vector spaces and linear relations form a category.

A 2-morphism in $\mathsf{LinRel}$ from a relation R:V\leavevmode\hbox to18pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 8.99997pt\lower-3.09982pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.99997pt}{-2.99998pt}\pgfsys@invoke{ }\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip 3.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 3.0pt\hfil&\hfil\hskip 14.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 3.0pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-5.79997pt}{-0.49998pt}\pgfsys@lineto{5.40005pt}{-0.49998pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.60004pt}{-0.49998pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{1.22223pt}\pgfsys@lineto{0.0pt}{-2.2222pt}\pgfsys@stroke\pgfsys@invoke{ } }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}W to a relation S:V\leavevmode\hbox to18pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 8.99997pt\lower-3.09982pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.99997pt}{-2.99998pt}\pgfsys@invoke{ }\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip 3.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 3.0pt\hfil&\hfil\hskip 14.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 3.0pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-5.79997pt}{-0.49998pt}\pgfsys@lineto{5.40005pt}{-0.49998pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.60004pt}{-0.49998pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{1.22223pt}\pgfsys@lineto{0.0pt}{-2.2222pt}\pgfsys@stroke\pgfsys@invoke{ } }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}W is a linear inclusion $R\hookrightarrow S$. We define the vertical composition of an inclusion $R_{1}\subset R_{2}\subset Y\times X$ followed by the inclusion $R_{2}\subset R_{3}\subset Y\times X$ to be the inclusion $R_{1}\subset R_{3}$. It is easy to check that if $S\subset S^{\prime}\subset Z\times Y$ and $R\subset R^{\prime}\subset Y\times X$ are two pairs of inclusions of linear relations then

[TABLE]

Consequently vector spaces, linear relations and inclusions form a (strict) 2-category. We denote it by $\mathsf{LinRel}$.

Remark 2.24**.**

There is a functor $\mathsf{graph}$ from the category $\mathsf{Vect}$ of vector spaces and linear maps to the underlying 1-category of $\mathsf{LinRel}$. The functor does nothing on objects and is defined on arrows by

[TABLE]

$\Box$

Given two composible morphisms of submersions $f:a\to b$ and $g:b\to c$ it is easy to see that

[TABLE]

In general there is no reason for the inclusion to be an equality of linear subspaces. Hence $\mathsf{Crl}$ is a lax 2-functor. Here is an example.

Example 2.25**.**

Recall that an open system on a submersion of the form $id_{M}:M\to M$ is a vector field on the manifold $M$. Now consider the pair of embeddings

[TABLE]

and

[TABLE]

The constant vector field $\frac{d}{dx}$ on $\mathbb{R}$ is $(g\circ f)$-related to any vector field $v$ on $\mathbb{R}^{3}$ with $v(x,0,0)=\frac{\partial}{\partial x}$. However, such a vector field $v$ need not be tangent to the $xy$-plane and thus need not be $g$-related to any vector field on $\mathbb{R}^{2}$. Thus in this case

[TABLE]

Remark 2.26**.**

The lax functor $\mathsf{Crl}:\mathsf{SSub}\to\mathsf{LinRel}$ carries all the essential information of the forgetful functor $u:\mathsf{OS}\to\mathsf{SSub}$. That is, for each object $a\in\mathsf{SSub}$, the fiber $u^{-1}(a)$ is isomorphic to $\mathsf{Crl}(a)$, and to each morphism $f:a\to b$ we can associate the linear relation $\mathsf{Crl}(f)$. Thus the functor $u$ is in some ways akin to the Grothendieck construction of $\mathsf{Crl}$, but $u$ is not a fibration of categories.

3. Interconnections and networks

We take the point of view that a network is a pattern of interconnection of a collection of open systems. The goal of this section is to make the previous sentence precise. The idea is to start with a finite unordered list of surjective submersions. Formally such a list is a map $\tau:X\to\mathsf{SSub}$, where $X$ is a finite set and $\mathsf{SSub}$ is the category of surjective submersions defined above. That is, $\tau(x)$ is a surjective submersion for every $x\in X$. A pattern of interconnection is then an appropriate map of surjective submersions $\psi:a\to\prod_{x\in X}\tau(x)$. To explain what maps are appropriate and the intuition behind this definition we start with two examples.

Example 3.1**.**

Let $M,U,V$ be manifolds. Then the projection on the first factor

[TABLE]

is a surjective submersions. Consider an open system $F:M\times U\times V\to TM$. Let $\phi:M\to V$ be a smooth map. Then the map $G:M\times U\to TM$ defined by

[TABLE]

is an open system on the submersion $M\times U\to M$.

Note also that

[TABLE]

where $\varphi:M\times U\to M\times U\times V$ is given by

[TABLE]

We therefore view $\varphi$ as defining a pattern of interconnection of opens systems. Namely $\varphi$ gives rise to the linear map

[TABLE]

Example 3.2**.**

Example 3.1 above gives us a way to view a vector field on a product of two manifold as a pair of interconnected open systems, that is, as a network. Namely let $M,N$ be two manifolds and let $X:M\times N\to TM\times TN$ be a vector field on their product. Then $X$ is of the form

[TABLE]

where $F:M\times N\to TM$ and $G:N\times M\to TN$ are the appropriate smooth maps. In fact it is easy to see that $F$ and $G$ are open systems. Moreover it should be intuitively clear that $X$ is obtained from $F$ and $G$ by plugging the states of the open system $F$ into the inputs of the open system $G$ and the states of $G$ into the inputs of $F$. More precisely consider the product open system

[TABLE]

Then

[TABLE]

where

[TABLE]

is defined by

[TABLE]

$\Box$

The notion of an interconnection map informally introduced above easily generalizes to maps between more general submersions.

Definition 3.3**.**

A morphism $\varphi=(\varphi_{\mathsf{tot}},\varphi_{\mathsf{st}}):a\to b$ between two submersions is a interconnection morphism if $\varphi_{\mathsf{st}}$ is a diffeomorphism.

Remark 3.4**.**

We are interested in interconnection morphisms $\varphi$ with the property that $\varphi_{\mathsf{st}}$ is the identity map. We fear, however, that requiring $\varphi_{\mathsf{st}}$ to be the identity outright may cause trouble.

Remark 3.5**.**

It may be that Definition 3.3 of an “interconnection map” is a bit too general. For example, one may want to additionally insist that $\varphi_{\mathsf{tot}}:a_{\mathsf{tot}}\to b_{\mathsf{tot}}$ is an embedding, though we will not need this restriction in what follows. The additional requirement that $\varphi_{\mathsf{tot}}$ is an embedding would capture the idea that after plugging states into inputs the total space of the systems (inputs and states) should be smaller.

Remark 3.6**.**

An interconnection morphism $\varphi:a\to b$ of Definition 3.3 gives rise to a linear map

[TABLE]

It is given by

[TABLE]

Note that

[TABLE]

(q.v. Remark 2.24).

Definition 3.7** (The category $\mathsf{SSub}^{\mathsf{int}}$ of submersions and interconnection maps).**

The collection of interconnection maps is closed under compositions. Consequently the collections of surjective submersions and interconnection maps forms a subcategory of the category $\mathsf{SSub}$. We denote it by $\mathsf{SSub}^{\mathsf{int}}$.

Remark 3.8**.**

The subcategory $\mathsf{SSub}^{\mathsf{int}}$ of $\mathsf{SSub}$ has the same objects as the category $\mathsf{SSub}$. For any two composible morphisms $c\xleftarrow{\psi}b\xleftarrow{\varphi}a$ of $\mathsf{SSub}^{\mathsf{int}}$ we have

[TABLE]

Therefore we have another way to extend the assignment

[TABLE]

to a functor. Namely we have an evident functor

[TABLE]

which is defined on arrows by

[TABLE]

(see Remark 3.6). At the risk of causing a temporary confusion we will also denote this functor by $\mathsf{Crl}$. We will see later in the paper (Lemma 8.12) that the functors

[TABLE]

are components of a single morphism of double categories.

We are now in position to define a network of open systems.

Definition 3.10**.**

A network of open systems is an unordered list of submersions $\tau:X\to\mathsf{SSub}$ indexed by a finite set $X$ together with an interconnection morphism $\psi:b\to\prod_{x\in X}\tau(x)$.

Example 3.11**.**

Examples 3.1 and 3.2 are both examples of networks of open systems in the sense of Definition 3.10.

In Example 3.1 the “list” of submersions consists of the single submersion $M\times U\times V\to M$, which we can think of a map $\tau:\{*\}\to\mathsf{SSub}$ with $\tau(*)=(M\times U\times V\to M)$. The submersion $b$ is $(M\times U\to M)$. The interconnection map $\varphi:b\to\tau(*)$ is defined by $\varphi_{st}=id_{M}$,

[TABLE]

In Example 3.2 the list of submersions is the map $\tau:\{1,2\}\to\mathsf{SSub}$ with

[TABLE]

The submersion $b$ is $id:M\times N\to M\times N$. The interconnection map $\varphi:b\to\tau(1)\times\tau(2)$ is defined by $\varphi_{\mathsf{st}}=id_{M\times N}$ and

[TABLE]

Remark 3.12** **(Networks as patterns of

interconnections).

A network is a pattern of interconnection of open systems in the following sense. Let $a_{1},\ldots,a_{n}$ be a collection of surjective submersions (so here $X=\{1,\ldots,n\}$ and $\tau:\{1,\ldots,n\}\to\mathsf{SSub}$ is given by $\tau(i)=a_{i}$). Let $\psi:b\to\prod_{i=1}^{n}a_{i}$ be an interconnection morphism. Then given a collection of open systems $\{F_{i}\in\mathsf{Crl}(a_{i})\}_{i=1}^{n}$ we get an open system $F\in\mathsf{Crl}(b)$ which is defined by

[TABLE]

We will say more about networks as patterns in the next section.

We finish the section with one more example of a network. We’ll come back to this example in Example 9.3 and Example 10.6.

Example 3.14**.**

Let $M,U$ be two smooth manifolds, $p:M\times U\to M$ a trivial fiber bundle and $\phi:M\to U$ a smooth map. Let $X$ be a three element set, $X=\{1,2,3\}$, and $\tau:X\to\mathsf{SSub}$ the constant map with $\tau(1)=\tau(20=\tau(3)=(p:M\times U\to M).$ We choose $b$ to be the trivial submersion $\mathsf{id}_{M^{3}}:M^{3}\to M^{3}$ and

[TABLE]

to be the interconnection morphism with $\psi_{\mathsf{tot}}$ given by

[TABLE]

Then $(\tau:X\to\mathsf{SSub},\psi:b\to\prod_{x\in X}\tau(x))$ is a network in the sense of Definition 3.10.

Note that for any three open systems $F_{1},F_{2},F_{3}\in\mathsf{Crl}(M\times U\to M)$ the system

[TABLE]

is a vector field on $M^{3}$ given by

[TABLE]

4. Networks as morphisms in a colored operad

We now give networks in the sense of Definition 3.10 an operadic interpretation. Recall that if $\mathsf{C}$ is a symmetric monoidal category with the monoidal product $\otimes$ the corresponding (representable colored) operad ${\mathcal{O}}\mathsf{C}$ has the same objects as $\mathsf{C}$. For any objects $a_{1},\ldots,a_{n},b$ of $\mathsf{C}$ the set $\operatorname{\mathsf{Hom}}_{{\mathcal{O}}\mathsf{C}}(a_{1},\ldots,a_{n}:b)$ of morphisms in the operad ${\mathcal{O}}\mathsf{C}$ from $a_{1},\ldots,a_{n}$ to $b$ is defined to be $\operatorname{\mathsf{Hom}}_{\mathsf{C}}(a_{1}\otimes\cdots\otimes a_{n},b)$:

[TABLE]

Since $\mathsf{SSub}$ has finite products, it is a Cartesian symmetric monoidal category. It is easy to see that the product of two interconnection morphisms is again an interconnection morphisms. Consequently $\mathsf{SSub}^{\mathsf{int}}$ inherits from $\mathsf{SSub}$ the structure of a symmetric monoidal category.111Note that $\mathsf{SSub}^{\mathsf{int}}$ is not Cartesian. The issue is that if $a_{1}$, $a_{2}$ are two surjective submersions then the two projection maps $\pi_{i}:a_{1}\times a_{2}\to a_{i}$ ($i=1,2$) are not (in general) morphisms in $\mathsf{SSub}^{\mathsf{int}}$, so $\mathsf{SSub}^{\mathsf{int}}$ does not have products. Consequently the opposite category $(\mathsf{SSub}^{\mathsf{int}})^{\mathsf{op}}$ is also symmetric monoidal. Now for any $n+1$ objects $a_{1},\ldots,a_{n},b\in\mathsf{SSub}^{\mathsf{int}}$ a morphism in the operad ${\mathcal{O}}((\mathsf{SSub}^{\mathsf{int}})^{\mathsf{op}})$ is exactly a network in the sense of Definition 3.10.

We next interpret (3.13) in terms of an algebra over the operad ${\mathcal{O}}((\mathsf{SSub}^{\mathsf{int}})^{\mathsf{op}})$. In order to carry this out we view the category $\mathsf{Vect}$ of (real) vector spaces as a symmetric monoidal category with the monoidal product being the direct sum $\oplus$.

Lemma 4.1**.**

The functor

[TABLE]

defined by (3.9) is a lax monoidal functor.

Proof.

For any two surjective submersions $a,b$ we have a linear map

[TABLE]

which is given by

[TABLE]

for all $(F,G)\in\mathsf{Crl}(a)\oplus\mathsf{Crl}(b)$. It is easy to see that for any two interconnection morphisms $f:a\to a^{\prime}$, $g:b\to b^{\prime}$ the diagram

[TABLE]

commutes. ∎

Since the functor $\mathsf{Crl}:(\mathsf{SSub}^{\mathsf{int}})^{\mathsf{op}}\to\mathsf{Vect}$ is monoidal, it induces a map of colored operads

[TABLE]

Therefore for any morphism

[TABLE]

we get a morphism

[TABLE]

The linear map ${\mathcal{O}}\mathsf{Crl}(\psi):\oplus_{i}\mathsf{Crl}(a_{i})\to\mathsf{Crl}(b)$ is given by

[TABLE]

for any $(F_{1},\ldots,F_{n})\in\oplus_{i=1}^{n}\mathsf{Crl}(a_{i})$.

We would next like to give and justify a meaningful notion of a map between networks. Our strategy is to first discuss morphisms between lists of submersions. We carry this out in the next section. Maps between networks themselves will be defined in Section 9 after further preparation.

5. Categories of lists

Think of sets as discrete categories. Then for any category $\mathscr{C}$ we have the category $\mathsf{FinSet}/\mathscr{C}$. By definition its objects are functors of the form $\tau:X\to\mathscr{C}$, where $X$ is a finite set (i.e., a finite discrete category). Morphisms are strictly commuting triangles of the form

[TABLE]

where $\varphi$ is a map of finite sets. We think of an object $(X\xrightarrow{\tau}\mathscr{C})$ of $\mathsf{FinSet}/\mathscr{C}$ as unordered list $\{\tau(a)\}_{a\in X}$ of objects of the category $\mathscr{C}$ indexed by the finite set $X$. The composition of morphisms in $\mathsf{FinSet}/\mathscr{C}$ is given by pasting triangles together.

Remark 5.1**.**

In [VSL] the category $\mathsf{FinSet}/\mathscr{C}$ is called the category of typed finite sets of type $\mathscr{C}$.

Remark 5.2**.**

If the category $\mathscr{C}$ has all finite products there is a canonical functor

[TABLE]

On objects the functor $\mathbb{P}_{\mathscr{C}}$ is defined by

[TABLE]

On a morphism

[TABLE]

the functor $\mathbb{P}_{\mathscr{C}}$ is defined by requiring that the diagram

[TABLE]

commutes for all $a\in X$. Here and elsewhere $\pi_{a}:\mathbb{P}_{\mathscr{C}}(\tau)=\prod_{x\in X}\tau(x)\to\tau(a)$ is the projection on the $a$th factor (i.e., one of the structure maps of the categorical product) and $\pi_{\varphi(a)}$ is defined similarly.

Since the objects of the category $\mathsf{FinSet}/\mathscr{C}$ are functors it is natural to modify the morphisms by allowing the triangles to be 2-commutative rather than strictly commutative. There are two choices for the direction of the 2-arrow. If $\mathscr{C}$ has finite products it is natural to choose 2-commuting triangles of the form

[TABLE]

as morphisms between lists. We denote this variant of $\mathsf{FinSet}/\mathscr{C}$ by $(\mathsf{FinSet}/\mathscr{C})^{\Leftarrow}$. The reason why this is “natural” is that if $\mathscr{C}$ has finite products we again have a functor

[TABLE]

which is defined on objects by

[TABLE]

To extend the definition of $\mathbb{P}_{\mathscr{C}}$ to morphisms, we define

[TABLE]

by requiring that the diagram

[TABLE]

commutes for all $a\in X$.

Here is an example of $(\mathsf{FinSet}/\mathscr{C})^{\Leftarrow}$ that we very much care about: take $\mathscr{C}=\mathsf{SSub}$, the category of surjective submersions. Since $\mathsf{SSub}$ has finite products we have a contravariant functor

[TABLE]

Other categories of interest are $\mathscr{C}=\mathsf{Man}$, the category of manifolds and $\mathscr{C}=\mathsf{Vect}$, the category of vector spaces and linear maps.

Remark 5.5**.**

There is a canonical faithful functor $\jmath:\mathsf{FinSet}/\mathscr{C}\hookrightarrow(\mathsf{FinSet}/\mathscr{C})^{\Leftarrow}$ which is identity on objects. The functor $\jmath$ sends an arrow $\varphi:(X\xrightarrow{\tau}\mathscr{C})\to(Y\xrightarrow{\mu}\mathscr{C})$ in $\mathsf{FinSet}/\mathsf{SSub}$ to the arrow

[TABLE]

in $(\mathsf{FinSet}/\mathsf{SSub})^{\Leftarrow}$ (note that $\tau=\mu\circ\varphi$). If the category $\mathscr{C}$ has finite products, the diagram

[TABLE]

commutes.

Remark 5.6**.**

Note also that given two objects $X\xrightarrow{\tau}\mathscr{C}$ and $Y\xrightarrow{\mu}\mathscr{C}$ in $\mathsf{FinSet}/\mathscr{C}$ (or in $(\mathsf{FinSet}/\mathscr{C})^{\Leftarrow}$) we have the new object

[TABLE]

which is defined by taking disjoint unions (i.e., coproducts). Moreover

[TABLE]

where $\mathbb{P}_{\mathscr{C}}(\tau)\times\mathbb{P}_{\mathscr{C}}(\mu)$ is the categorical product of $\mathbb{P}_{\mathscr{C}}(\tau)$ and $\mathbb{P}_{\mathscr{C}}(\mu)$ in $\mathscr{C}$.

5.1. The functor $\odot:((\mathsf{FinSet}/\mathsf{LinRel})^{\Leftarrow})^{\mathsf{op}}\to\mathsf{LinRel}$

If $\mathscr{C}=\mathsf{LinRel}$, the (2-)category of vector spaces and linear relations, then the category $(\mathsf{FinSet}/\mathsf{LinRel})^{\Leftarrow}$ of finite unordered lists of vector spaces still makes sense. However the existence of an extension of the assignment

[TABLE]

to a functor is a bit more delicate since the direct sum $\oplus$ is not a product in $\mathsf{LinRel}$. This said, given a finite list $\tau:X\to\mathsf{LinRel}$ of vector spaces and an arbitrary vector space $Z$ there is a canonical map

[TABLE]

It assigns to a collection of the subspaces $\{R_{a}\subset\tau(a)\times Z\}_{a\in X}$ (that is, to a collection \{R_{a}:Z\leavevmode\hbox to18pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 8.99997pt\lower-3.09982pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.99997pt}{-2.99998pt}\pgfsys@invoke{ }\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip 3.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 3.0pt\hfil&\hfil\hskip 14.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 3.0pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-5.79997pt}{-0.49998pt}\pgfsys@lineto{5.40005pt}{-0.49998pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.60004pt}{-0.49998pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{1.22223pt}\pgfsys@lineto{0.0pt}{-2.2222pt}\pgfsys@stroke\pgfsys@invoke{ } }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\tau(a)\}_{a\in X} of arrows in $\mathsf{LinRel}$) the intersection

[TABLE]

Here $\pi_{a}:\oplus_{x\in X}\tau(x)\to\tau(a)$ are the canonical projections.

Proposition 5.8**.**

The assignment

[TABLE]

extends to a lax functor

[TABLE]

Proof.

Given a list $\tau\in(\mathsf{FinSet}/\mathsf{LinRel})^{\Leftarrow}$ we define

[TABLE]

Given a 2-commuting triangle

[TABLE]

we set

[TABLE]

where

[TABLE]

are the projections and the relations \Phi(a):\mu(\varphi(a)\leavevmode\hbox to18pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 8.99997pt\lower-3.09982pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.99997pt}{-2.99998pt}\pgfsys@invoke{ }\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip 3.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 3.0pt\hfil&\hfil\hskip 14.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 3.0pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-5.79997pt}{-0.49998pt}\pgfsys@lineto{5.40005pt}{-0.49998pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.60004pt}{-0.49998pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{1.22223pt}\pgfsys@lineto{0.0pt}{-2.2222pt}\pgfsys@stroke\pgfsys@invoke{ } }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\tau(a)) are the component relations of the natural transformation $\Phi:\mu\circ\varphi\Rightarrow\tau$. It remains to check that given a pair of triangles

[TABLE]

that can be composed (pasted together) we have

[TABLE]

This is a computation. By definition (see (5.7) and (5.9)) we have

[TABLE]

Hence

[TABLE]

The left hand side, on the other hand, is a subspace of

[TABLE]

which is given by

[TABLE]

Since the right hand side of (5.10) is contained in the right hand side of (5.11), the result follows. ∎

Example 5.12**.**

We use the notation of Proposition 5.8 above. Suppose that $X=\{1,2,3\}$, $\tau:X\to\mathsf{LinRel}$ is a function that assigns to each $i\in X$ the same vector space $W$, $Y$ is a one point set $\{*\}$, $\mu(*)$ is some vector space $V$, and $\Phi(i)\subseteq\tau(i)\times\mu(*)=W\times V$, $i=1,2,3$, are some linear relations. Then $\odot(\varphi,\Phi)\subseteq W^{3}\times V$ is the relation

[TABLE]

$\Box$

The following lemma will prove useful in computing examples.

Lemma 5.13**.**

If the components \Phi(a):\mu(\varphi(a))\leavevmode\hbox to18pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 8.99997pt\lower-3.09982pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.99997pt}{-2.99998pt}\pgfsys@invoke{ }\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip 3.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 3.0pt\hfil&\hfil\hskip 14.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 3.0pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-5.79997pt}{-0.49998pt}\pgfsys@lineto{5.40005pt}{-0.49998pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.60004pt}{-0.49998pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{1.22223pt}\pgfsys@lineto{0.0pt}{-2.2222pt}\pgfsys@stroke\pgfsys@invoke{ } }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\tau(a) of the natural transformation $\Phi:\mu\circ\varphi\Rightarrow\tau$ in the morphism

[TABLE]

in the category $\mathsf{FinSet}/\mathsf{LinRel})^{\Leftarrow}$ are graphs of linear maps $\phi(a):\mu(\varphi(a))\to\tau(a)$, i.e., $\Phi(a)=\mathsf{graph}(\phi(a))$, then the relation \odot(\varphi,\Phi):\odot(\mu)\leavevmode\hbox to18pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 8.99997pt\lower-3.09982pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.99997pt}{-2.99998pt}\pgfsys@invoke{ }\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip 3.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 3.0pt\hfil&\hfil\hskip 14.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 3.0pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-5.79997pt}{-0.49998pt}\pgfsys@lineto{5.40005pt}{-0.49998pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.60004pt}{-0.49998pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{1.22223pt}\pgfsys@lineto{0.0pt}{-2.2222pt}\pgfsys@stroke\pgfsys@invoke{ } }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\odot(\tau) is the graph of a linear map.

Proof.

The linear map

[TABLE]

in question is uniquely defined by requiring that the diagrams

[TABLE]

commute for all $a\in X$. The map $\oplus(\varphi,\Phi)$ is the required map since

[TABLE]

∎

5.2. The algebra ${\mathcal{O}}\mathsf{Crl}:{\mathcal{O}}((\mathsf{SSub}^{\mathsf{int}})^{\mathsf{op}})\to{\mathcal{O}}\mathsf{Vect}$ in

terms of lists

In discussing colored operads in Section 4 we swept a few details under the rug. We now revisit the discussion and revise Remark 3.12.

Given a symmetric monoidal category $(\mathsf{C},\otimes)$ a morphism in the corresponding colored operad ${\mathcal{O}}\mathsf{C}$ has more generally as its source a finite list $X\xrightarrow{\tau}\mathsf{C}$. A morphism in ${\mathcal{O}}\mathsf{C}$ with the source $X\xrightarrow{\tau}\mathsf{C}$ and target $b\in\mathsf{C}$ is a morphism in $\mathsf{C}$ from the product $x_{1}\otimes(x_{2}\otimes(\cdots\otimes x_{n})\ldots)$ for some choice of a bijection $\nu:\{1,\ldots,n\}\to X$ (where we set $x_{i}:=\tau(\nu(i))$). We make such a choice for each list in $\mathsf{FinSet}/\mathsf{C}$. The choices don’t matter thanks to Mac Lane’s coherence theorem.

Since the functor $\mathsf{Crl}:((\mathsf{SSub}^{\mathsf{int}})^{\mathsf{op}},\times)\to(\mathsf{Vect},\oplus)$ is lax monoidal (see Lemma 4.1) for any ordered list $a:\{1,\ldots,n\}\to\mathsf{SSub}$ of submersions we have a canonical linear map

[TABLE]

It is given by

[TABLE]

Lemma 5.16**.**

For any unordered list $\tau:X\to\mathsf{SSub}$ we have a canonical linear map

[TABLE]

so that if $X=\{1,\ldots,n\}$ then $\mathsf{Crl}_{\tau}$ is given by (5.15).

Proof.

We start by introducing notation. For each $a\in X$ we have a submersion $\tau(a)$ which we write as

[TABLE]

We denote the projections from $\bigoplus_{x\in X}\mathsf{Crl}(\tau(x))$ to $\mathsf{Crl}(\tau(a))$ by $\varpi_{a}$:

[TABLE]

Since $\mathbb{P}_{\mathsf{SSub}}(\tau)=\prod_{a\in X}\tau(a)$ we have canonical projections

[TABLE]

Each $\pi_{a}$ is a pair of maps of manifolds:

[TABLE]

We set

[TABLE]

We define the projections

[TABLE]

by

[TABLE]

for all $\psi\in\mathsf{Crl}(\mathbb{P}_{\mathsf{SSub}}(\tau))$. Since $T(\prod_{x\in X}\tau(x)_{\mathsf{st}})=\prod_{x\in X}T\,\tau(x)_{\mathsf{st}}$, the projections $\Xi_{a}$ make $\mathsf{Crl}(\mathbb{P}_{\mathsf{SSub}}(\tau))$ into a direct sum:

[TABLE]

Finally we have pull-back maps

[TABLE]

[TABLE]

or all $\psi_{a}\in\mathsf{Crl}(\tau(a)_{\mathsf{tot}}\xrightarrow{\upsilon_{a}}\tau(a)_{\mathsf{st}})$. By the universal property of products the family of maps

[TABLE]

uniquely define a linear map $\mathsf{Crl}_{\tau}$ making the diagram

[TABLE]

of vector spaces and linear maps commute. Note that by definition of $\mathsf{Crl}_{\tau}$

[TABLE]

for any $(F_{x})_{x\in X}\in\bigoplus_{x\in X}\mathsf{Crl}(\tau(x))$ and any $a\in X$. ∎

Remark 5.21**.**

It follows from Lemma 5.16 that given a list $X\xrightarrow{\tau}\mathsf{SSub}$ and an interconnection morphism $b\xrightarrow{\psi}\prod_{x\in X}\tau(x)=\mathbb{P}_{\mathsf{SSub}}(\tau)$ we have a linear map

[TABLE]

which is given by

[TABLE]

6. Wiring diagrams

In section 4 we constructed the colored operad ${\mathcal{O}}((\mathsf{SSub}^{\mathsf{int}})^{\mathsf{op}})$ and the algebra

[TABLE]

This algebra is similar to the algebra

[TABLE]

over the operad of ${\mathcal{O}}\mathbf{W}$ of wiring diagrams defined in [VSL]. We now contrast and compare the two operads and the two algebras.

To make the comparison easier we recall the definition of the monoidal category $\mathbf{W}$ of wiring diagrams and the functor $\mathcal{G}:{\bf W}\to\mathsf{Set}$. Note first that in [VSL] open continuous time dynamical systems are viewed differently from the way we have been viewing them in this paper. There an open system consists of three manifolds $M,U^{\mathrm{inp}},U^{\mathrm{out}}$, a smooth map $f^{\mathrm{out}}:M\to U^{\mathrm{out}}$ and an open system $f^{\mathrm{inp}}\in\mathsf{Crl}(M\times U^{\mathrm{inp}}\to M)$. To distinguish the two approaches we will refer to the tuple

[TABLE]

as a factorized open system $f^{\mathrm{inp}}$ with output $f^{\mathrm{out}}$. The manifolds $M,U^{\mathrm{inp}},U^{\mathrm{out}}$ are, respectively, the spaces of states, inputs and outputs of the system $(M,U^{\mathrm{inp}},U^{\mathrm{out}},f^{\mathrm{inp}},f^{\mathrm{out}})$. Factorized open systems with outputs form a category, which in [VSL] is called $\mathsf{ODS}$ (for Open Dynamical Systems). By definition a morphism from $(M_{1},U_{1}^{\mathrm{inp}},U_{1}^{\mathrm{out}},f_{1}^{\mathrm{inp}},f_{1}^{\mathrm{out}})$ to $(M_{2},U^{\mathrm{inp}}_{2},U^{\mathrm{out}}_{2},f^{\mathrm{inp}}_{2},f^{\mathrm{out}}_{2})$ is a triple of maps $\zeta=(\zeta^{\mathsf{st}}:M_{1}\to M_{2},\zeta^{\mathrm{inp}}:U_{1}^{\mathrm{inp}}\to U_{2}^{\mathrm{inp}},\zeta^{\mathrm{out}}:U_{1}^{\mathrm{out}}\to U_{2}^{\mathrm{out}})$ so that the following diagram

[TABLE]

commutes. This category has finite products. The symmetric monoidal category $\mathbf{W}$ is defined as follows. The objects of $\mathbf{W}$ are pairs of unordered lists of manifolds (or, equivalently, pairs of typed finite sets of type “manifold”). Thus by definition an object $X$ of $\mathbf{W}$ is an ordered pair $(\tau^{\mathrm{inp}}:X^{\mathrm{inp}}\to\mathsf{Man},\tau^{\mathrm{out}}:X^{\mathrm{out}}\to\mathsf{Man})$ of objects of $\mathsf{FinSet}/\mathsf{Man}$. The objects of $\mathbf{W}$ are called boxes. The morphisms in $\mathbf{W}$ are called wiring diagrams. A wiring diagram is a triple $(X,Y,\varphi)$ where $X,Y$ are boxes and

[TABLE]

is an isomorphism in $\mathsf{FinSet}/\mathsf{Man}$ (here and below we are suppressing maps to $\mathsf{Man}$) so that

[TABLE]

Condition (6.1) allows us to decompose $\varphi$ into a pair $\varphi=(\varphi^{\mathrm{inp}},\varphi^{\mathrm{out}})$:

[TABLE]

Defining composition of morphisms in $\mathbf{W}$ and proving that composition is associative, that is, proving that $\mathbf{W}$ is actually a category, requires work (see [VSL]). Compare that with the construction of the category $\mathsf{SSub}^{\mathsf{int}}$.

A wire in a wiring diagram $(X,Y,\varphi)$ is a pair $(a,b)$, where $a\in X^{\mathrm{inp}}\sqcup Y^{\mathrm{out}}$, $b\in X^{\mathrm{out}}\sqcup Y^{\mathrm{inp}}$, and $\varphi(a)=b$. The monoidal product on $\mathbf{W}$ is disjoint union:

[TABLE]

The semantics of $\mathbf{W}$ is obtained by filling in the boxes in the following sense. Given a box $X=(X^{\mathrm{inp}},X^{\mathrm{out}})\in(\mathsf{FinSet}/\mathsf{Man})^{2}$ we have a pair of manifolds $(\mathbb{P}X^{\mathrm{inp}},\mathbb{P}X^{\mathrm{out}})$, where as before the functor $\mathbb{P}=\mathbb{P}_{\mathsf{Man}}:(\mathsf{FinSet}/\mathsf{Man})^{\mathsf{op}}\to\mathsf{Man}$ is defined on objects by taking products:

[TABLE]

Therefore a choice of a manifold $M$ defines a product fiber bundle

[TABLE]

We then can further choose an output map $f^{\mathrm{out}}:M\to\mathbb{P}X^{\mathrm{out}}$ and a factorized open system $f^{\mathrm{inp}}:M\times\mathbb{P}X^{\mathrm{inp}}\to TM$. This is the consideration behind the definition of the functor $\mathcal{G}:\mathbf{W}\to\mathsf{Set}$. Its value on an object $X$ of $\mathbf{W}$ is, by definition, the collection

[TABLE]

where $f=f^{\mathrm{inp}}\times f^{\mathrm{out}}$ are factorized open systems with outputs. (To make sure that $\mathcal{G}(X)$ is actually a set and not a bigger collection we should, strictly speaking, replace the category $\mathsf{Man}$ of manifolds by an equivalent small category. For example we can redefine $\mathsf{Man}$ to consist of manifolds that are embedded in the disjoint union $\sqcup_{n\in\mathbb{N}}\mathbb{R}^{n}$.)

We now see that the monoidal category $\mathbf{W}$ is set up so that an object is a kind of black box with wires sticking out. The wires are partitioned into two sets. The first set of wires receive inputs. The other set of wires report outputs. The box is filled with open dynamical systems. By design we have no direct access to the state spaces of these systems. Compare this with the category of $\mathsf{SSub}$ where the objects specify the spaces of states of the systems.

Note also that the functor $\mathcal{G}:\mathbf{W}\to\mathsf{Set}$ is very coarse. For example if we start with a box $X$ whose inputs $X^{\mathrm{inp}}$ and $X^{\mathrm{out}}$ are both singletons $\{*\}$ and $\tau^{\mathrm{inp}}(*)=\tau^{\mathrm{out}}(*)=$ a point then $\mathcal{G}(X)$ is (in bijective correspondence with) the set of all possible continuous time closed dynamical systems.

To conclude our discussion of wiring diagrams, the differences between the algebra ${\mathcal{O}}\mathsf{Crl}:{\mathcal{O}}((\mathsf{SSub}^{\mathsf{int}})^{\mathsf{op}})\to{\mathcal{O}}\mathsf{Vect}$ of this paper and ${\mathcal{O}}\mathcal{G}:{\mathcal{O}}\mathbf{W}\to{\mathcal{O}}\mathsf{Set}$ of [VSL] are differences in philosophy and in intended applications. The approach of [VSL] is to treat an open system as a black box with the space of internal states as completely unknown with the algebra supplying all possible choices of state spaces. By contrast in this paper we treat the space of internal states (and the total space) as known and have the algebra supply the possible choices of open systems (“dynamics”) that live on a given surjective submersion.

7. Extension of the functor $\mathsf{Crl}$ to the category $(\mathsf{FinSet}/\mathsf{SSub})^{\Leftarrow}$ of lists of submersions

This section and the next one are technical. The main result of this section is Lemma 7.1. It will be reformulated as Lemma 8.8 in the next section once double categories are introduced. Lemma 7.1 is, in effect, half the proof of the main theorem of the paper, Theorem 9.5.

In Lemma 5.16 we extended the object part of the functor $\mathsf{Crl}:\mathsf{SSub}\to\mathsf{LinRel}$ to finite unordered lists of submersions, which are objects of the category $(\mathsf{FinSet}/\mathsf{SSub})^{\Leftarrow}$. We would like to extend $\mathsf{Crl}$ to maps between lists, that is, to morphisms in the category $(\mathsf{FinSet}/\mathsf{SSub})^{\Leftarrow}$ of lists of submersions. To this end consider a map

[TABLE]

between two lists of submersions. We then have a pair of linear maps

[TABLE]

We also have a map of submersions

[TABLE]

hence a linear relation

[TABLE]

On the other hand we have a morphism of lists of vector spaces

[TABLE]

which give rise to a linear relation

[TABLE]

(see subsection 5.1). These four maps are related in the following way.

Lemma 7.1**.**

Suppose

[TABLE]

is a morphism in the category of lists of submersions. The linear map

[TABLE]

sends the linear relation $\odot(\varphi,\mathsf{Crl}\circ\Phi)$ to a subspace of the relation $\mathsf{Crl}(\mathbb{P}_{\mathsf{SSub}}(\varphi,\Phi))$. That is, for any pair

[TABLE]

of lists of open systems, the open system $\mathsf{Crl}_{\mu}((G_{y}))$ (see (5.17)) is $\mathbb{P}(\varphi,\Phi)$ related to the open system $\mathsf{Crl}_{\tau}((F_{x}))$.

Proof.

Recall that the map $\mathbb{P}_{\mathsf{SSub}}(\varphi,\Phi):\mathbb{P}_{\mathsf{SSub}}(\mu)\to\mathbb{P}_{\mathsf{SSub}}(\tau)$ is defined by requiring that the diagrams

[TABLE]

commutes for all $a\in X$ (compare with (5.4)). By definition of the relation $\mathsf{Crl}(\mathbb{P}(\varphi,\Phi))$,

[TABLE]

if and only the two open systems are $\mathbb{P}(\varphi,\Phi)$ related. That is, if and only if the diagram

[TABLE]

commutes. By definition of the relation \odot(\varphi,\Phi):\oplus_{y\in Y}\mathsf{Crl}(\mu(y))\leavevmode\hbox to18pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 8.99997pt\lower-3.09982pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.99997pt}{-2.99998pt}\pgfsys@invoke{ }\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip 3.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 3.0pt\hfil&\hfil\hskip 14.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 3.0pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-5.79997pt}{-0.49998pt}\pgfsys@lineto{5.40005pt}{-0.49998pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.60004pt}{-0.49998pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{1.22223pt}\pgfsys@lineto{0.0pt}{-2.2222pt}\pgfsys@stroke\pgfsys@invoke{ } }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\oplus_{x\in X}\mathsf{Crl}(\tau(x)) a pair

[TABLE]

of lists of open systems belongs to the relation $\odot(\varphi,\Phi)$ if and only if $F_{x}$ and $G_{\varphi(x)}$ are $\Phi(x)$ related for all $x\in X$. That is, if and only if

[TABLE]

commutes for all $x\in X$. The tangent bundle of a product of manifolds is a product of tangent bundles

[TABLE]

Consequently the diagram (7.3) commutes if and only if

[TABLE]

for all $a\in X$. Here, as above, $\pi_{a}^{\mathsf{st}}:\prod_{x\in X}\tau(x)_{\mathsf{st}}\to\tau(a)_{\mathsf{st}}$ is the projection on the $a^{th}$ factor. By definition of $\mathsf{Crl}_{\tau}$ (see (5.20))

[TABLE]

for any $a\in X$. By (7.2)

[TABLE]

Hence the left hand side of (7.5) is

[TABLE]

We next compute the right hand side. Using the commutativity of (7.2) again we see that

[TABLE]

Thus (7.5) holds.

∎

Lemma 7.1 has a nice formulation in terms of double categories — see Lemma 8.8. We will reformulate Lemma 7.1 after recalling the notions of a double category in the next section.

8. Double categories

The goal of this section is to introduce double categories, to reformulate Lemma 7.1 in terms of double categories as Lemma 8.8, and to prove Lemma 8.12. Lemmas 8.8 and 8.12 are used in the next section to prove Theorem 9.5, which is the main theorem of the paper.

Recall that one can define (strict) double categories as categories internal to the category $\mathsf{Cat}$ of categories.

Definition 8.1**.**

A double category $\mathbb{D}$ consists of two categories $\mathbb{D}_{1}$ (of arrows) and $\mathbb{D}_{0}$ (of objects) together with five structure functors:

[TABLE]

so that

[TABLE]

for all arrows $\mu,\gamma$ of $\mathbb{D}_{1}$ and

[TABLE]

for all arrows $\mu,\gamma,\nu$ of $\mathbb{D}_{1}$.

Notation 8.2**.**

We call the objects of $\mathbb{D}_{0}$ [math]-cells or objects and the morphisms of $\mathbb{D}_{0}$ the (vertical) 1-morphisms. We depict 1-morphisms as $a\xleftarrow{f}b$. The objects of $\mathbb{D}_{1}$ are called the (horizontal) 1-cells (or the horizontal 1-morphisms). A morphism $\alpha:\mu\to\nu$ of $\mathbb{D}_{1}$ with $\mathfrak{s}(\alpha)=f$ and $\mathfrak{t}(\alpha)=g$ is a 2-morphism or a 2-cell (we will use the two terms interchangeably, pace Shulman [Sh2]). We depict the two cell $\alpha$ as

[TABLE]

Remark 8.4**.**

A 2-morphism in a double category $\mathbb{D}$ of the form

[TABLE]

is called globular. Associated to every strict double category $\mathbb{D}$ there is a horizontal 2-category $\mathscr{H}(\mathbb{D})$ consisting of objects, (horizontal) 1-cells and globular 2-morphisms.

The main double category of interest for us is the double category $\mathsf{RelVect}$ of vector spaces, linear maps and linear relations which we presently define.

Definition 8.5** **(The double category $\mathsf{RelVect}$ of vector spaces,

linear maps and linear relations).

The category $\mathsf{RelVect}_{0}$ is the category of real vector spaces and linear maps. The objects of the category $\mathsf{RelVect}_{1}$ are linear relations. Recall that we use the convention that a linear relation $R$ from a vector space $V$ to a vector space $W$ is a subspace of $W\times V$, which we write as W\stackrel{{\scriptstyle R}}{{\leavevmode\hbox to18pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 8.99997pt\lower-3.09982pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.99997pt}{-2.99998pt}\pgfsys@invoke{ }\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip 3.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 3.0pt\hfil&\hfil\hskip 14.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 3.0pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{5.80002pt}{-0.49998pt}\pgfsys@lineto{-5.40001pt}{-0.49998pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-5.59999pt}{-0.49998pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.00005pt}{-2.2222pt}\pgfsys@lineto{0.00005pt}{1.22223pt}\pgfsys@stroke\pgfsys@invoke{ } }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}V. A morphism

[TABLE]

in $\mathsf{RelVect}_{1}$, that is, a 2-morphism in $\mathsf{RelVect}$, is a pair of linear maps $f:V\to Y$, $g:W\to Y$ so that $(g\times f)(R)\subset S$ which we picture as

[TABLE]

Note that the 2-cell $\alpha$ in (8.6) is completely determined by its “edges” $g$, $f$, $R$ and $S$.

The composition of morphisms in $\mathsf{RelVect}_{1}$ is the compositions of pairs of linear maps: given

[TABLE]

[TABLE]

Note that we do not require the 2-morphisms in $\mathsf{RelVect}$ to be commutative diagrams of relations: the condition $(g,f)(R)\subset S$ does not imply that $\mathop{\mathrm{missing}}{graph}(f)\circ R=S\circ\mathop{\mathrm{missing}}{graph}(f^{\prime})$ (where $\mathop{\mathrm{missing}}{graph}(f)$, $\mathop{\mathrm{missing}}{graph}(g)$ are the graphs of $f$ and $g$ respectively and $\circ$ stands for the composition of relations).

To finish the description of the double category $\mathsf{RelVect}$ we need to list the functors $\mathfrak{u},\mathfrak{s},\mathfrak{t}$ and $\mathfrak{m}$. This is not difficult. The functor $\mathfrak{u}$ sends a linear map $f:V\to W$ to the 2-morphism

[TABLE]

The result of applying the functor $\mathfrak{s}$ to the 2-morphism (8.6) is the linear map $f$ and the result of applying the functor $\mathfrak{t}$ is the linear map $g$. The horizontal composition $\mathfrak{m}$ is given by

[TABLE]

Remark 8.7**.**

The horizontal category of the double category $\mathsf{RelVect}$ is the 2-category $\mathsf{LinRel}$ (see Definition 2.23):

[TABLE]

We can now restate Lemma 7.1 in terms of the double categories:

Lemma 8.8**.**

A morphism

[TABLE]

in the category of lists of submersions gives rise to the 2-cell

[TABLE]

in the double category $\mathsf{RelVect}$ of vector spaces, linear maps and linear relations.

Next we clarify Remark 3.8.

Definition 8.10** **(The double category $\mathsf{SSub}^{\Box}$ of

submersions).

The objects of the double category $\mathsf{SSub}^{\Box}$ are surjective submersions. The (horizontal) 1-cells/1-morphisms are maps of submersions. The vertical 1-morphisms are interconnection morphisms (see Definition 3.3). The 2-cells of $\mathsf{SSub}^{\Box}$ are squares of the form

[TABLE]

where $\mu,\nu$ are maps of submersions, $f$, $g$ are interconnection morphisms and

[TABLE]

in the category $\mathsf{SSub}$ of submersions.

Remark 8.11**.**

Any category $\mathscr{C}$ trivially defines a double category $\mathrm{db}\mathscr{C}$: the 2-cells of $\mathrm{db}\mathscr{C}$ are commuting squares in $\mathscr{C}$. We can also restrict the morphisms in these commuting squares by requiring that say vertical morphisms belong to a subcategory $\mathscr{D}$ of $\mathscr{C}$. In Definition 8.10 above we took $\mathscr{C}$ to be $\mathsf{SSub}$ and its subcategory $\mathscr{D}$ to be $\mathsf{SSub}^{\mathsf{int}}$.

It will be useful in understanding maps of networks to view the functor $\mathsf{Crl}$ as a map of double categories.

Lemma 8.12**.**

A 2-cell

[TABLE]

in the double category $\mathsf{SSub}^{\Box}$ of submersions (where $\mu,\nu$ are submersions and $\psi,\varphi$ are interconnection maps) gives rise to the 2-cell

[TABLE]

in the double category $\mathsf{RelVect}$. In other words if the open systems $F\in\mathsf{Crl}(a)$ and $G\in\mathsf{Crl}(b)$ are $\mu$-related then $\varphi^{*}F$ and $\psi^{*}G$ are $\nu$-related.

Proof.

Since $F$ and $G$ are $\mu$-related

[TABLE]

(see Notation 2.15). By definition of $\varphi^{*}$ (see Remark 3.6)

[TABLE]

Similarly

[TABLE]

We compute:

[TABLE]

∎

9. Maps between networks of open systems

Recall (Definition 3.10) that a network of open systems is an unordered list of submersions $\tau:X\to\mathsf{SSub}$ indexed by a finite set $X$ together with an interconnection morphism $\psi:b\to\prod_{x\in X}\tau(x)\equiv\mathbb{P}_{\mathsf{SSub}}(\tau)$.

Definition 9.1** (Maps between networks of open systems).**

A map from a network $(\tau:X\to\mathsf{SSub},\psi:b\to\mathbb{P}_{\mathsf{SSub}}(\tau)\equiv\prod_{x\in X}\tau(x))$ of open systems to a network $(\mu:Y\to\mathsf{SSub},\nu:c\to\mathbb{P}_{\mathsf{SSub}}(\mu)\equiv\prod_{y\in Y}\mu(y))$ is a morphism

[TABLE]

in the category $(\mathsf{FinSet}/\mathsf{SSub})^{\Leftarrow}$ of lists of submersions together with a map of submersions f:c\leavevmode\hbox to18pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 8.99997pt\lower-3.09982pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.99997pt}{-2.99998pt}\pgfsys@invoke{ }\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip 3.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 3.0pt\hfil&\hfil\hskip 14.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 3.0pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-5.79997pt}{-0.49998pt}\pgfsys@lineto{5.40005pt}{-0.49998pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.60004pt}{-0.49998pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{1.22223pt}\pgfsys@lineto{0.0pt}{-2.2222pt}\pgfsys@stroke\pgfsys@invoke{ } }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}b so that the diagram of submersions

[TABLE]

defines a 2-cell in the double category $\mathsf{SSub}^{\Box}$. That is, we require that

[TABLE]

Example 9.3**.**

Here is an example of a map between two networks of open systems.

Our first network is the network $(\tau:X\to\mathsf{SSub},\psi:b\to\prod_{x\in X}\tau(x))$ of Example 3.14, where $M,U$ are smooth manifolds, $p:M\times U\to M$ is a trivial fiber bundle, $\phi:M\to U$ a smooth map, $X=\{1,2,3\}$, $\tau:X\to\mathsf{SSub}$ is constant map with $\tau(1)=\tau(2)=\tau(3)=(p:M\times U\to M)$, $b$ is the trivial submersion $\mathsf{id}_{M^{3}}:M^{3}\to M^{3}$ and

[TABLE]

is the interconnection morphism with $\psi_{\mathsf{tot}}$ given by

[TABLE]

Our second network is $(\mu:Y\to\mathsf{SSub},\nu:c\to\prod_{y\in Y}\mu(y))$ where $Y=\{*\}$, $\mu(*)=(p:M\times U\to M)$, $c$ is the trivial submersion $\mathsf{id}_{M}:M\to M$ and $\nu:c\to\mu(*)$ is the interconnection morphism with $\nu_{\mathsf{tot}}$ given by

[TABLE]

We now write down a map of networks. We take $\varphi:X\to Y$ to be the only possible map and set $\Phi_{i}:\mu(*)\to\tau(i)$ to be the identity map for all $i\in X$. That is, consider the morphism of lists

[TABLE]

In this case the induced map

[TABLE]

is the diagonal map

[TABLE]

We take $f:c\to b$ to be the diagonal map as well:

[TABLE]

Since

[TABLE]

for all $m\in M$, \left(\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 35.98732pt\hbox{{\hbox{\kern-35.98732pt\raise 17.07156pt\hbox{\hbox{\kern 3.0pt\raise-3.41666pt\hbox{\textstyle{X}}}}}}{\hbox{\kern 21.43872pt\raise 17.07156pt\hbox{\hbox{\kern 3.0pt\raise-3.41666pt\hbox{\textstyle{Y}}}}}}{\hbox{\kern-14.11115pt\raise-17.07156pt\hbox{\hbox{\kern 3.0pt\raise-3.47223pt\hbox{\textstyle{\mathsf{SSub}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-23.28638pt\raise-3.74992pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{\scriptstyle{\tau}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-5.38507pt\raise-10.59935pt\hbox{\hbox{\kern 0.0pt\raise 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}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip 3.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 3.0pt\hfil&\hfil\hskip 14.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { 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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.60004pt}{-0.49998pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{1.22223pt}\pgfsys@lineto{0.0pt}{-2.2222pt}\pgfsys@stroke\pgfsys@invoke{ } }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}b\right) is a morphism of networks.

Note a curious feature of the example described above. Consider an arbitrary open system $F\in\mathsf{Crl}(M\times U\to M)=\mathsf{Crl}(\mu(*))$. The interconnection map $\nu:(M\to M)\to(M\times U\to M)$ pulls $F$ back to a vector field $v:M\to TM$ which is given by

[TABLE]

On the other hand

[TABLE]

is an open system on $\prod_{x\in X}\tau(x)=(M\times U\to M)^{3}$ and the interconnection map $\psi:b\to\prod_{x\in X}\tau(x)$ pulls $F\times F\times F$ to a vector field $u:M^{3}\to TM^{3}$. This vector field is given by

[TABLE]

Note that the diagonal

[TABLE]

is an invariant submanifold of the vector field $u$. We can rephrase the invariance of the diagonal by saying that the vector fields $v$ and $u$ are $f:M\to M^{3}$-related, where as before $f$ denotes the diagonal map. Theorem 9.5 below shows that the invariance of the diagonal $\Delta_{M^{3}}$ for all vector fields on $M^{3}$ of the form (9.4) is not just a lucky accident.

Theorem 9.5**.**

A map

[TABLE]

of networks of open systems gives rise to a 2-cell

[TABLE]

in the double category $\mathsf{RelVect}$ of vector spaces, linear maps and linear relations.

Proof.

By Lemma 8.8 the map

[TABLE]

in the category of lists of submersions gives rise to the 2-cell

[TABLE]

in $\mathsf{RelVect}$. By Lemma 8.12 the 2-cell

[TABLE]

in $\mathsf{SSub}^{\Box}$ gives rise to the 2-cell

[TABLE]

in $\mathsf{RelVect}$. The vertical composite of the 2-cell (9.7) followed by the 2-cell (9.8) is the 2-cell (9.6). ∎

We now illustrate Theorem 9.5 in several more examples. Before we do that We remind the reader that the space

[TABLE]

of open systems on the identity map $\mathsf{id}:M\to M$ for some manifold $M$ is the space of vector fields $\mathscr{X}(M)$ on the manifold $M$.

Remark 9.9**.**

Note that a map $f$ from a trivial fiber bundle $p:M\times U\to M$ to a trivial fiber bundle $q:N\times W\to N$ is completely determined by the map $f_{\mathsf{tot}}:M\times U\to N\times W$ between their total spaces (cf. Definition 2.13). Indeed, since the diagram

[TABLE]

commutes, the map $f_{\mathsf{tot}}$ has to be of the form

[TABLE]

for some smooth map $h:M\times U\to W$.

Example 9.10**.**

Consider the map of lists

[TABLE]

where $\varphi:X\to Y$ is the only possible map.and set $\tau(1)=\tau(2)=\mu(*)=(p:\mathbb{R}\times\mathbb{R}\to\mathbb{R})$, where $p$ is the projection on the first factor. We define $\Phi=\{\Phi_{i}:\mu(\varphi(i))\to\tau(i)\}_{i=1}^{2}$ as follows. We choose

[TABLE]

to be the map

[TABLE]

and

[TABLE]

to be the map

[TABLE]

Then $\mathbb{P}(\varphi,\Phi):(p:\mathbb{R}\times\mathbb{R}\to\mathbb{R})\to(p:\mathbb{R}\times\mathbb{R}\to R)^{2}$ is given by

[TABLE]

We choose $s:M\to U$ to be the identity map. This gives us an interconnection map

[TABLE]

We choose $\psi:(\mathsf{id}:\mathbb{R}\to\mathbb{R})^{2}\to(p:\mathbb{R}\times\mathbb{R}\to\mathbb{R})^{2}$ to be

[TABLE]

We choose $f:M=\mathbb{R}\to\mathbb{R}^{2}=M^{2}$ to be the map

[TABLE]

It is easy to see that

[TABLE]

In this case Theorem 9.5 tells us that given any three open systems $G,F_{1},F_{2}\in\mathsf{Crl}(p:\mathbb{R}^{2}\to\mathbb{R})$ so that $G$ is $\Phi_{i}$-related to $F_{i}$ ($i\in X$), the vector field $\nu^{*}G$ is $f$-related to the vector field $\psi^{*}(F_{1},F_{2})$. Consequently since the image of $f$ is the parabola

[TABLE]

the parabola is an invariant submanifold of the vector field $\psi^{*}(F_{1},F_{2})$. Such an invariant submanifold can never arise in the coupled cell networks formalism.

A reader may wonder the the vector space of triples $G,F_{1},F_{2}\in\mathsf{Crl}(p:\mathbb{R}^{2}\to\mathbb{R})$ so that $G$ is $\Phi_{i}$-related to $F_{i}$ ($i\in X$) is non-zero. It is not hard to see that the space of such triples is at least as big as the space $C^{\infty}(\mathbb{R}^{2})$. Indeed, given a function $g\in C^{\infty}(\mathbb{R}^{2})$ let

[TABLE]

Then

[TABLE]

and

[TABLE]

Consequently the parabola $\{x_{1}=(x_{2})^{2}\}$ is an invariant submanifold for any vector field $v:\mathbb{R}^{2}\to\mathbb{R}^{2}$ of the form

[TABLE]

for any function $g\in C^{\infty}(\mathbb{R}^{2})$. $\Box$.

Example 9.11**.**

In Examples 9.3 and 9.10 we started with two collections of open systems and ended up with two related closed systems. It is easy to modify Example 9.10 so that the end result are two related open systems.

We now carry out the modification. As before let $X=\{1,2\}$, $Y=\{*\}$, and let $\varphi:X\to Y$ be the only possible map. Now consider the surjective submersion

[TABLE]

Set $\mu(*)=\tau(1)=\tau(2)=(q:\mathbb{R}^{3}\to\mathbb{R})$. Choose $\Phi_{i}:\mu(*)\to\tau(i)$, $i=1,2$, to be the maps

[TABLE]

This defines a morphism

[TABLE]

In the category of lists of submersions. It is easy to see that

[TABLE]

Choose the interconnection maps $\psi,\nu$ as follows:

[TABLE]

Let $f:(p:\mathbb{R}^{2}\to\mathbb{R})\to(p:\mathbb{R}^{2}\to\mathbb{R})^{2}$ be the map

[TABLE]

It is again easy to check that the equality

[TABLE]

holds with our choices of $\psi,\nu,f,p$ and $\Phi$. Therefore, by Theorem 9.5, given three open systems $G,F_{1},F_{2}\in\mathsf{Crl}(q:\mathbb{R}^{3}\to\mathbb{R})$ so that $G$ is $\Phi_{1}$-related to $F_{1}$ and $\Phi_{2}$-related to $F_{2}$, the open systems $\nu^{*}G$ is $f$-related to the open system $\psi^{*}(F_{1},F_{2})$.

10. Networks of manifolds

The first result of this section shows that the networks of manifolds defined in [DL1] (hence the coupled cell networks of Golubitsky, Stewart et al.) are a special case of the networks of open systems (Definition 3.10). In particular they are morphisms in the colored operad ${\mathcal{O}}((\mathsf{SSub}^{\mathsf{int}})^{\mathsf{op}}$. We then show that Theorem 3 of [DL1] (which is the main result of that paper) is a direct consequence of Theorem 9.5 above. In particular we show that fibrations of networks of manifolds (Definition 10.7) give rise to maps of networks of open systems in the sense of Definition 9.1. To give credit where it is due, Definition 9.1 and Theorem 9.3 were directly inspired by Theorem 3 of [DL1]. We start by setting up notation, which differs somewhat from the notation of [DL1].

Definition 10.1**.**

A finite directed graph $G$ is a pair of finite sets $G_{1}$ (arrows/edges), $G_{0}$ (nodes/vertices) and two maps $s,t:G_{1}\to G_{0}$ (source and target). We write: $G=\{G_{1}\rightrightarrows G_{0}\}$.

Definition 10.2**.**

A network of manifolds is a pair $(G,{\mathcal{P}})$ where $G=\{G_{1}\rightrightarrows G_{0}\}$ is a finite graph and ${\mathcal{P}}:G_{0}\to\mathsf{Man}$ is a list of manifolds (i.e., $G_{0}\xrightarrow{{\mathcal{P}}}\mathsf{Man}$ is an object of the category $\mathsf{FinSet}/\mathsf{Man}$).

A map of networks of manifolds $\varphi:(G,{\mathcal{P}})\to(G^{\prime},{\mathcal{P}}^{\prime})$ is a map of graphs $\varphi:G\to G^{\prime}$ so that ${\mathcal{P}}^{\prime}\circ\varphi={\mathcal{P}}$.

Recall that since the category of manifolds $\mathsf{Man}$ has finite products, we have a functor

[TABLE]

which assigns to a list $\mu:X\to\mathsf{Man}$ the corresponding product:

[TABLE]

(cf. Section 5). In particular to every network $(G,{\mathcal{P}})$ of manifolds in the sense of Definition 10.2 the functor $\mathbb{P}$ assigns the manifold $\mathbb{P}({\mathcal{P}})=\mathbb{P}(G_{0}\xrightarrow{{\mathcal{P}}}\mathsf{Man})$ which we think of as the total phase space of the network. And to every map of networks of manifolds $\varphi:(G,{\mathcal{P}}))\to(G^{\prime},{\mathcal{P}}^{\prime}))$ the functor $\mathbb{P}$ assigns a map

[TABLE]

between their total phase spaces.

Proposition 10.3**.**

A network of manifolds $(G,{\mathcal{P}})$ encodes

1. (1)

a list of submersions $I:G_{0}\to\mathsf{SSub}$ (i.e., an object of $\mathsf{FinSet}/\mathsf{SSub}$) and 2. (2)

an interconnection morphism ${\mathcal{I}}_{G}:(\mathbb{P}_{\mathsf{Man}}({\mathcal{P}})\xrightarrow{\mathsf{id}}\mathbb{P}_{\mathsf{Man}}({\mathcal{P}}))\to\mathbb{P}_{\mathsf{SSub}}(I)$.

Consequently a network of manifolds in the sense of Definition 10.2 does give rise to a network of open systems in the sense of Definition 3.10.

Remark 10.4**.**

Recall (Remark 2.10) that for any manifold $M$ the space $\mathsf{Crl}(M\xrightarrow{id}M)$ of control systems is the space $\mathscr{X}(M)$ of vector fields on the manifold $M$. Hence the interconnection map ${\mathcal{I}}_{G}:(\mathbb{P}_{\mathsf{Man}}({\mathcal{P}})\xrightarrow{\mathsf{id}}\mathbb{P}_{\mathsf{Man}}({\mathcal{P}}))\to\mathbb{P}_{\mathsf{SSub}}(I)$ gives rise to the linear map

[TABLE]

from the space of the open systems on the product $\mathbb{P}_{\mathsf{SSub}}(I)=\prod_{a\in G_{0}}I(a)$ of submersions to the space of vector fields on the manifold $\mathbb{P}_{\mathsf{Man}}({\mathcal{P}})$.

Remark 10.5**.**

Recall that by Lemma 5.16 a list of submersions $\tau:X\to\mathsf{SSub}$ gives rise to a canonical linear map $\mathsf{Crl}_{\tau}:\bigoplus_{x\in X}\mathsf{Crl}(\tau(x))\to\mathsf{Crl}(\prod_{x\in X}\tau(x))$. Consequently for any network of manifolds $(G,{\mathcal{P}})$ we get a linear map

[TABLE]

hence the composite map

[TABLE]

Proof of Proposition 10.3.

Given a node $a$ of a graph $G$ we associate two maps of finite sets:

• •

$\iota_{a}:\{a\}\hookrightarrow G_{0}$ and

• •

$s|_{t^{-1}(a)}:t^{-1}(a)\to G_{0}$ (recall that $s,t:G_{1}\to G_{0}$ are the source and target maps of the graph $G$).

The set $t^{-1}(a)$ is the collection of arrows of $G$ with target $a$ and $s$ sends this collection to the sources of the arrows. The composition with ${\mathcal{P}}:G_{0}\to\mathsf{Man}$ gives us two lists of manifolds:

[TABLE]

and

[TABLE]

Applying the functor $\mathbb{P}$ gives us two manifolds: $\mathbb{P}({\mathcal{P}}\circ\iota_{a})$, which is just ${\mathcal{P}}(a)$, and $\mathbb{P}(\xi_{a})=\prod_{\gamma\in t^{-1}(a)}{\mathcal{P}}(s(\gamma))$. We define the submersion $I(a)$ to be the projection on the first factor

[TABLE]

This gives us the desired list of submersions

[TABLE]

Since

[TABLE]

[TABLE]

By definition

[TABLE]

Therefore

[TABLE]

It follows that in order to construct an interconnection map

[TABLE]

it suffices to construct a map

[TABLE]

This map $f$ too comes from a map of finite sets. Namely, the family $\{s|_{t^{-1}(a)}:t^{-1}(a)\to G_{0}\}_{a\in G_{0}}$ defines

[TABLE]

and the diagram

[TABLE]

commutes. So set $f:=\mathbb{P}_{\mathsf{Man}}(\sqcup s|_{t^{-1}(a)})$. ∎

Example 10.6**.**

Let $G$ be the graph

123

.

Let ${\mathcal{P}}$ be the function that assigns to every node the same manifold $M$. Then

[TABLE]

for every $a\in G_{0}$,

[TABLE]

and the interconnection map ${\mathcal{I}}_{G}:M^{3}\to(M^{2})^{3}$ is given by

[TABLE]

for all $(x_{1},x_{2},x_{3})\in M^{3}$. Finally

[TABLE]

(see Remark 10.5) is given by

[TABLE]

$\Box$

In [DL1, DL2] a class of maps of networks of manifolds was singled out.

Definition 10.7** (Fibration of networks of manifolds).**

A map $\varphi:G\to G^{\prime}$ of directed graphs is a graph fibration if for any vertex $a$ of $G$ and any edge $e^{\prime}$ of $G^{\prime}$ ending at $\varphi(a)$ (i.e., the target $t(e^{\prime})$ of the edge $e^{\prime}$ is $\varphi(a)$) there is a unique edge $e$ of $G$ ending at $a$ with $\varphi(e)=e^{\prime}$.

A map of networks of manifolds $\varphi:(G,{\mathcal{P}})\to(G^{\prime},{\mathcal{P}}^{\prime})$ is a fibration if $\varphi:G\to G^{\prime}$ is a graph fibration.

Remark 10.8**.**

A graph fibration $\varphi:G\to G^{\prime}$ is in general neither injective nor surjective on vertices. However, for every vertex $a\in G_{0}$ it induces a bijection between the set $t^{-1}(a)$ of arrows of $G$ with target $a$ and the set $t^{-1}(\varphi(a))$ of arrows of $G^{\prime}$ with target $\varphi(a)$.

The reason for singling out fibrations of networks of manifold is that they give rise to maps of dynamical systems. This is the main result of [DL1] and of [DL2]. (In [DL2] only the so called “groupoid invariant” vector fields were considered. The requirement of the groupoid symmetry turned out to be irrelevant and was dropped in [DL1].) As we have seen above, a network of manifolds is a morphism in the operad ${\mathcal{O}}((\mathsf{SSub}^{\mathsf{int}})^{\mathsf{op}})$ whose target happens to be a fibration of the form $M\xrightarrow{id_{M}}M$. With the benefit of hindsight, the results of [DL1] can be reformulated as the two lemmas and the theorem below.

Lemma 10.9**.**

Let $\varphi:(G,{\mathcal{P}})\to(G^{\prime},{\mathcal{P}}^{\prime})$ be a fibration of networks of manifolds. Then for each vertex $a\in G_{0}$ we have isomorphism of submersions

[TABLE]

Hence a fibration $\varphi$ gives rise to a morphism

[TABLE]

in the category $(\mathsf{FinSet}/\mathsf{SSub})^{\Leftarrow}$.

Proof.

Since $\varphi:G\to G^{\prime}$ is a graph fibration, the restriction

[TABLE]

is a bijection for each vertex $a$ of the graph $G$. Since ${\mathcal{P}}^{\prime}\circ\varphi={\mathcal{P}}$

[TABLE]

is an isomorphism in $\mathsf{FinSet}/\mathsf{Man}$. Here, as in the proof of Proposition 10.3,

[TABLE]

and $\xi_{\varphi(a)}$ is defined similarly. Consequently

[TABLE]

is an isomorphism in $\mathsf{Man}$. Thus for each $a\in G_{0}$ we have an isomorphism of surjective submersions

[TABLE]

(note that ${\mathcal{P}}(a)={\mathcal{P}}^{\prime}(\varphi(a))$). The family of isomorphisms $\{\Phi(a)\}_{a\in G_{0}}$ together with the map $\varphi$ define a morphism

[TABLE]

in $(\mathsf{FinSet}/\mathsf{SSub})^{\Leftarrow}$. ∎

Example 10.11**.**

Let $\varphi\colon G\to G^{\prime}$ be the graph fibration

123*****$\varphi$

.

Define ${\mathcal{P}}^{\prime}:G_{0}^{\prime}\to\mathsf{Man}$ by setting ${\mathcal{P}}^{\prime}(*)=M$ for some manifold $M$ on the single vertex $*$ of $G^{\prime}$. Define ${\mathcal{P}}:G_{0}\to\mathsf{Man}$ by setting ${\mathcal{P}}(i)=M$ for $i=1,2,3$. Then

[TABLE]

is a fibration of networks of manifolds. As in Example 10.6 we have a list of submersions $I=I_{G}:G_{0}\to\mathsf{SSub}$ with

[TABLE]

for every $a\in G_{0}$.

Similarly we have $I^{\prime}:G_{0}^{\prime}=\{*\}\to\mathsf{SSub}$ given by

[TABLE]

Tracing through the proof of Lemma 10.9 we see that the maps

[TABLE]

of submersions are the identity maps for all $a\in G_{0}$. Consequently the diagram

[TABLE]

commutes.

Remark 10.12**.**

Note that for a fibration $\varphi:(G,{\mathcal{P}})\to(G^{\prime},{\mathcal{P}}^{\prime})$ of networks of manifolds, the components $\Phi(a):I(a)\to I^{\prime}(\varphi(a))$ of the natural transformation $\Phi:I^{\prime}\circ\varphi\Rightarrow I$ are all isomorphisms of submersions. Consequently the relations

[TABLE]

are graphs of linear maps. It follows from Lemma 5.13 that the relation

[TABLE]

is a graph of the linear map $\oplus(\varphi,\mathsf{Crl}\circ\Phi)$.

Remark 10.13**.**

We identify the category $\mathsf{Man}$ of manifolds with a subcategory of the category $\mathsf{SSub}$ of surjective submersions. Namely we identify a manifold $M$ with the submersion $id_{M}:M\to M$ and a map $f:M\to N$ with the map of submersions $(f,f):(M\xrightarrow{id_{M}}M)\to(N\xrightarrow{\mathsf{id}_{N}}N)$. In particular given a morphism of lists of manifolds

[TABLE]

we have a map of submersions

[TABLE]

Lemma 10.14**.**

A fibration of networks of manifolds $\varphi:(G,{\mathcal{P}})\to(G^{\prime},{\mathcal{P}}^{\prime})$ gives rise to a map of networks of open systems

[TABLE]

where, as in Remark 10.13 the category of manifolds is identified with a subcategory of submersions.

Proof.

By Lemma 10.9 the fibration $\varphi$ gives rise to a map to a morphism

[TABLE]

in the category $(\mathsf{FinSet}/\mathsf{SSub})^{\Leftarrow}$. Consequently we have a map of submersions

[TABLE]

We need to check that the diagram

[TABLE]

commutes in the category $\mathsf{SSub}$ of submersions, hence defines a 2-cell in the double category $\mathsf{SSub}^{\Box}$.

Since $\varphi$ is a fibration of networks of manifolds, for each node $a$ of the graph $G$ we have a commuting square in $\mathsf{FinSet}/\mathsf{Man}$:

[TABLE]

Here as before $\xi_{a}={\mathcal{P}}\circ s|_{t^{-1}(a)}$ and $\xi_{\varphi(a)}={\mathcal{P}}^{\prime}\circ s|_{t^{-1}(\varphi(a))}$. We now drop the maps to $\mathsf{Man}$ to reduce the clutter and only keep track of maps of finite sets. By the universal property of coproducts the diagrams 10.15 define a unique map

[TABLE]

in $\mathsf{FinSet}/\mathsf{Man}$ so that the diagram

[TABLE]

commutes for all nodes $a\in G_{0}$. Applying the functor $\mathbb{P}=\mathbb{P}_{\mathsf{Man}}$ gives the commuting diagram in $\mathsf{Man}$:

[TABLE]

The fact that $\mathbb{P}$ takes coproducts to products and the universal properties ensure that

[TABLE]

∎

The main result of [DL1] can now be stated as follows.

Theorem 10.16** ([DL1, Theorem 3]).**

A fibrations of networks of manifolds $\varphi:(G,{\mathcal{P}})\to(G^{\prime},{\mathcal{P}}^{\prime})$ gives rise to a 2-cell

[TABLE]

in $\mathsf{RelVect}$. Here as before $\Phi:I^{\prime}\circ\varphi\Rightarrow I$ is the natural transformation of Lemma 10.9.

Consequently for every family open systems $(w_{b})_{b\in G_{0}^{\prime}}\in\bigoplus_{b\in G_{0}^{\prime}}\mathsf{Crl}(I^{\prime}(b))$ we have a map of dynamical systems

[TABLE]

(q.v. Remark 10.12).

Proof.

By Lemma 10.14 a fibration of networks of manifolds $\varphi:(G,{\mathcal{P}})\to(G^{\prime},{\mathcal{P}}^{\prime})$ gives rise to a map of networks of open systems

[TABLE]

By Theorem 9.5 the map of networks of open systems give rise to the 2-cell (10.17) in the double category $\mathsf{RelVect}$.

By Remark 10.12 the relation $\odot(\varphi,\mathsf{Crl}\circ\Phi)$ is the graph of the linear map $\oplus(\varphi,\mathsf{Crl}\circ\Phi)$. The fact that (10.17) is 2-cell in $\mathsf{RelVect}$ translates into the conditions that for any tuple $(w_{b})_{b\in G_{0}^{\prime}}\in\bigoplus_{b\in G_{0}^{\prime}}\mathsf{Crl}(I^{\prime}(b))$ the vector fields $({\mathcal{I}}_{G^{\prime}}^{*}\circ\mathsf{Crl}_{I^{\prime}})\,((w_{b}))$ and $({\mathcal{I}}_{G})^{*}\circ\mathsf{Crl}_{I}\circ\oplus(\varphi,\mathsf{Crl}\circ\Phi))\,((w_{b}))$ are $\mathbb{P}(\varphi)$-related. ∎

Example 10.19**.**

We illustrate Theorem 10.16 by considering the fibration $\varphi:(G,{\mathcal{P}})\to(G^{\prime},{\mathcal{P}}^{\prime})$ of networks of manifold of Example 10.11:

123*****$\varphi$

.

with ${\mathcal{P}}^{\prime}(*)=M$ for some manifold $M$ and with ${\mathcal{P}}(a)=M$ for all $a\in G_{0}=\{1,2,3\}$. As we have seen in Example 10.11 the list $I:G_{0}\to\mathsf{SSub}$ is given by

[TABLE]

for every $a\in G_{0}$ and $I^{\prime}:G_{0}^{\prime}=\{*\}\to\mathsf{SSub}$ given by

[TABLE]

Consequently

[TABLE]

The interconnection map ${\mathcal{I}}_{G}:M^{3}\to(M^{2})^{3}$ is given by

[TABLE]

for all $(x_{1},x_{2},x_{3})\in M^{3}$. The map

[TABLE]

is given by

[TABLE]

Finally $({\mathcal{I}}_{G})^{*}\circ\mathsf{Crl}_{I}:\mathsf{Crl}(M\times M\to M)^{\oplus 3}\to\mathscr{X}(M^{3})$ is given by

[TABLE]

for all $(x_{1},x_{2},x_{3})\in M^{3}$ and all $(w_{1},w_{2},w_{3})\in\mathsf{Crl}(M\times M\to M)^{\oplus 3}$.

On the other hand

[TABLE]

and the interconnection map ${\mathcal{I}}_{G^{\prime}}:M\to M^{2}$ is given by

[TABLE]

for all $x\in M$. Consequently

[TABLE]

is given by

[TABLE]

for all $w\in\mathsf{Crl}(M\times M\to M)=\oplus_{b\in G_{0}^{\prime}}\mathsf{Crl}(I^{\prime}(b))$ and all $x\in M$.

Observe first that

[TABLE]

is the diagonal map. Next note that the linear map

[TABLE]

(the graph of which is the relation $\odot(\varphi,\mathsf{Crl}\circ\mathsf{id})$) is given by

[TABLE]

By Theorem 10.16 for any open system $w\in\mathsf{Crl}(M\times M\to M)$ the vector fields

[TABLE]

and

[TABLE]

are $\mathbb{P}_{\mathsf{Man}}(\varphi)$-related. The fact that $v$ is $\mathbb{P}_{\mathsf{Man}}(\varphi)$-related to $u$ for any choice of an open system $w\in\mathsf{Crl}(M\times M\to M)$ can also be checked directly.