an algebra of open continuous time dynamical systems and networks · 2019. 5. 7. · an algebra of...

AN ALGEBRA OF OPEN CONTINUOUS TIME DYNAMICAL

SYSTEMS AND NETWORKS

EUGENE LERMAN AND DAVID I. SPIVAK

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 interact-

ing 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

[Sp], [RS], [VSL].

It is also very useful to consider maps between dynamical systems. This amounts to

viewing dynamical systems as objects in an appropriate category. This is the point taken

by DeVille and Lerman in the study of dynamics on networks [DL1], [DL2], [DL3]. . The

goal of this paper is to describe an algebraic structure that encompasses both approaches

to systems of systems. To this end we replace the monoidal category of wiring diagrams

by a monoidal double category whose objects are surjective submersions. We show that

the assignment of the vector space of open systems to a surjective submersion extends

naturally to a lax monoidal functor with values in a monoidal double category RelVect�

of vector spaces, linear maps and linear relations. 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. As a special case we recover the results of DeVille

and Lerman on fibrations of networks of manifolds.

Contents

1. Introduction 2

Organization of the paper 4

Acknowledgments 5

2. Continuous time dynamical systems 5

3. Open systems and interconnection 7

3.1. The category of open systems 7

3.2. Interconnection maps 10

4. The extension of Crl to (FinSet/SSub)⇐ and the main result 134.1. The category of lists (FinSet/C )⇐ in a category C 134.2. The functor � : ((FinSet/RelVect)⇐)op → RelVect 155. Networks of manifolds 23

6. Wiring diagrams 29

References 31

Spivak was supported by AFOSR grant FA9550–14–1–0031 and NASA grant NNH13ZEA001N.

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2 EUGENE LERMAN AND DAVID I. SPIVAK

1. Introduction

Many systems of interest in science and engineering are made up of interacting subsys-

tems. These subsystems, in turn, could be made up of collections of smaller interacting

subsystems and so on. These kind of holarchic structures [Koe] (systems of systems) 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.

One of the fundamental problems in the theory of (closed) dynamical systems is con-

structing maps between dynamical systems or, failing that, proving their existence. 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 desirable to have a systematic way of con-

structing maps of dynamical systems out of appropriate maps between collections of open

systems. It is these kinds of considerations that underlie the development of the groupoid

formalism for coupled cell networks of Golubitsky, Stewart, and their collaborators (e.g.,

[SGP, GST, GS]). The approach of Golubitsky et al. was reinterpreted and generalized

[DL1, DL2] by DeVille and Lerman.

In the approach of Spivak et al. systems of systems are studied with an emphasis on

“blackboxing:” ones starts with several collections of open systems, puts each collection

into a “box,” interconnects the subsystems in the box, and then treats the box as a new

open system to be interconnected with other systems.

In the approach of DeVille and Lerman, there is also a certain amount of blackboxing:

one starts with a collection of open systems encoded by a decorated directed graphs and

puts them together into a closed system, at which point the process of interconnection stops.

The emphasis there is on the combinatorial construction of maps between the resulting black

boxes. DeVille and Lerman use these maps, among other things, to generalize the synchrony

result of Golubitsky et al. from systems of ODEs to nonlinear systems on manifolds.

A language encompassing both approaches should be able to deal with

(1) interconnection of open systems and “blackboxing;”

(2) dynamically meaningful relations between systems.

As demonstrated by Spivak and collaborators (1) is modeled well by algebras over rep-

resentable colored operads, i.e., by a certain set-valued functor on a certain symmetric

monoidal category. Item (2) seems to require viewing open systems as a category in a

completely different way. Consequently a category suitable for handling (1) and a category

suitable for handling (2) would have roughly the same objects but completely different mor-

phisms. Therefore, to have both (1) and (2) we consider a structure that has two different

types of morphisms plus a compatibility condition between them. One can see a germ of

this idea in [DL1]: there directed graphs encode interconnections and graph fibrations en-

code relations between dynamical systems compatible with interconnections (the groupoid

symmetries of Golubitsky et al. turned out to be superfluous and were abandoned).

Here is a brief intuitive description of what the present paper is about. Given an open

system, one can plug outputs into inputs. This is a little boring. If we have several open

AN ALGEBRA OF OPEN CONTINUOUS TIME DYNAMICAL SYSTEMS AND NETWORKS 3

systems, we can take their product first and then plug the outputs into inputs. This is

more interesting — it constructs a network. Once we have a network, we would like to

understand its behavior. One way to do it is by studying maps from other networks into

our network and maps out of our network. In general, as we mentioned above, existence

of maps between dynamical systems is a hard problem. However, given two compatible

networks of relatively simple systems, perhaps one could solve the problem locally and find

a map subsystem by subsystem. Interconnecting them, we would like to ensure that the

collections of maps we started with induce maps between the two resulting networks. The

main theorem of the paper, Theorem 4.16 tells us that this is indeed the case.

To describe Theorem 4.16 more precisely we need to to introduce some terminology.

We view open systems geometrically: for us an open system on a surjective submersion

p : Q → M is a map F : Q → TM with π ◦ F = p, where π : TM → M is the canonicalprojection. The space of all open systems on a submersion p : Q→M forms a vector space

Crl(p : Q→M) := {F : Q→ TM | π ◦ F = p}.

A map f : (Q→M)→ (Q′ →M ′) between two surjective submersions (see Definition 3.7)gives rise to a linear subspace Crl(f) ⊆ Crl(Q′ →M ′)× Crl(Q→M). It consists of pairs ofopen systems (F ′, F ) that are f -related (see Definition 3.11). We view the space Crl(f) asa linear relation

Crl(f) : Crl(Q→M) Crl(Q′ →M ′).

More precisely Crl(f) is a morphism in the category of vector spaces and linear relations.

We single out a class of morphisms of submersions that we call interconnection maps (see

Definition 3.21). An interconnection map I : (Q → M) → (P → M) gives rise to a linearmap (and not just a linear relation)

I∗ : Crl(Q→M)→ Crl(P →M).

For an open system F ∈ Crl(Q→M) the system I∗F may be viewed as a system resultingfrom plugging outputs of F into some of the inputs of F (see subsection 3.2).

The main theorem of the paper, Theorem 4.16, says the following. Start with two finite

collections of surjective submersions {µ(y)}y∈Y and {τ(x)}x∈X (Y and X are finite sets), amap of sets ϕ : X → Y , and a collection

Φx : µ(ϕ(x))→ τ(x), x ∈ X

of maps of submersions. The data (ϕ,Φ) give rise to a map of submersions P(ϕ,Φ):∏y∈Y µ(y)→∏

x∈X τ(x) (see subsection 4.1). Suppose further we have two interconnection maps I′ : (Q′ →

M ′) → ∏y∈Y µ(y), I : (Q → M) →∏x∈X τ(x) and a map f : (Q

′ → M ′) → (Q → M) so


that the diagram of maps of submersions

∏x∈X τ(x) (Q→M)

∏y∈Y µ(y) (Q′ →M ′)

oo IOO

P(ϕ,Φ)

ooI′

OO

f

commutes. The interconnection map I gives rise to the linear map

I∗ : ⊕x∈X Crl(τ(x))→ Crl(Q→M)

(see subsection 4.2). Similarly I ′ gives rise to the linear map

(I ′)∗ : ⊕y∈Y Crl(µ(y))→ Crl(Q′ →M ′).

Then for any two tuples (Fx) ∈ ⊕x∈XCrl(τ(x)) and (Gy) ∈ ⊕y∈Y Crl(µ(y)) of open systemsso that for all x ∈ X the system Gϕ(x) is Φx-related to Fx, the open systems (I ′)∗(Gy) andI∗(Fx) are f -related.

Our proof of Theorem 4.16 involves turning the category of surjective submersions (also

known as fibered manifolds) into a monoidal double category SSub�. We then construct alax monoidal functor Crl from SSub� to the monoidal double category RelVect� of vectorspaces, linear maps and linear relations; the functor Crl assigns to a submersion p : Q→Mthe vector space of Crl(p : Q→M) open systems on the submersion.

We keep the category theory in the background, as an organizing principle. In particular

the reader need not know more than a definition of a double category. Our references for

monoidal double categories are [BMM] and [Sh2].

Organization of the paper. In Section 2 we recall that continuous time dynamical sys-

tems, which we think of as vector fields on manifolds, form a category DS. We show

that assigning the space of vector fields X (M) to a manifold M extends to a functor

X : Man → RelVect from the category of manifolds Man to the 2-category of (real infinitedimensional) vector spaces and linear relations. We also introduce the (pseudo-) double

category RelVect� (Definition 2.9). We start Section 3 by recalling a geometric definitionof an open system on a surjective submersion. Open systems are also known as control

systems. We define the category OS of open systems and provide reasons for our choice

of the notion of morphisms. By analogy with the functor X of Section 2 we introduce a

functor Crl (Definition 3.17) from the category SSub of surjective submersions (i.e., fibered

manifolds) to the 2-category of RelVect. The functor happens to be lax. We single out a

special class of morphisms between submersions which we call the interconnection maps.

These morphisms provide a geometric way of thinking about the act of plugging outputs of

an open system into its inputs. Lemma 3.24 provides the first piece of evidence that Crl is

a functor of double categories. Lemma 3.28 shows that Crl is indeed a lax monoidal functor

of monoidal double categories.


We start Section 4 by introducing the category FinSet/C of unordered lists of objects

in a category C and its variants (FinSet/C )⇐) and (FinSet/C )⇒). If C has finite prod-ucts then we have a canonical functor P((FinSet/C )⇐)op → C . Even though the di-rect sum is not a product in the category RelVect we still have a canonical (lax) func-

tor � : ((FinSet/RelVect)⇐)op → RelVect (Proposition 4.5). We construct a linear mapCrl(X,τ) : ⊕a∈X Crl(τ(a)) → Crl(P(X, τ)) for every finite list τ : X → SSub of submersions(i.e., for every object (X, τ) ∈ (FinSet/SSub)⇐) in Construction 4.9. We then prove themain result of the paper, Theorem 4.16. In Section 5 we recall the constructions of [DL1]

and explain how the main result of that paper is a special case of Theorem 4.16. In the last

section of the paper, Section 6, we compare the functor Crl : SSub� → RelVect� with thealgebra of open systems on the operad of wiring diagrams of [VSL].

Acknowledgments. We thank Tobias Fritz, Joachim Kock and John Baez for a number

of helpful comments.

2. Continuous time dynamical systems

The purpose of this section is to recall some elementary facts about continuous time

dynamical systems from a category-theoretic point of view. We start with a definition.

Definition 2.1. A continuous time dynamical system is a pair (M,X), where M is a man-

ifold and X is a vector field on M .

A map from a dynamical system (M,X) to a dynamical system (N,Y ) is a smooth map

f : M → N that intertwines the two vector fields:Tf ◦X = Y ◦ f, (2.2)

where Tf : TM → TN denotes the differential of f . Equivalently the diagram

M TM

N TN.

f��

X //

Tf��

Y//

(2.3)

commutes. We say that the vector fields X and Y are f -related.

Remark 2.4. It is a standard fact that if f : (M,X)→ (N,Y ) is a morphism of dynamicalsystem then for any integral curve γ of the vector field X, the composite f ◦γ is an integralcurve of the vector field Y .

Definition 2.5 (The category DS of continuous time dynamical systems). It is easy to see

that maps of dynamical systems can be composed. Consequently dynamical systems and

their morphisms form a category, which we denote by DS.

The space X (M) of all vector fields on a manifold M is an infinite dimensional vector

space. However, given a map f : M → N of manifolds there is no induced linear map fromX (M) to X (N). There is instead a linear relation X (f) ⊆X (N)×X (M). It is definedby

X (f) :={

(Y,X) ∈X (N)×X (M) | Y ◦ f = Tf ◦X}. (2.6)


Consequently if we want to extend the assignment

M 7→X (M)

to a functor, we are forced to use the category RelVect of vector spaces and linear relations

as our target category. Note that RelVect is a 2-category: a 2-arrow from a relation to a

relation is an inclusion map. We now record a formal definition.

Definition 2.7 (The 2-category RelVect of real vector spaces and linear relations). The

objects of the category RelVect are (real) vector spaces. A 1-morphism from a vector space

V to a vector space W is a subspace R ⊆W × V . We write R : V W and say that R isa linear relation from V to W .

Given two linear relations S ⊆ Z × Y and R ⊆ Y ×X we define their composite S ◦ Rto be the linear relation

S ◦R := {(z, x) ∈ Z ×X | there exist (z, y) ∈ S and (y′, x) ∈ R with y = y′}.

It is easy to check that the composition of relations is associative; hence vector spaces and

linear relations form a category. A 2-morphism in RelVect from a relation R : V W to a

relation S : V W is a linear inclusion R ↪→ S.

Remark 2.8. Note that the functor X : Man→ RelVect defined in (2.6) is lax: for a pairof composible maps M

f−→ N g−→ Q of manifolds, there is an inclusion

X (g) ◦X (f) ⊆X (g ◦ f),

and the inclusion may be proper. For example consider the pair of embeddings

f : R ↪→ R2, f(x) = (x, 0)

and

g : R2 ↪→ R3, g(x, y) = (x, y, 0).The constant vector field ddx on R is (g◦f)-related to any vector field v on R3 with v(x, 0, 0) =∂∂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 R2.

Later on we will need to think of vector spaces, linear maps, and linear relations as

a double category. Recall that a double category is a category internal to the category of

categories. See, for example, [Sh1, Sh2]. Thus a double category D has a category of objects

D0, a category of morphisms D1 and all the structure maps are functors. Equivalently a

double category has two types of 1-morphisms: “vertical” and “horizontal.” While choosing

which is which is arbitrary in general, we will denote our choice of horizontal morphisms

with tick-marks, . A double category additionally has 2-cells that are usually pictured

as squares with “vertical” and “horizontal” morphisms as boundaries. We will mark the

2-cells with double arrows ⇒ to distinguish them from commuting diagrams.

Definition 2.9 (The double category RelVect� of vector spaces, linear maps, and linearrelations). The double category RelVect� is defined as follows. Its objects are (real) vector


spaces, its “horizontal” 1-arrows are linear relations, its “vertical” 1-arrows are linear maps,

and its 2-cells are diagrams

X X ′

Y Y ′

R

f f ′

S

⇓

where f and f ′ are linear maps, R ⊆ X ′ ×X and S ⊆ Y ′ × Y are linear relations, and forany r = (x′, x) ∈ R we have (f ′(x′), f(x)) ∈ S

In our diagrams of 2-cells in RelVect� the double arrow will always point from a relation(a “horizontal” morphism) to a relation.

Remark 2.10. Note that we do not require the 2-cells in RelVect� to be commutativediagrams: the condition (f ′, f)(R) ⊆ S does not imply that graph(f) ◦R = S ◦ graph(f ′).

Remark 2.11. The double category RelVect� is a framed bicategory, in the sense of Shulman[Sh1], also known as an equipment. We do not seem to use this structure in the paper.

3. Open systems and interconnection

We first define the category of open systems and maps between them, which we call

dynamical morphisms. Later in this section we will discuss a different kind of morphism,

called interconnection.

3.1. The category of open systems. Following Brockett [Bro] we define a (continuous-

time) open system as follows:

Definition 3.1 (open system). A continuous time open system on a surjective submersion

p : Q→M is a smooth map F : Q→ TM so that F (q) ∈ Tp(q)M for all q ∈ Q. That is, thefollowing diagram commutes:

Q TM

Mp $$

F //

π�� (3.2)

Thus an open system is a pair (Qp−→ M,F ) where p is a surjective submersion and

F : Q→ TM is a smooth map satisfying (3.2). We think of the space of points of Q as thespace of total variables and of the points of M as state variables.

Remark 3.3. Suppose the manifold Q in the definition above is a product M ×U and thesurjective submersion p : M × U → M is the projection on the first factor. We then thinkof U as the space of input variables and say that the open system F : M × U → TM is anopen system with a choice of factorization of the total variables into inputs and states.

Note that in general even if a surjective submersion p : Q → M is a trivial fiber bundlewith a typical fiber U there is no preferred choice of a factorization Q

'−→M ×U . The lackof natural factorization of variables of open systems into inputs and states is something that

has been emphasized by Willems [W].


Notation 3.4. We denote the space of all open systems on a surjective submersion p : Q→M by Crl(Q→M). In other words

Crl(Qp−→M) := {F : Q→ TM | π ◦ F = p}, (3.5)

where, as before, π : TM →M is the canonical projection.

Remark 3.6. If the surjective submersion Q → M is the identity map id: M → M thenthe space of open systems Crl(M

id−→M) is the space X (M) of vector fields on the manifoldM . As a result we have an unfortunate clash of terminologies: a closed system is a special

kind of an open system.

There are two kinds of morphisms between open systems. The first one, given in Defini-

tion 3.11, is analogous to a map between two (continuous time closed) dynamical systems.

It is helpful to first introduce the category SSub of surjective submersions.

Definition 3.7 (The category SSub of surjective submersions). The objects of the category

SSub are surjective submersions p : Q→M , where Q and M are manifolds. A morphism ffrom a submersion p : Q→M to a submersion p′ : Q′ →M ′ is a pair of maps ftot : Q→ Q′,fst : M →M ′ so that the following square commutes:

Q Q′

M M ′

ftot

p p′

fst

Remark 3.8. Surjective submersions are also known as fibered manifolds. The term “fibered

manifold” goes back to Seifert and Whitney and has been in use since early 1930’s.

Remark 3.9. The category SSub of surjective submersions has finite products: if Qp−→M ,

Er−→ N are two submersions then so is the map p× r : Q×E →M ×N and p× r has the

correct universal property. The categorical product gives SSub the structure of a Cartesian

monoidal category.

Remark 3.10. Let Man2 = Man×Man. There is a faithful functor ı : Man2 → SSub, fromthe category of pairs of manifolds and pairs of smooth maps to the category of surjective

submersions. It is defined on objects and morphisms, respectively, by

ı(U,M) = (U ×M →M) ı(ftot, fst) = (ftot × fst, fst)

Definition 3.11. A dynamical morphism from an open system (Q → M,F ) to an opensystem (Q′ →M ′, F ′) is a morphism of submersions f = (ftot, fst) : (Q→M)→ (Q′ →M ′)for which the following diagram commutes:

Q Q′

TM TM ′

ftot

F F ′

Tfst

In this case, we say that the open systems (Q→M,F ) and (Q′ →M ′, F ′) are f -related.


Definition 3.12 (The category OS of open systems). We define the category OS of open

systems as follows. The objects of OS are open systems (Q → M,F ) as in Definition 3.1.Morphisms of OS are dynamical morphisms (q.v. Definition 3.11 above).

Remark 3.13. There are several reasons why we think that our definition of dynamical

morphisms in OS is reasonable.

(1) As we noted above any vector field X : M → TM “is” an open system (M idM−−→M,X). In other words our definition of dynamical morphisms is chosen so that the

natural inclusion DS ↪→ OS of dynamical systems into open systems is a functor.(2) Suppose f1 : M1 → N1, f2 : M2 → N2 are two maps of manifolds and X ∈X (M1×

M2), Y ∈X (N1×N2) are a pair of vector fields that are f = (f1, f2)-related. Eachof these vector fields defines a pair of open systems:

X(m1,m2) =(X1(m1,m2), X2(m1,m2)

)

Y (n1, n2) =(Y1(n1, n2), Y2(n1, n2)

).

It is easy to see that X1 is (f, f1)-related to Y1 and X2 is (f, f2)-related to Y2.

(3) An analogue of an integral curve for an open system F : Q→ TM on a submersionp : Q→M is curve γ : (a, b)→ Q so that

d

dt(p ◦ γ) (t) = F (γ(t)).

We will call any such curve γ a trajectory of the open system F ; it is a dynamical

morphism, if (a, b) is considered a dynamical system with unit vector field.

It is easy to see that f : (Q → M,F ) → (Q′ → M ′) is a dynamical morphism ofan open systems and γ is a trajectory of F then f ◦ γ is a trajectory of F ′.

Remark 3.14. The category OS of open dynamical systems and dynamical morphisms has

all finite products. Given open systems (Q → M,F ) and (Q′ → M ′, F ′), their product is(Q×Q′ →M ×M ′, F × F ′), where

(F × F ′)(q, q′) :=(F (q), F ′(q′)

)(3.15)

for all (q, q′) ∈ Q×Q′.

Remark 3.16. One can additionally require that an open system (Q → M,F ) has aspecified output/readout map g : M → O. This is the point of view taken in [VSL]. In thepresent paper the readout maps are absorbed in the interconnection maps. Alternatively one

can view our setup as requiring that all the readout maps are the identity maps id : M →M .

Definition 3.17 (The functor Crl : SSub→ RelVect).The assignment (Q → M) 7→ Crl(Q → M) of vector spaces of open systems to surjectivesubmersions (see 3.4) extends to a functor

Crl : SSub→ RelVectas follows. Given a map f = (ftot, fst) : (Q

p−→M)→ (Q′ p′−→M ′) between two submersions

we have a linear relation Crl(f) : Crl(p) Crl(p′):

Crl(f) := {(F ′, F ) ∈ Crl(Q′ →M ′)× Crl(Q→M) | Tfst ◦ F = F ′ ◦ ftot}


As in the case of X : Man→ RelVect the functor Crl is lax.

Remark 3.18. We note that either of two conditions on a map f = (ftot, fst) : (Q→M)→(E → N) of surjective submersions will guarantee that the relation Crl(f) is a graph of alinear map:

(1) ftot has an inverse g, or

(2) fst has an inverse h.

The reasons are simple. Assume (1). Then for any F ∈ Crl(Q→ M), G ∈ Crl(E → N) wehave

Tfst ◦ F = G ◦ ftot ⇔ Tfst ◦ F ◦ g = G.Hence in this case

Crl(f) = {(G,F ) | G = Tfst ◦ F ◦ g}.Assume (2). Then for any F ∈ Crl(Q→M , G ∈ Crl(E → N) we have

Tfst ◦ F = G ◦ ftot ⇔ F = Th ◦G ◦ ftot.Hence in this case

Crl(f) = {(G,F ) | F = Th ◦G ◦ ftot}.

Remark 3.19. We have the evident forgetful functor u : OS→ SSub from open systems tosubmersions that forgets the dynamics. That is, on objects u is given by

u(Q→M,F ) = (Q→M).This functor preserves products.

Remark 3.20. The lax functor Crl : SSub→ RelVect carries all the essential information ofthe functor u : OS → SSub. That is, for each object (Q p−→ M) ∈ SSub, the fiber u−1(Q p−→M) is isomorphic to Crl(Q

p−→ M), and to each morphism f : (Q p−→ M) → (Q′ p′−→ M ′) we

can associate the linear relation Crl(f). Thus the functor u is in some ways akin to the

Grothendieck construction of Crl, but it is not a fibration of categories.

3.2. Interconnection maps. Suppose F : M × U × V → TM is an open system withthe input space U × V , and suppose that φ : M → V is a smooth map. Then the mapG : M × U → TM defined by

G(m,u) = F (m,u, φ(u))

is an open system on M × U → M . We think of G as being obtained from F by pluggingthe output φ : M → V into the inputs of F , that is, by interconnecting the outputs withthe inputs.

Note also that G = F ◦ ϕ where ϕ : M × U →M × U × V is given byϕ(m,u) = (m,u, φ(m)).

We therefore can view this interconnection as a linear map

ϕ∗ : Crl(M × U × V →M)→ Crl(M × U →M), ϕ∗F := F ◦ ϕ.The construction easily generalizes to maps between submersions.


Definition 3.21. In the category SSub of surjective submersion, we say that a morphism

ϕ = (ϕtot, ϕst) : (Qp−→M)→ (Q′ p

′−→M ′) is a interconnection morphism if ϕst is a diffeomor-

phism.

Remark 3.22. It may be that the definition above is a bit too general to be called ”inter-

connection maps”. For example, one might want to additionally insist that ϕtot : Q → Q′is an embedding, though we will not need this restriction in what follows. The additional

requirement that ftot is an embedding would capture the idea that after plugging outputs

into inputs the total space of the systems (controls and states) should be smaller.

Remark 3.23. An interconnection morphism ϕ : (Qp−→M)→ (Q′ p

′−→M ′) defines a linear

map

ϕ∗ : Crl(Q′p′−→M ′)→ Crl(Q p−→M).

It is given by

ϕ∗F := T (ϕst)−1 ◦ F ◦ ϕtot.Compare with Remark 3.18.

Lemma 3.24. The functor Crl sends a commuting square of morphisms in the category of

SSub of the form

(Q′ →M) (E′ → N)

(Q→M) (E → N)

f ′

f

ϕ ψ (3.25)

where ϕ,ψ are interconnection morphisms (i.e., ϕst, ψst are diffeomorphisms) into a 2-cell

Crl(Q′ →M) Crl(E′ → N)

Crl(Q→M) Crl(E → N)

Crl(f)′

ϕ∗ ψ∗

Crl(f)

⇓

in the double category RelVect�. In other words if the open systems F ∈ Crl(Q′ →M) andG ∈ Crl(E′ → N) are f ′-related then ϕ∗F and ψ∗G are f -related.

Proof. Suppose that open systems F ′ : Q′ → TM ′ and G′ : E′ → TN ′ are f ′-related, i.e.,that (G,F ) ∈ Crl(f ′). Then

Tf ′st ◦ ϕ∗F ′ = Tfst ◦ T (ϕst)−1 ◦ F ′ ◦ ϕtot= T (ψst)

−1 ◦ Tf ′st ◦ F ′ ◦ ϕtot= T (ψst)

−1 ◦G′ ◦ f ′tot ◦ ϕtot (since (3.25) commutes)= T (ψst)

−1 ◦G ◦ ψtot ◦ ftot= ψ∗G′ ◦ ftot.

�


Remark 3.26. One can interpret Lemma 3.24 in terms of functors of double categories.

Namely, any category C trivially defines a double category dbC : the 2-cells of dbC are

commuting squares in C . We can also restrict the morphisms in these commuting squares

by requiring that say vertical morphisms belong to a subcategory of C . Hence we can (and

should and will) turn the category SSub of submersions into a double category SSub� asfollows: the “horizontal” morphisms are the dynamical morphisms in SSub, the “vertical”

morphisms are interconnection morphisms and 2-cells are commuting squares. Lemma 3.24

then says that Crl : SSub� → RelVect� is a functor of double categories.

The category RelVect of vector spaces and linear relations has a monoidal structure: given

two relations R : V W and R′ : V ′ W ′ we define

R⊕R′ : V ⊕ V ′ W ⊕W ′

by ((w,w′), (v, v′)

)∈ R⊕R′ ⇔ (w, v) ∈ R and (w′, v′) ∈ R′.

The category SSub of surjective submersions has a monoidal structure as well; the monoidal

structure on SSub is Cartesian. Given two submersions Qipi−→Mi, i = 1, 2, we have a linear

map

j : Crl(Q1 →M1)⊕ Crl(Q2 →M2)→ Crl(Q1 ×Q2 →M1 ×M2) (3.27)which sends a pair of open systems (F1, F2) to their product open system

F1 × F2 : Q1 ×Q2 → TM1 × TM2,as in (3.15). However the functor Crl : SSub → RelVect is not lax monoidal; instead thedouble category structure on RelVect� has to be taken into account:

Lemma 3.28. A pair of (dynamical) morphisms

fi : (Qipi−→Mi)→ (Ei ri−→ Ni), i = 1, 2

in the category SSub of surjective submersions, gives rise to a 2-cell

⊕2i=1 Crl(Qi →Mi)

⊕2i=1 Crl(Ei → Ni)

Crl(∏2

i=1(Qi →Mi))

Crl(∏2

i=1(Ei → Ni))

⊕Crl(fi)

j k

Crl(∏fi)

⇓ (3.29)

in the double category RelVect�. Here j is the linear map in (3.27) and k is defined similarly.

Proof. Recall that each morphism fi is a pair of compatible maps (fi)tot : Qi → Ei and(fi)st : Mi → Ni. Suppose that for each i we have (Gi, Fi) ∈ Crl(fi). Then T (fi)st ◦ Fi =Gi ◦ (fi)tot, i = 1, 2. Therefore for all (q1, q2) ∈ Q1 ×Q2

(T (f1)st × T (f2)st) ◦ (F1 × F2)(q1, q2) = (T (f1)st ◦ F1(q1), T (f2)st ◦ F2(q2)= (G1 ◦ (f1)tot(q1), G2 ◦ (f2)tot(q2))= (G1 ×G2) ◦ ((f1)tot × (f2)tot)(q1, q2).


Hence

(k × j)(⊕Crl(fi)) ⊆ Crl(∏

fi).

�

The above lemma will be generalized to arbitrary sets of dynamical morphisms in Con-

struction 4.9.

Remark 3.30. Recall that the notion of a monoidal functor F : S → R from a monoidalcategory (S ,⊗S ) to a monoidal category (R,⊗R) includes a natural transformation

PR ◦ (F ,F)⇒ F ◦ PS .If R is a double category, K,H : S ⇒ R is a pair of functors, then a natural transformationα from K to H assigns to each object of S a “vertical” arrow in R and to every morphism

of S a 2-cell in R. Therefore Lemma 3.28 again argues that the target of the functor Crl

has to be a double category.

4. The extension of Crl to (FinSet/SSub)⇐ and the main result

4.1. The category of lists (FinSet/C )⇐ in a category C . Think of sets as discretecategories. Then for any category C we have the category FinSet/C . By definition its

objects are functors of the form τ : X → C , where X is a finite set (i.e., a finite discretecategory). Morphisms are strictly commuting triangles of the form

X Y

C

τ ��

p //

µ�� .

We think of an object (X, τ) of FinSet/C as unordered list {τ(a)}a∈X of objects of thecategory C indexed by the finite set X. The composition of morphisms in FinSet/C is

given by pasting triangles together.

Remark 4.1. In [VSL] the category FinSet/C is called the category of typed finite sets of

type C .

Remark 4.2. If the category C has all finite products there is a canonical functor

P : (FinSet/C )op → C .On objects the functor P is defined by

P(X, τ) :=∏

x∈Xτ(x).

We define

P(p) ≡ P

X Y

C

τ ��

p //

µ��

: P(Y, µ)→ P(X, τ)


by requiring that the diagram

P(Y, µ) P(X, τ)

µ(p(a)) τ(a)

P(p)

πp(a) πa

id

commutes for all a ∈ X. Here and elsewhere πa : P(X, τ) =∏x∈X τ(x) → τ(a) is the

projection on the ath factor and πp(a) is defined similarly.

Since the objects of the category FinSet/C are functors it is natural to modify the mor-

phisms by allowing the triangles to be 2-commutative rather than strictly commutative.

There are two choices for the direction of the 2-arrow. If C has finite products it is natural

to choose 2-commuting triangles of the form

X Y

C

τ ��

p //

µ��

Φrz

(4.3)

as morphisms between lists. We denote this variant of FinSet/C by (FinSet/C )⇐. Thereason why this is “natural” is that if C has finite products we have a functor

P : ((FinSet/C )⇐)op → C .

which is defined on objects by

P(X, τ) :=∏

a∈Xτ(a).

We define

P(p,Φ) := P

X Y

C

τ ��

p //

µ��

Φrz

by requiring that the diagram

P(Y, µ) P(X, τ)

µ(p(a)) τ(a)

P(p,Φ)

πp(a) πa

Φ(a)

(4.4)

commutes for all a ∈ X.Here is an example of (FinSet/C )⇐ that we very much care about: take C = SSub, the

category of surjective submersions. Since SSub has finite products we have a contravariant

functor

P : ((FinSet/SSub)⇐)op → SSub.


4.2. The functor � : ((FinSet/RelVect)⇐)op → RelVect.If C = RelVect, the (2-)category of vector spaces and linear relations, then the category

(FinSet/RelVect)⇐ of finite unordered lists of vector spaces still makes sense. However theexistence of an extension of the assignment

(FinSet/RelVect) 3 (X, τ) 7→ ⊕a∈Xτ(a) ∈ RelVectto a functor is a bit more delicate since the direct sum ⊕ is not a product in RelVect. Thissaid, given a finite list τ : X → RelVect of vector spaces and an arbitrary vector space Zthere is a canonical map

⊕

a∈XHomRelVect(Z, τ(a))→ HomRelVect(Z,⊕a∈Xτ(a)).

It assigns to a collection of the subspaces

{Ra ⊆ τ(a)× Z}a∈X = {Ra : Z τ(a)}a∈Xthe intersection ⋂

a∈X(πa × idZ)−1(Ra) ⊆

⊕

a∈Xτ(a))× Z.

Here πa : ⊕a′∈X τ(a′)→ τ(a) are the canonical projections.

Proposition 4.5. The assignment

(FinSet/RelVect) 3 (X, τ) 7→ ⊕a∈Xτ(a) ∈ RelVectextends to a lax functor

� : (FinSet/RelVect)⇐)op → RelVect

Proof. Given a list (X, τ) ∈ (FinSet/RelVect)⇐ we define�(X, τ) := ⊕a∈Xτ(a).

Given a 2-commuting triangle

X Y

RelVect

τ ��

p //

µ��

Φrz

we set

�(p,Φ) :=⋂

a∈X(πa × πp(a))−1(Φ(a)),

where

πa × πp(a) : ⊕a′∈X τ(a′)×⊕b∈Y µ(b)→ τ(a)× µ(p(a))are the projections and Φ(a) ⊆ τ(a) × µ(p(a)) are the component relations of the naturaltransformation Φ: µ ◦ p⇒ τ . It remains to check that given a pair of triangles

X Y Z

RelVect

τ""

p //

µ

��

q //

ν||

Ψw

Φw


that can be composed (pasted together) we have

�(pq,Φ ◦ (Ψp)) ⊇ �(p,Φ) ◦ �(q,Ψ).This is a computation. By definition we have

�(q,Ψ) = {((wb)b∈Y , (vc)c∈Z) | (wb, vq(b)) ∈ Ψ(b) for all b ∈ Y }�(p,Φ) = {((ua)a∈X , (wb)b∈Y ) | (ua, wp(a)) ∈ Φ(a) for all a ∈ X}.

Hence

�(p,Φ) ◦ �(q,Ψ) =

((ua)a∈X , (vc)c∈Z)

∣∣∣∣∣∣

∃(wb) ∈ ⊕b∈Y µ(b) so that∀a, ((ua), (wp(a))) ∈ Φ(a) and∀b, ((wb), (vq(b))) ∈ Ψ(b)

. (4.6)

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

�(X, τ)×�(Z, ν) = ⊕a∈Xτ(a)×⊕c∈Zν(c)which is given by

�(p,Φ)◦ � (q,Ψ) := (4.7){((ua)a∈X , (vc)c∈Z) | (ua, vq(p(a))) ∈ Φ(a) ◦Ψ(φ(a)) for all a ∈ X}.

Since (4.6) is contained in (4.7), the result follows. �

Example 4.8. We use the notation of Proposition 4.5 above. Suppose that X = {1, 2, 3},τ : X → RelVect is a function that assigns to each i ∈ X the same vector space W , that Y isa one point set {∗}, that µ(∗) is some vector space V , and that Φ(i) ⊆ τ(i)×µ(∗) = W ×Vare some linear relations. Then �(p,Φ) ⊆W 3 × V is the relation

�(p,Φ) = {(w1, w2, w3, v) | (wi, v) ∈ Φ(i), i = 1, 2, 3}.�

We now generalize Lemma 3.28. We start by constructing a linear map

Crl(X,τ) : ⊕a∈X Crl(τ(a))→ Crl(P(X, τ))for every finite list τ : X → SSub of submersions (i.e., for every object (X, τ) ∈ (FinSet/SSub)⇐).

Construction 4.9 (Construction of Crl(X,τ) : ⊕a∈X Crl(τ(a))→ Crl(P(X, τ)) ).To carry out the construction we need to introduce a fair amount of notation. For each

a ∈ X we have a submersion τ(a) which we write asτ = (τ(a)tot

υa−→ τ(a)st).We denote the projections from

⊕a∈X Crl(τ(a)) to Crl(τ(a) by $a:

$a :⊕

a′∈XCrl(τ(a′))→ Crl(τ(a)).

Since P(X, τ) =∏a∈X τ(a) we have canonical projections

πa :∏

a′∈Xτ(a′)→ τ(a).


Each πa is a pair of maps of manifolds:

πtota :∏

a′∈Xτ(a′)tot → τ(a)tot and πsta :

∏

a′∈Xτ(a′)st → τ(a)st.

We set

ϑa := υa ◦ πtota :∏

a′∈Xτ(a′)tot → τ(a)st.

We define the projections

πa : Crl(P(X, τ)) = Crl

(∏

a′∈Xτ(a′)

)→ Crl

(∏

a′∈Xτ(a′)tot

ϑa−→ τ(a)st)

by

πa(G) := Tπsta ◦G.

Since T (∏a′∈X τ(a

′)st) =∏a′∈X Tτ(a

′)st, the projections πa make Crl(P(X, τ)) into a directsum:

Crl(P(X, τ)) =⊕

a∈XCrl

(∏

a′∈Xτ(a′)tot

ϑa−→ τ(a)st).

Finally we have pull-back maps

(πtota )∗ : Crl(τ(a)tot

υa−→ τ(a)st)→ Crl(∏

a′∈Xτ(a′)tot

ϑa−→ τ(a)st),

(πtota )∗Fa := Fa ◦ πtota .

By the universal property of products the family of maps

{(πtota )

∗ ◦$a : ⊕a′∈X Crl(τ(a′))→ Crl(∏

a′∈Xτ(a′)tot

ϑa−→ τ(a)st)}

uniquely define a linear map Crl(X,τ) making the diagram

⊕a∈X Crl(τ(a)) Crl(P(X, τ))

Crl(τ(a)) Crl(∏

a′∈X τ(a′)tot

ϑa−→ τ(a)st)

Crl(X,τ) //

$a

��

πa

��

(πtota )∗//

(4.10)

of vector spaces and linear maps commute. �.


Lemma 4.11. A morphisms

X Y

SSub

τ

��

p //

µ

��

Φ

v~in (FinSet/SSub)⇐ gives rise to a 2-cell

�(Y,Crl ◦ µ) = ⊕b∈Y Crl(µ(b)) �(X,Crl ◦ τ) = Crl(P(X, τ))

Crl(P(Y, µ)) Crl(P(X, τ))

�(p,Crl◦Φ)

Crl(Y,µ) Crl(X,τ)

Crl(P(p,Φ))

⇓ (4.12)

in RelVect�. Here as before

�(p,Crl ◦ Φ) = �

X Y

RelVect

Crl◦τ��

p //

Crl◦µ��

Crl◦Φv~

is a linear relation.

Proof. We want to show that for any pair pair

((Fa)a∈X , (Gb)b∈Y ) ∈ ⊕a∈XCrl(τ(a))×⊕b∈Y Crl(µ(b))

of lists of open systems that lies in the relation �(p,Crl ◦Φ), the pair (Crl((Fa)),Crl((Gb)))lies in Crl(P(p,Φ)). By definition of the relation Crl(P(p,Φ)),

(Crl((Fa)),Crl((Gb))) ∈ Crl(P(p,Φ))

if and only if the diagram

P(Y, µ)tot T (P(Y, µ)st)

P(X, τ)tot T (P(X, τ)st)

Crl(Y,µ)(Gb) //

Crl((Fa))//

T (P(p,Φ)st)

��

T (P(p,Φ)tot)

��

(4.13)

commutes. By definition of �(p,Φ): ⊕b∈Y Crl(µ(b))→ ⊕a∈XCrl(τ(a)) a pair

((Fa)a∈X , (Gb)b∈Y ) ∈ ⊕a∈XCrl(τ(a))×⊕b∈Y Crl(µ(b))


of lists of open systems belongs to the relation �(p,Φ) if and only if

µ(p(a))tot Tµ(p(a))st

τ(a)tot Tτ(a)st

Gp(a) //

Fa

//

T (Φ(a)st)

��

Φ(a)tot

��

(4.14)

commutes for all a ∈ X. Since we have

T (P(X, τ)st) =∏

a∈X(Tτ(a)st),

the diagram (4.13) commutes if and only if

Tπsta ◦ Crl(X,τ)((Fa)) ◦ P(p,Φ)tot = Tπsta ◦ T (P(p,Φ)st) ◦ Crl(Y,µ)((Gb)) (4.15)

for all a ∈ X. Here, as above, πsta :∏a′∈X τ(a

′)st → τ(a)st is the projection on the athfactor. Recall that the map P(p,Φ) of fibrations is uniquely defined by the commutativityof (4.4) for all a ∈ X. Hence the left hand side of (4.15) is

Fa ◦ Φ(a)tot ◦ πtotp(a).

We next compute the right hand side. Using the commutativity of (4.4) again we see that

Tπsta ◦ T (P(p,Φ)st) ◦ Crl(Y,µ)((Gb)) =

= T (Φ(a)st) ◦ Tπstp(a) ◦ Crl(Y,µ)((Gb))= T (Φ(a)st) ◦ (πtotp(a))∗ ◦$p(a)((Gb)) by definition of Crl(Y,µ), c.f. (4.10)= T (Φ(a)st) ◦ ◦Gp(a) ◦ πtotp(a)= Fa ◦ Φ(a)tot ◦ πtotp(a) since (4.14) commutes

Thus (4.15) holds. �

Theorem 4.16 (Main theorem). A morphism

X Y

SSub

τ ��

p //

µ��

Φrz

in (FinSet/SSub)⇐ together with a 2-cell

(Q′ →M ′) (Q→M)

P(Y, µ) P(X, τ)

f

ψ ϕ

P(p,Φ)

⇓ (4.17)


in the double category of submersions SSub� (where ψ and ϕ are interconnection morphisms)give rise to the 2-cell

Crl(Q′ →M ′) Crl(Q→M))

⊕b∈Y Crl(µ(b))

⊕a∈X Crl(τ(a))

Crl(f)

⇑

�(p,Crl◦Φ)

ψ∗ ϕ∗ (4.18)

in the pseudo-double category RelVect�.

Proof. We construct the 2-cell (4.18) by pasting together the two 2-cells

⊕b∈Y Crl(µ(b))

⊕a∈X Crl(τ(a))

Crl(P(Y, µ)) Crl(P(X, τ)))

Crl(Q′ →M ′) Crl(Q→M))

�(p,Crl◦Φ)

Crl(Y,µ) Crl(X,τ)

Crl(P(p,Φ))

ψ∗ ϕ∗

⇓

Crl(f)

⇓

The left 2-cell is the 2-cell (4.12) of Lemma 4.11. Apply the functor Crl to the commuting

diagram (4.17). By Lemma 3.24 this gives us the desired bottom 2-cell. Now paste together

the two 2-cells to obtain (4.18). �

We now illustrate Theorem 4.16 in several examples. We note first that the space

Crl(id : M →M)of open systems on the identity map id: M → M for some manifold M is the space ofvector fields X (M) on the manifold M . Thus vector fields, which are closed systems,

are a special kind of open systems. Note next that a map f from a trivial fiber bundle

p : M × U → M to a trivial fiber bundle q : N ×W → W is completely determined by themap ftot : M × U → N ×W between their total spaces (cf. Definition 3.7). Indeed, sincethe diagram

M × U N ×W

M N

ftot //

��

fst

//

commutes, the map ftot has to be of the form

ftot(m,u) = (fst(m), h(m,u))

for some smooth map h : M × U →W .

Example 4.19. Let M and U be two manifolds and p : M × U → M the projection onthe first factor. Let X = {1, 2, 3} be a three element set and Y = {∗} a singleton. Defineτ(i) = (p : M × U → M) for all i ∈ X and set µ(∗) = (p : M × U → M) as well. We take


p : X → Y to be the only possible map and set Φi : µ(∗)→ τ(i) to be the identity map forall i ∈ X. In this case the map P(φ,Φ): (M × U →M)→ (M × U →M)3 is the diagonalmap

∆: (M × U →M)→ (M × U →M)3 ∆(m,u) = ((m,u), (m,u), (m,u)).Choose a map s : M → U . This choice gives rise to the interconnection map

I ′ : (id : M →M)→ (p : M × U →M), I ′(m) = (m, s(m)).Given an open system F : M × U → TM its pullback (I ′)∗F ∈ X (M) is the vector fielddefined by

(I ′)∗F (m) = F (m, s(m)).

Next consider the interconnection map

I : (id : M →M)3 → (p : M×U →M)3, I(m1,m2,m3) = ((m1, s(m2)), (m2, s(m1)), (m3, s(m2)).For any triple of open systems (F1, F2, F3) ∈ ⊕i∈XCrl(τ(i)) ↪→ Crl(

∏i∈X τ(i)) the pullback

I∗(F1, F2, F3) ∈X (M3) is given by(I∗(F1, F2, F3)) (m1,m2,m3) = (F1(m1, s(m2)), F2(m2, s(m1)), F (m3, s(m2))

for all (m1,m2,m3) ∈ M3. We choose f : (id : M → M) → (id : M → M)3 to be thediagonal map:

f(m) = (m,m,m).

Since Φi : µ(∗) → τ(i) is the identity map for all i, G ∈ Crl(µ(∗)) is Φi-related to Fi ∈Crl(τ(i)) if and only if G = Fi (for all i). Theorem 4.16 tells us that in this case for any

open system G ∈ Crl(M × U →M) the vector field (I ′)∗G on M is f -related to the vectorfield I∗(G,G,G) on M × M × M . Of course in this example this fact is easy to checkdirectly.

The example above is a simple variation on the examples that can be found in the coupled

cell networks literature. The next example is different.

Example 4.20. In this example X is a two element set {1, 2}, Y is again a singleton andp : X → Y is the only possible map. We again consider the surjective submersions of theform M × U → M and set τ(i) = (p : M × U → M) = µ(∗) for all i. To make thingsmore concrete we now take M = U = R, the real line. We choose Φ1 : (p : R × R → R) →(p : R× R→ R) to be the map

Φ2(x, u) = (x2, u)

and Φ2 : (p : R× R→ R)→ (p : R× R→ R) to be the mapΦ2(x, u) = (x, u

2).

Then P(p,Φ): (p : R× R→ R)→ (p : R× R→ R)2 is given byP(p,Φ)(x, u) = ((x2, u), (x, u2)).

We choose s : M → U to be the identity map. This gives us an interconnection mapI ′ : (id : R→ R)→ (p : R× R→ R), I ′(x) = (x, x).


We choose I : (id : R→ R)2 → (p : R× R→ R)2 to beI(x1, x2) = ((x1, x2), (x2, x1)).

We choose f : M = R→ R3 = M3 to be the mapf(x) = (x2, x).

It is easy to see that

P(p,Φ) ◦ I ′ = I ◦ f.In this case Theorem 4.16 tells us that given any three open systems G,F1, F2 ∈ Crl(p : R2 →R) so that G is Φi-related to Fi (i ∈ X), the vector field (I ′)∗G is f -related to the vectorfield I∗(F1, F2). Consequently since the image of f is the parabola

{(x1, x2) ∈ R2 | x21 = x2},the parabola is an invariant submanifold of the vector field I∗(F1, F2). Such an invariantsubmanifold could never arise in the coupled cell networks formalism.

A reader may wonder the the vector space of triples G,F1, F2 ∈ Crl(p : R2 → R) so thatG is Φi-related to Fi (i ∈ X) is non-zero. It is not hard to see that the space of such triplesis at least as big as the space C∞(R2). Indeed, given a function g ∈ C∞(R2) let

G(x, u) =1

2xg(x2, u2)

F1(v, u) = vg(v, u2)

F2(x,w) =1

2xg(x2, w).

Then F2◦Φ2(x, u) = G(x, u) and (F1◦Φ1)(x, u) = x2g(x2, u2) = 2xG(x, u) = (TΦ1◦G)(x, u).

Example 4.21. In Examples 4.19 and 4.20 we started with two collections of open systems

and ended up with two related closed systems. It is easy to change the examples so that

the end result are two related open systems. We now modify Example 4.20. As before

let X = {1, 2}, Y = {∗}, and let p : X → Y be the only possible map. Now consider thesurjective submersion

q : R3 → R q(x, u, v) = x.Set µ(∗) = τ(1) = τ(2) = (q : R3 → R). Choose Φi : µ(∗)→ τ(i), i = 1, 2, to be the maps

Φ1(x, u, v) = (x2, u, v), Φ2(x, u, v) = (x, u

2, v).

Then

P(p,Φ) (x, u, v) = ((x2, u, v), (x, u2, v)).Choose the interconnection maps I, I ′ as follows:

I ′ : (p : R2 → R)→ (q : R3 → R) I ′(x, v) = (x, x, v);I : (p : R2 → R)2 → (q : R3 → R)2 I((x1, v1), (x2, v2)) = ((x1, x2, v), (x2, x1, v)).

Let f : (p : R2 → R)→ (p : R2 → R)2 be the mapf(x, v) = ((x2, v), (x, v)).


It is again easy to check that the equality

P(p,Φ) ◦ I ′ = I ◦ f

holds with our choices of I, I ′, f, p and Φ. Therefore, by Theorem 4.16, given three opensystems G,F1, F2 ∈ Crl(q : R3 → R) so that G is Φ1-related to F1 and Φ2-related to F2, theopen systems (I ′)∗G is f -related to the open system I∗(F1, F2).

5. Networks of manifolds

The purpose of this section is to explain how the main results of [DL1] follow from

Theorem 4.16. We start by recalling some definitions and notation.

Definition 5.1. A finite directed graph G is a pair of finite sets G1 (arrows, edges), G0(nodes, vertices) and two maps s, t : G1 → G0 (source and target). We write: G = {G1 ⇒G0}.

Definition 5.2. A network of manifolds is a pair (G, (G0,P)) where G = {G1 ⇒ G0} isa finite graph and P : G0 → Man is a list of manifolds (i.e., (G0,P)) is an object of thecategory FinSet/Man).

A map of networks of manifolds ϕ : (G, (G0,P)) → (G′, (G′0,P ′) is a map of graphsϕ : G→ G′ so that P ′ ◦ ϕ = P.

Since the composite of two maps of networks of manifolds is again a map of networks,

networks of manifolds and their maps form a category. However, it turns out that for our

purposes this is not quite the right category. A better one additionally requires that a map

between networks is a fibration. We now define this notion.

Definition 5.3 (Fibration of networks of manifolds). A map ϕ : G→ G′ of directed graphsis a graph fibration if for any vertex a of G and any edge e′ of G′ ending at ϕ(a) there is aunique edge e of G ending at a with ϕ(e) = e′.

A map of networks of manifolds ϕ : (G, (G0,P))→ (G′, (G′0,P ′)) is a fibration if ϕ : G→G′ is a graph fibration.

Remark 5.4. A graph fibration ϕ : G→ G′ is in general neither injective nor surjective onvertices. However, for every vertex a ∈ G0 it induces a bijection between the set t−1(a) ofarrows of G with target a and the set t−1(ϕ(a)) of arrows of G′ with target ϕ(a).

Remark 5.5. The terminology “graph fibration” follows Boldi and Vigna [BV]. The term

“fibration of networks of manifolds” was introduced in [DL1].

Since the category of manifolds Man has finite products, we have a functor

P : (FinSet/Man)op → Man,

which assigns to a list τ : X → Man the corresponding product:

P(X, τ) =∏

a∈Xτ(a)


(cf. Subsection 4.1). In particular to every network (G, (G0,P)) the functor P assigns themanifold P(G0,P) which we think of as the total phase space of the network. And to everymap of networks of manifolds ϕ : (G, (G0,P))→ (G′, (G′0,P ′)) the functor P assigns a map

P(ϕ) : P(G′0,P ′)→ P(G0,P).between their total phase spaces.

Proposition 5.6. A network of manifolds (G, (G0,P)) encodes(1) a list of submersions I : G0 → SSub (which is an object of FinSet/SSub) and(2) an interconnection morphism IG : P(G0, I)→ (P(G0,P) id−→ P(G0,P)).

Remark 5.7. The data I : G0 → SSub and IG : P(G0, I) → (id : P(G0,P) → P(G0,P)together define a linear map

I∗G ◦ Crl(G0,I) :⊕

a∈G0Crl(I(a))→X (P(G0,P))

from the direct sum of vector spaces of control systems supported by the submersions I(a)

to the space of vector fields on the manifold P(G0,P).

Proof of Proposition 5.6. Note first that if

X Y

Man

τ ��

ϕ //

µ��

is a monomorphism in FinSet/Man, i.e., if ϕ : X → Y is injective, then Pϕ : P(Y, µ) →P(X, τ) is a surjective submersion.

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

• ιa : {a} ↪→ G0 and• s|t−1(a) : t−1(a)→ G0 (recall that s, t : G1 → G0 are source and target maps).

The set t−1(a) is the collection of arrows of G with target a and s sends this collection to thesources of the arrows. The composition with P : G0 → Man gives us two lists of manifolds:

P ◦ ιa : {a} → Manand

ξa := P ◦ s|t−1(a) : t−1(a)→ Man.Applying the functor P gives us two manifolds P(a) = P({a},P ◦ ιa) and P(t−1(a), ξa). Wedefine the submersion I(a) to be the projection on the second factor from P(t−1(a), ξa)×P(a)to P(a):

I(a) := (P(t−1(a), ξa)× P(a) pr2−−→ P(a)).(Recall that we have a faithful functor ı : Man2 → SSub. So I(a) = ı(P(a),P(t−1(a), ξa).)Since by definition Pa∈G0I(a) =

∏a∈G0 I(a), and since∏

a∈G0({a},P ◦ ιa) =

∏

a∈G0P(a) = P(G0,P),


the submersion Pa∈G0I(a) is the projection on the second factor∏

a∈G0P(t−1(a), ξa)

× P(G0,P) → P(G0,P).

Equivalently the submersion Pa∈G0I(a) is obtained by applying the functor P to the monomor-phism (G0,P)→ (G0 t

⊔a∈G0 t

−1(a),P t (⊔a∈G0 ξa)) in FinSet/Man:

Pa∈G0I(a) = P

(G0,P) ↪→ (G0 t

⊔

a∈G0t−1(a),P t

⊔

a∈G0ξa))

(5.8)

To construct a map of submersions IG : P(G0, I)→ (P(G0,P) id−→ P(G0,P)) it suffices toconstruct a map f : P(G0,P)) →

∏a∈G0 P(t

−1(a), ξa). This map too comes from a map offinite sets. Namely, the family {s|t−1(a) : t−1(a)→ G0}a∈G0 defines

ts|t−1(a) :⊔t−1(a)→ G0,

and the diagram

⊔t−1(a) G0

Man

tξa��

t(s|t−1(a)) //

P��

commutes. So let f = P(ts|t−1(a)). �

Example 5.9. Let G be the graph

1 2 3

.

Let P be the function that assigns to every node the same manifold M . Then

I(a) = (M ×M pr1−−→M)

for every a ∈ G0, P(G0, I) = (M ×M →M)3 ' (M3 ×M3 →M3) and I : M3 → (M2)3 isgiven by

I(x1, x2, x3) = ((x1, x2), (x2, x1), (x3, x2))for all (x1, x2, x3) ∈M3. Finally (IG)∗ ◦ Crl(G0,I) : Crl(M ×M →M)⊕3 →X (M3) is givenby

((IG)∗ ◦ Crl(G0,I))(w1, w2, w3)

)(x1, x2, x3) =

(w1(x1, x2), w2(x2, x1), w3(x3, x2)

).

Proposition 5.10. Suppose ϕ : (G, (G0,P)) → (G′, (G′0,P ′)) is a fibration of networks ofmanifolds. Then


(1) For each a ∈ G0 we have an isomorphism Φ(a) : I(a) → I ′(ϕ(a)) of submersions.That is, ϕ gives rise to the morphism

G0 G′0

SSubI ��

ϕ //

I′��

Φrz

in (FinSet/SSub)⇐.(2) The diagram

P(G0, I)(P(G0,P) id−→ P(G0,P)

)

P(G′0, I ′)(P(G′0,P ′)

id−→ P(G′0,P ′))

oo IGOO

P(ϕ,Φ)

ooIG′

OO

(P(ϕ),P(ϕ))

commutes in SSub.

Proof of Proposition 5.10.

(1) Since ϕ : G→ G′ is a graph fibration, the restriction

ϕ|t−1(a) : t−1(a)→ t−1(ϕ(a))

is a bijection for each vertex a of the graph G. Since P ′ ◦ ϕ = P

φ(a) := ϕ|t−1(a) : (t−1(a), ξa)→ (t−1(ϕ(a)), ξϕ(a))

is an isomorphism in FinSet/Man. Here, as in the proof of Proposition 5.6,

ξa = P ◦ s|t−1(a)and ξϕ(a) is defined similarly. Consequently

P(ϕ|t−1(a)) : P(t−1(a), ξa)→ P(t−1(ϕ(a)), ξϕ(a))

is an isomorphism in Man. Thus for each a ∈ G0 we have an isomorphism of surjectivesubmersions

Φ(a) :(P(t−1(a), ξa)× P(a)→ P(a)

)−−→

(P(t−1(ϕ(a)), ξϕ(a))× P ′(ϕ(a))→ P ′(ϕ(a))

)

(note that P(a) = P”(ϕ(a))). The family of isomorphisms {Φ(a)}a∈G0 together with themap ϕ define a morphism

G0 G′0

SSubI ��

ϕ //

I′��

Φrz

in (FinSet/SSub)⇐.


(2) Since ϕ is a fibration of networks, for each node a of the graph G we have a commuting

square in FinSet/Man:

t−1(a) G0

t−1(ϕ(a)) G′0

Man

s|t−1(a) //

φ(a)��

ϕ��

s|t−1(ϕ(a)) //

ξa

��

P

��ξϕ(a) $$ P ′

. (5.11)

We now drop the maps to Man to reduce the clutter and only keep track of maps of finite

sets. By the universal property of coproducts the diagrams 5.11 define a unique map

Ψ:⊔

a∈G0t−1(a)→

⊔

b∈G′0

t−1(b)

in FinSet/Man so that the diagram

t−1(a)⊔a′∈G0 t

−1(a′) G0

t−1(ϕ(a))⊔b∈G′0 t

−1(b) G′0

//

φ(a)

��//

Ψ

��

ϕ

��

tb∈G′0s|t−1(b)//

ta′∈G0s|t−1(a′) //

commutes for all nodes a ∈ G0. Applying the functor P gives the commuting diagram inMan:

P(⊔a∈G0 t

−1(a)) P(G0)

P(⊔b∈G′0 t

−1(b)) P(G′0)

OO

P(Ψ)

OO

P(ϕ)

ooP(tb∈G′0s|t−1(b))

ooP(ta′∈G0s|t−1(a′))

.

The fact that P takes coproducts to products and the universal properties ensure that

(P(Ψ)× P(ϕ),P(ϕ)) = P(ϕ,Φ).

�

We are now in position to recover Theorem 3 of [DL1]:


Theorem 5.12. A fibration of networks of manifolds ϕ : (G, (G0,P))→ (G′, (G′0,P ′)) givesrise to a 2-cell

⊕b∈G′0 Crl(I

′(b))⊕

a∈G0 Crl(I(a))

X (P(G′0,P ′)) X (P(G0,P))

�(ϕ,Crl◦Φ)

(IG′ )∗◦Crl(G′0,I′) (IG)∗◦Crl(G0,I)

X (P(ϕ))

⇓

in RelVect�. That is, for every family open systems (wb)b∈G′0 ∈⊕

b∈G′0 Crl(I′(b)) we have a

map of dynamical systems

P(ϕ) :(

(P(G′0,P ′), ((I ′G)∗ ◦ Crl(G′0,I′)) ((wb)))−−→

((P(G0,P), ((IG)∗ ◦ Crl(G0,I) ◦ �(ϕ,Crl ◦ Φ)) ((wb))

). (5.13)

Proof. Combine Proposition 5.10 and Theorem 4.16. �

We end the section with an example. It is essentially Example 4.19 of the introduction

expressed in the language of fibrations of networks of manifolds.

Example 5.14. Let ϕ : G→ G′ be the graph fibration

1 2 3*ϕ

.

Define P ′ : G′0 → Man by setting P ′(∗) = M for some manifold M on the single vertex ∗ ofG′. Define P : G0 → Man by setting P(i) = M for i = 1, 2, 3. Then

ϕ : (G, (G0,P))→ (G′, (G′0,P ′))is a fibration of networks of manifolds. As in Example 5.9 we have a list of submersions

I : G0 → SSub withI(a) = (M ×M pr1−−→M)

for every a ∈ G0, P(G0, I) = (M ×M →M)3 ' (M3 ×M3 →M3) and I : M3 → (M2)3 isgiven by

IG(x1, x2, x3) = ((x1, x2), (x2, x1), (x3, x2))for all (x1, x2, x3) ∈M3. Finally (IG)∗ ◦ Crl(G0,I) : Crl(M ×M →M)⊕3 →X (M3) is givenby

((IG)∗ ◦ Crl(G0,I))(w1, w2, w3)

)(x1, x2, x3) =

(w1(x1, x2), w2(x2, x1), w3(x3, x2)

).

Similarly we have I ′ : G′0 = {∗} → SSub given by

I ′(∗) = (M ×M pr1−−→M),P(G′0, I ′) = (M ×M →M) and IG′ : M →M ×M is given by

IG′(x) = (x, x)


for all x ∈M . Hence (IG′)∗ ◦ Crl(G′0,I) : Crl(M ×M →M)→X (M) is given by

((IG′)∗ ◦ Crl(G′0,I)(w)

)(x) = w(x, x).

With these definitions the diagram

G0 G′0

SSubI ��

ϕ //

I′��

idrz

commutes. It is not hard to see that

P(ϕ, id) : (M ×M →M)→ (M ×M →M)3

is the diagonal map and

�(ϕ,Crl ◦ id) : Crl(M ×M →M)→ Crl(M ×M →M)⊕3

is given by

�(ϕ,Crl ◦ id)(w) = (w,w,w).The map Pϕ : M = P(G′0,P ′) → P(G0, (G0,P)) = M3 is the diagonal map. By Theo-rem 5.12 for any open systems w ∈ Crl(M ×M →M) the vector fields

v : M → TM, v(x) = w(x, x)and

u : M3 → TM3, u(x1, x2, x3) = ((w(x1, x2), w(x2, x1), w(x3, x2))are Pϕ-related.

6. Wiring diagrams

In order to prove our main result we constructed a lax monoidal functor

Crl : SSub� → RelVect�

between two monoidal double categories. This functor does two things: it keeps track

of relations between dynamical systems (dynamical morphisms) and it encodes plugging

outputs into inputs (interconnection morphisms). If we only want to focus on building

more complicated systems out of subsystems we may forget dynamical morphisms SSub�

and linear relations in RelVect�. We are then left with a monoidal contravariant functor

Crl : (SSubI)op → Vect,where the objects SSubI are surjective submersions, the morphisms are the interconnec-tion morphisms and the monoidal product is the Cartesian product of submersions. The

monoidal product in Vect is the direct sum (which is both a categorical product and a

coproduct). On objects it is given by

Crl(p : Q→M) = {f : Q→ TM | π ◦ f = p},where, as before, π : TM →M is the canonical projection.


Since every symmetric monoidal category C defines a colored operad OC we get an algebraOCrl : O(SSubI)op → Vect.

This algebra looks very similar to the algebra

OG : OW→ Setover the operad of OW of wiring diagrams defined in [VSL].

There are however some fundamental differences. To make the comparison easier let us

now recall the definition of the monoidal category W of wiring diagrams and the functor

G : W → Set. To begin with open continuous time dynamical systems are viewed in [VSL]somewhat differently from the way we have been viewing them in this paper. There an

open system consists of three manifolds M,U inp, Uout and two smooth maps

f inp : M × U inp → TM, fout : M → Uout

with f inp being an open system on the product fiber bundle M ×U inp →M in the sense ofDefinition 3.1. To distinguish the two approaches we will refer to the tuple

(M,U inp, Uout, f inp, fout)

as a factorized open system f inp with output fout. The manifolds M,U inp, Uout are, re-

spectively, the spaces of states, inputs and outputs of the system (M,U inp, Uout, f inp, fout).

Factorized open systems with outputs form a category, which in [VSL] is called ODS. By def-

inition a morphism from (M1, Uinp1 , U

out1 , f

inp1 , f

out1 ) to (M2, U

inp2 , U

out2 , f

inp2 , f

out2 ) is a triple

of maps ζ = (ζst : M1 → M2, ζ inp : U inp1 → U inp2 , ζout : Uout1 → Uout2 ) so that the followingdiagram

M1 × U inp1f inp1 ×fout1//

ζst×ζinp��

TM1 × Uout1Tζst×ζout��

M2 × U inp2f inp2 ×fout2

// TM2 × Uout2

commutes. This category has finite products.

The symmetric monoidal category W is defined as follows. The objects of W are pairs of

unordered lists of manifolds (or, equivalently, pairs of typed finite sets of type “manifold”).

Thus an object X of W is a pair (X inp, Xout) = (τ inp : Ainp → Man, τout : Aout → Man).The objects of W are called boxes. The morphisms in W are called wiring diagrams. A

wiring diagram is a triple (X,Y, ϕ) where X,Y are boxes and

ϕ : X inp t Y out → Xout t Y inp

is an isomorphism in FinSet/Man so that

ϕ(Y out) ⊆ Xout. (6.1)Condition (6.1) allows us to decompose ϕ into a pair ϕ = (ϕinp, ϕout):

{ϕinp : X inp → Xout t Y inpϕout : Y out → Xout . (6.2)


A wire in a wiring diagram (X,Y, ϕ) is a pair (a, b), where a ∈ X inptY out, b ∈ XouttY inp,and ϕ(a) = b. The monoidal product on W is disjoint union:

(τ inp : Ainp → Man, τout : Aout → Man) t (µinp : Binp → Man, µout : Bout → Man) :=(τ inp t µinp : Ainp tBinp → Man, τout t µout : Aout tBout → Man).

The semantics of W is obtained by filling in the boxes in the following sense. Given a box

X = (X inp, Xout) ∈ (FinSet/Man)2 we have a pair of manifolds (PX inp,PXout), where asbefore the functor P : (FinSet/Man)op → Man is given on objects by taking products:

P(τ : A→ Man) :=∏

a∈Aτ(a).

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

M × PX inp →M.We then can further choose an output map fout : M → PXout and a factorized open systemf inp : M × PX inp → TM . This is the consideration behind the definition of the functorG : W→ Set. Its value on an object X of W is, by definition, the collection

G(X) := {(S, f) | S ∈ FinSet/Man, f inp × fout : PS × PX inp → TPS × PXout}where f = f inp×fout are factorized open systems with outputs. (To make sure that G(X) isactually a set and not a bigger collection we should, strictly speaking, replace the category

Man of manifolds by an equivalent small category. For example we can redefine Man to

consist of manifolds that are embedded in the disjoint union tn∈NRn.)We now see that the monoidal category 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 SSub where the objects specify the spaces of states of the systems.

Note also that the functor G : W → Set is a bit coarse. For example if we start with abox X whose inputs X inp and Xout are both singletons of type “point” then G(X) is theset of all possible continuous time dynamical systems. One can upgrade G to a functor withvalues in the category Cat of small categories and keep track of the dynamical morphisms in

this setting. At the cost of adding some complexity, one can then modify W to a monoidal

double category and G to a functor thereof. This would allow us to express an additionallayer of interactions. We may address this elsewhere.

To conclude, the approaches of [VSL] and the present paper differ significantly in their

philosophies and offer distinct perspectives on the open continuous time dynamical systems.

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Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green

Street Urbana, IL 61801

Massachusetts Institute of Technology, Department of Mathematics, E17-332, 77 Mas-

sachusetts Avenue, Cambridge, MA 02139-4307

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