aa r X i v : . [ m a t h . C T ] A ug UNIVERSITY OF CALIFORNIARIVERSIDEOpen Systems: A Double Categorical PerspectiveA Dissertation submitted in partial satisfactionof the requirements for the degree ofDoctor of PhilosophyinMathematicsbyKenny Allen CourserMarch 2020Dissertation Committee:
Professor John Baez, ChairpersonProfessor Jacob GreensteinProfessor David Weisbartopyright byKenny Allen Courser2020 cknowledgments
First, I would like to thank my advisor John Baez for taking me on to be his student. Without hispatience and humor, this thesis would not exist. I would like to thank Mike Shulman, ChristinaVasilakopoulou, Daniel Cicala, Blake Pollard and David Weisbart for their help and guidance overthe past several years. I would like to thank the other members of our category theory researchgroup during my time here at University of California, Riverside—Brandon Coya, Jason Erbele,Joe Moeller, Jade Master and Christian Williams—for helpful discussions and feedback. I wouldlike to thank my grandparents Valerie and Lup´e for their love and support and for allowing me tofocus on school. I would like to thank my little brother, Christian, my little sister, Catherine, myolder sister, Emily, my older brother, Andrew, and my dad, Frank, for all the memories createdwhile growing up, and of course, Buster. I would like to thank my cohort—Adam and Bryansito >< , Eddie, Josh, Kevin, Mikahl, Ryan, Tim and James—for their companionship while traversingthe gauntlet of grad school. I would also like to thank my hometown friends—Jimmy, Richie, Tj,Dombot, Kurt, Daniel and Austin—for all the memories in middle school, high school and earlycollege. I would like to thank the community over at the nLab; whenever I needed to look up aparticular concept or idea, the nLab was one of the first places that I would look. And lastly, Iwould like to thank my mom, Jodi, for her love and support and for bringing me into this world.ivo my mother, Jodi.v BSTRACT OF THE DISSERTATION
Open Systems: A Double Categorical PerspectivebyKenny Allen CourserDoctor of Philosophy, Graduate Program in MathematicsUniversity of California, Riverside, March 2020Professor John Baez, ChairpersonFong developed ‘decorated cospans’ to model various kinds of open systems: that is, systems withinputs and outputs. In this framework, open systems are seen as the morphisms of a category andcan be composed as such, allowing larger open systems to be built up from smaller ones. Muchwork has already been done in this direction, but there is a problem: the notion of isomorphismbetween decorated cospans is often too restrictive. Here we introduce and compare two waysaround this problem: structured cospans, and a new version of decorated cospans. Structuredcospans are very simple: given a functor L : A → X , a ‘structured cospan’ is a diagram in X ofthe form L ( a ) → x ← L ( b ). If A and X have finite colimits and L is a left adjoint, there is asymmetric monoidal category whose objects are those of A and whose morphisms are isomorphismclasses of structured cospans. However, this category arises from a more fundamental structure:a symmetric monoidal double category. Under certain conditions this symmetric monoidal doublecategory is equivalent to one built using our new version of decorated cospans. We apply theseideas to symmetric monoidal double categories of open electrical circuits, open Markov processesand open Petri nets. vi ontents A.1 Everyday categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146A.1.1 Monoidal categories and monoidal functors . . . . . . . . . . . . . . . . . . . 150A.1.2 Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153A.2 Double categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156A.2.1 Monoidal double categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163A.2.2 Monoidal double functors and transformations . . . . . . . . . . . . . . . . . 168A.3 Bicategories and 2-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170A.3.1 Pseudofunctors and pseudonatural transformations . . . . . . . . . . . . . . . 172
Bibliography 174 viii hapter 1
Introduction
This is a thesis about compositional frameworks for describing ‘open networks’, which are net-works with prescribed ‘inputs’ and ‘outputs’. One well-known type of network is a ‘Petri net’. Petrinets are important in computer science, chemistry and other subjects. For example, the chemicalreaction that takes two atoms of hydrogen and one atom of oxygen and produces a molecule ofwater can be represented by this very simple Petri net: HO α H O Here we have a set of ‘places’ (or in chemistry, ‘species’) drawn in yellow and a set of ‘transitions’(or ‘reactions’) drawn in blue. The disjoint union of these two sets then forms the vertex set of adirected bipartite graph, which is one description of a Petri net.Networks can often be seen as pieces of larger networks. This naturally leads to the idea of an open
Petri net, meaning that the set of places is equipped with inputs and outputs. We can dothis by prescribing two functions into the set of places that pick out these inputs and outputs. Forexample: HO α H O a b The inputs and outputs let us compose open Petri nets. For example, suppose we have another openPetri net that represents the chemical reaction of two molecules of water turning into hydronium1nd hydroxide: H O β OH − H O + cb Since the outputs of the first open Petri net coincide with the inputs of the second, we can composethem by identifying the outputs of the first with the inputs of the second: HO α H O β OH − H O +
123 56 a c
Similarly we can ‘tensor’ two open Petri nets by placing them side by side: HO α H O
123 4 H O β OH − H O + b + ca + b The compositional nature of these open Petri nets, and of open networks in general, is suggestiveof an underlying categorical structure. Moreover, the ability to tensor these open networks nat-urally leads to a symmetric monoidal structure on these categories. In this thesis we study twoframeworks for constructing and working with symmetric monoidal categories whose morphisms areopen networks. The first, ‘decorated cospans’, was introduced by Brendan Fong [7]. The second,‘structured cospans’, is new. Here we study both frameworks using symmetric monoidal double categories, which have 2-morphisms that describe maps between open networks.The outline of the thesis is as follows. In Chapter 2, we present Fong’s decorated cospans andgive some examples in which they have been applied: graphs, electrical circuits, Markov processesand Petri nets. In Chapter 3, we introduce the framework of structured cospans. In Chapter 4, werevisit decorated cospans but at the level of double categories. In Chapter 5, we explore some of thesimilarities between double categories and bicategories, and in Chapter 6, we give an applicationof double categories to Markov processes and ‘coarse-grainings’ and show that coarse-graining iscompatible with black-boxing. This last application is constructed using neither structured cospans2or decorated cospans due to the complexity of its 2-morphisms, but is nevertheless a great exampleof how the rich structure of double categories and their appropriate maps can be used to modelcomplicated open dynamical systems.The first piece of work that this thesis is built upon,
A bicategory of decorated cospans [18],was an initial attempt at categorifying Fong’s theory of ‘decorated cospans’, which we introduce inChapter 2. Following a suggestion of Mike Shulman [37], this attempt made extensive use of doublecategories, and it was here that the current author’s journey into double categories began. Overthe course of this journey, John Baez noticed a flaw with the decorated cospans framework, whichwe explain in Section 2.2.1 and also at the beginning of Chapter 3. Thus, Baez conceived anotherframework which simultaneously corrected this flaw and was more convenient to use: ‘structuredcospans’. This is the main content of Chapter 3. This new framework also employs double cat-egories, and several applications which were previously illustrated using decorated cospans wereexplored using structured cospans in a recent paper with Baez,
Structured cospans [3]. Then,following along on this double categorical campaign, a more direct fix to decorated cospans wasintroduced by Baez, Vasilakopoulou and the author in
Structured versus decorated cospans [4].This material constitutes Chapter 4: the main result is that the new improved decorated cospansare equivalent to structured cospans under certain mild conditions. Tangential to all of this, Baezand the current author wrote
Coarse-graining open Markov processes [2]. While this work alsomakes use of double categories, it uses neither decorated nor structured cospans, due to some moresophisticated structure that is necessary. This material makes up Chapter 6.3 hapter 2
Decorated cospan categories
This chapter is devoted to Fong’s theory of decorated cospans and a few of its applications.Fong’s theory of decorated cospans is well-suited to describing open networks: that is, networkswith prescribed inputs and outputs. We can build larger networks from smaller ones by attachingthe inputs of one to the outputs of another. This suggests that we should treat open networksas morphisms in a category. In addition to composing open networks, we can also put them sideby side in parallel, giving a monoidal category. Fong’s Theorem on decorated cospans provides aframework that captures all of this structure and more. Fong’s decorated cospan categories canthen serve as syntax categories for functors that describe the behavior of open networks, such asthe ‘black-box’ functors studied by Baez, Fong, Master and Pollard [2, 7, 8, 9, 10].In Section 2.1, we present Fong’s Theorem. For definitions of the terms used in this theorem, seeAppendix A. In Section 2.2, we present some previously studied applications of decorated cospanswhich will later be revisited in subsequent chapters from the perspective of other compositionalframeworks. These examples include open graphs, open electrical circuits, open Markov processesand open Petri nets.
Definition 2.1.1. A cospan in any category C is a diagram of the form ba a i o In other words, a cospan is an ordered pair of morphisms i and o in C whose target coincide.A result of Fong [23] which has been fundamental in the inspiration of a large portion of thisthesis is the following. Theorem 2.1.2 (Fong) . Let C be a category with finite colimts and F : ( C , + , → ( Set , × , asymmetric lax monoidal functor. Then there exists a symmetric monoidal category F Cospan whichhas:
1) objects as those of C and(2) morphisms as isomorphism classes of F -decorated cospans in C , which are pairs: ba a d ∈ F ( b ) i o Two F -decorated cospans are in the same isomorphism class if the following diagrams com-mute: ba a b ′ F ( b ) F ( b ′ ) i oi ′ o ′ f ∼ dd ′ F ( f ) for some isomorphism f . The composite of two composable F -decorated cospans ba a a b ′ a d ′ ∈ F ( b ′ ) d ∈ F ( b ) i o i ′ o ′ is given by ba a b ′ a b + b ′ b + a b ′ i o i ′ o ′ ψ ψjjψi jψo ′ λ − −−→ × d × d ′ −−−→ F ( b ) × F ( b ′ ) φ b,b ′ −−−→ F ( b + b ′ ) F ( j ) −−−→ F ( b + a b ′ ) where ψ is the natural map into a coproduct, j is the natural map from a coproduct into apushout, and φ b,b ′ : F ( b ) × F ( b ′ ) → F ( b + b ′ ) is the natural transformation coming from thestructure of the symmetric lax monoidal functor F : ( C , + , → ( Set , × , .The tensor product of two objects a and a is given by their binary coproduct a + a in C . he tensor product of two F -decorated cospans is given pointwise: ba a a ′ b ′ a ′ d ′ ∈ F ( b ′ ) d ∈ F ( b ) ⊗ = a + a ′ b + b ′ a + a ′ d + d ′ ∈ F ( b + b ′ ) i o i ′ o ′ i + i ′ o + o ′ d + d ′ := 1 λ − −−→ × d × d ′ −−−→ F ( b ) × F ( b ′ ) φ b,b ′ −−−→ F ( b + b ′ )We will also need a variant of Fong’s Theorem that gives a merely monoidal category: Theorem 2.1.3.
Let C be a category with finite colimts and F : ( C , + , → ( Set , × , a laxmonoidal functor. Then there exists a monoidal category F Cospan where the relevant structureis given as in Theorem 2.1.2.
The necessity of this weaker result was pointed out by an anonymous referee of Moeller andVasilakopoulou [35], which we explain in the introduction of Chapter 3 on ‘structured cospans’.
In this section we present some examples of applications of decorated cospans which have beenstudied in previous works [7, 8, 10, 12, 23].
Our first example is the category of ‘open graphs’. This makes clear some difficulties inFong’s approach to decorated cospans—problems that will be solved using our double categoryapproach. Let (
FinSet , + ,
0) denote the category of finite sets and functions made symmetricmonoidal using coproducts. To apply Fong’s Theorem, we seek a symmetric lax monoidal functor F : ( FinSet , + , → ( Set , × ,
1) that assigns to a finite set N the set of all graphs whose underlyingset of vertices is N . So, we define a graph structure on N to be a diagram in FinSet of thefollowing form.
E N st Here E is the set of edges of the graph while s, t : E → N are the source and target functions,respectively.If we naively try to take F ( N ) to be the set of graph structures on N , we immediately notice aproblem: this is not a set, but rather a proper class. Fong [23] gets around this by replacing FinSet with an equivalent small category, which by abuse of notation we shall call
FinSet . Using this smallversion of
FinSet in the definition of graph structure, we see that there is an actual set F ( N ) ofgraph structures on any N ∈ FinSet . Given a function f : N → N ′ we define F ( f ) : F ( N ) → F ( N ′ )6s follows. Given a graph structure on N , the function f induces a graph structure on N ′ if wedemand that the following diagrams commute: E NN ′ E NN ′ f fss ′ tt ′ This results in a graph structure on N ′ given by s ′ , t ′ : E → N ′ where s ′ = f s and t ′ = f t . In otherwords, we are pushing forward the set E of edges along the function f in such a way that sourcesand targets of edges are preserved. It is clear that this procedure is associative and preservesidentities, and thus defines a functor F : FinSet → Set .The next question is whether F is lax monoidal. For this, note that given a graph structure d on a finite set N and a graph structure d on another finite set N , there is a graph structure d + d on N + N , given by taking pointwise coproducts of the respective graph structures on N and N : E + E N + N s + s t + t One can check that there is a natural transformation µ N ,N : F ( N ) × F ( N ) → F ( N + N )mapping ( d , d ) to d + d , as one would expect if F were lax monoidal and µ were its laxator.Note the non-invertibility of the maps µ N ,N . For example, the figure below shows two graphs d ∈ F ( N ) and d ∈ F ( N ) in black; taking them together we get d + d ∈ F ( N + N ). If wealso include the red edge we obtain a graph that is not in the image of the laxator µ N ,N , but is aperfectly fine element of F ( N + N ). v v v w w w w Γ ∈ F ( N ) Γ ∈ F ( N ) e ′ e ′ e ′ e ′ e e e We also have a morphism µ : 1 → F ( ∅ ) which is, in fact, an isomorphism as the empty graph withno edges is the only possible graph structure on ∅ . However, as pointed out by the anonymousreferee of Moeller and Vasilakopoulou’s paper [35], µ does not obey the hexagon law required ofa lax monoidal functor! We explain why at the start of Chapter 3. To fix this, we can use Mac7ane’s Theorem to choose a small strict monoidal category equivalent to ( FinSet , + , FinSet , + ,
0) to denote this small strict monoidal category. Then we obtain the desired laxmonoidal functor F : ( FinSet , + , → ( Set , × , FinSet , + ,
0) for which the symmetries are identities. By Theorem 2.1.3, we have the following:
Corollary 2.2.1.
Let F : ( FinSet , + , → ( Set , × , be the lax monoidal functor described abovewhich assigns to N ∈ FinSet set of all graph structures whose underlying set of vertices is N . Thenthere exists a monoidal category F Cospan which has:(1) objects as those of ( FinSet , + , and(2) morphisms as isomorphism classes of open graphs , where an open graph is given by a pairof diagrams: NX Y E N sti o
Two open graphs are in the same isomorphism class if the following diagrams commute:
NX YN ′ f ∼ i oi ′ o ′ E NN ′ E NN ′ f fss ′ tt ′ for some isomorphism f . Composition and tensoring of objects and morphisms are given asin Theorem 2.1.2. Again we emphasize that in the above theorem we are using (
FinSet , + ,
0) to mean some smallstrict monoidal category equivalent to the usual category of this name. For any object N in thiscategory, F ( N ) is the set of all graph structures on N defined using this equivalent category. Thus,given graph structures on objects N , N and N : E N E N E N s t s t s t E + ( E + E ) N + ( N + N ) ( E + E ) + E N + ( N + N ) s + ( s + s ) t + ( t + t ) ( s + s ) + s ( t + t ) + t This strictification in the graph structures is necessary in order for the functor F of the previouscorollary to be lax monoidal. We will also employ this strictification of structures in the followingtwo applications. The remaining two applications, while taking on more of an applied flavor, are structurally verysimilar.
Definition 2.2.2.
Given a field k , a field with positive elements is a pair ( k, k + ) where k + ⊂ k is a subset such that r ∈ k + for every nonzero r ∈ k and such that k + is closed under addition,multiplication and division. Definition 2.2.3.
Let k be a field with positive elements. A k -graph is given by a diagram: k + E N str where r ( e ) ∈ k + is the resistance along the edge e ∈ E .Following the same ideas as in the previous example and using a small strict monoidally equiv-alent copy of FinSet , we see there is a lax monoidal functor that assigns to any N ∈ FinSet the setof all k -graph structures on N . Thus, by Theorem 2.1.3, we have the following. Theorem 2.2.4.
Let F : ( FinSet , + , → ( Set , × , be the lax monoidal functor which assigns toany N ∈ FinSet the set of all k -graph structures on N . Then there exists a monoidal category F Cospan which has:(1) objects as those of ( FinSet , + , and(2) morphisms as isomorphism classes of open k -graphs , where an open k -graph is given by apair of diagrams: NX Y k + E N r sti o wo open graphs are in the same isomorphism class if the following diagrams commute: NX YN ′ f ∼ i oi ′ o ′ k + E E NN ′ E NN ′ f fr ss ′ r ′ tt ′ for some isomorphism f . Composition and tensoring of objects and morphisms are given asin Theorem 2.1.2. An electrical circuit made of resistors can then be seen as a k -graph in which we take the field k tobe R and take k + to consist of the positive real numbers. Baez and Fong also consider more generalcircuits containing resistors, inductors and capacitors, using a larger field with positive elements[7]. They study the behavior of these circuits using a ‘black-boxing’ functor from F Cospan to acategory of linear relations.
Our final example involves Petri nets, which have been studied extensively by Baez and Masterin a recent work [9].
Definition 2.2.5. A Petri net is given by the following diagram in
Set . T N [ S ] st We call S the set of species and T the set of transitions ; N [ S ] stands for the free commutativemonoid on S .In this example, we wish to use Fong’s Theorem with a functor F that assigns to each set S theset F ( S ) of all Petri nets having S as their set of species. Unfortunately, if we do this, F ( S ) is nota set: it is a proper class. To avoid this problem, we invoke the axiom of universes and choose aGrothendieck universe U . We call sets in U small and arbitrary sets large .We let ( Set , + ,
0) be a strict monoidal category that is monoidally equivalent to the category ofsmall sets with coproduct as its monoidal structure. The category (
Set , + ,
0) is a large category:more precisely, it is a category with a large set of objects and a large set of morphisms. For any S ∈ ( Set , + , F ( S ) of Petri nets having S as its set of species and some T ∈ ( Set , + ,
0) as its set of transitions. We write (
SET , × ,
1) for the category of large sets with10roduct as its monoidal structure. We can make F : ( Set , + , → ( SET , × ,
1) into a lax monoidalfunctor where the natural transformation µ S ,S : F ( S ) × F ( S ) → F ( S + S )is obtained in the same way as the previous natural transformations in the last three examples,namely by considering two individual Petri nets in parallel as a single Petri net. By Fong’s Theorem2.1.3, we have the following. Theorem 2.2.6.
Let F : ( Set , + , → ( SET , × , be the lax monoidal functor that assigns to a set S the large set F ( S ) of all Petri nets whose set of species is given by the set S . Then there existsa monoidal category F Cospan which has:(1) objects as those of ( Set , + , and(2) morphisms as isomorphism classes of open Petri nets which are given by pairs of diagrams: SX Y T N ( S ) sti o Two open Petri nets are in the same isomorphism class if the following diagrams commute:
SX YS ′ f ∼ i oi ′ o ′ T N [ S ] N [ S ′ ] T N [ S ] N [ S ′ ] N [ f ] N [ f ] ss ′ tt ′ for some isomorphism f . Composition and tensoring of objects and morphisms is given as inTheorem 2.1.2. Following ideas similar to those in the last two examples, Baez and Master study the reachabilityrelation of states of open Petri nets via black-boxing [9]. They in fact go further and construct a‘double category’ of open Petri nets and a corresponding black box double functor which shows acertain compatibility relation between ‘maps of open Petri nets’ and their black-boxings. Doublecategories are at the heart of this thesis and we will begin using them in the next chapter.11 hapter 3
Structured cospan double categories
The present chapter is about a particular kind of double categories, namely ‘foot-replaced doublecategories’. The first main result of this chapter is the construction of foot-replaced double cate-gories in Theorem 3.1.1 and the corresponding symmetric monoidal versions of these in Theorem3.1.2. The most important kind of foot-replaced double categories are the ‘structured cospan dou-ble categories’, which are the content of Theorem 3.2.3. In Section 3.3 we revisit the applicationsof Section 2.2, but from the perspective of structured cospans. In Section 3.4 we define maps offoot-replaced double categories, of which maps between structured cospan double categories are aspecial case. But first, let us explain the need for some of these concepts. At this point it wouldbe fruitful for readers unfamiliar with double categories to read Appendix A.2.Recall the first example of Fong’s theory of decorated cospans introduced in the previous chapter.Let F : FinSet → Set be the symmetric lax monoidal functor that assigns to a finite set b the (large)set of all possible graph structures on the finite set b , where a graph structure on b is given by adiagram in Set of the form:
E b. st Let b = { v , v } be a two element set. Then one element of the (large) set F ( b ), which is thecollection of all graph structures on the finite set b , is given by a single edge e whose source andtarget are v and v , respectively. v v e Denote this element of F ( b ) as d : 1 → F ( b ). Let a = { } and a = { } and define functions i : a → b and o : a → b by i (1) = v and o (2) = v . Then we have an F -decorated cospan: a b a F ( b ) i o d which is given by this open graph: i ov v e a b a a b ′ a F ( b ) 1 F ( b ′ ) i o i ′ o ′ d d ′ For these two F -decorated cospans to be in the same isomorphism class, the following triangle isto commute: F ( b ) F ( b ′ ) dd ′ F ( f ) This commutative triangle in
Set in the context of the symmetric lax monoidal functor F : FinSet → Set says the following: given a decoration d ∈ F ( b ), which is a graph structure with underlyingset of vertices b , the function F ( f ) pushes forward the graph structure d to the graph structure d ′ ∈ F ( b ′ ) with underlying set of vertices b ′ , and precisely this graph structure. The graph structureis given by the set of edges of d . For example, take b = { v , v } as before and let d ∈ F ( b ) be givenby: i ov v e Let b ′ = { w , w } and define a bijection f : b → b ′ by f ( v i ) = w i for i = 1 ,
2. Then the requirement F ( f )( d ) = d ′ says that d ′ ∈ F ( b ′ ) must be given by: i ′ o ′ w w e The important point is that the single edge of d ′ must also be e . If we were to label it say, e ′ ,there is no bijection f : b → b ′ such that the triangle above commutes, and hence no isomorphismbetween these two F -decorated cospans. F ( b ) F ( b ′ )1 2 i oi ′ o ′ ∄ F ( f ) ⇓ v v w w dd ′ ∄ F ( f ) ee ′ Thus, these two F -decorated cospans constitute distinct isomorphism classes! This nuisance isamplified when viewed from a higher categorical perspective, as seen in the first attempt at buildinga bicategory of decorated cospans [18]. In the first proposed bicategory F Cospan ( C ), there is no13-morphism from the former single-edged graph to the latter, when clearly there ought to be. Thetheory of foot-replaced double categories serves to remedy this situation. Again, for an introductionto double categories, see Appendix A.2.Another obstacle with decorated cospans was pointed out by an anonymous referee of Moellerand Vasilakopoulou [35]. For the original incarnation of decorated cospans, we start with a sym-metric lax monoidal functor F : ( C , + , → ( Set , × ,
1) where C is a finitely cocomplete categorymade symmetric monoidal with chosen binary coproducts and an initial object. The anonymousreferee has pointed out that even in the simplest of examples, namely the example of open graphsin Section 2.2.1, the ‘laxator hexagon’ required to commute in the definition of symmetric laxmonoidal functor (Definition A.1.8) may do so only up to isomorphism. This can be seen explicitlywith the following example.Let F : ( FinSet , + , → ( Set , × ,
1) be the functor of Section 2.2.1. In order for F to actually bea lax monoidal functor, the following laxator hexagon must commute: ( F ( a ) ⊗ F ( b )) ⊗ F ( c ) F ( a ) ⊗ ( F ( b ) ⊗ F ( c )) F ( a ⊗ b ) ⊗ F ( c ) F ( a ) ⊗ F ( b ⊗ c ) F (( a ⊗ b ) ⊗ c ) F ( a ⊗ ( b ⊗ c )) α ′ µ a,b ⊗ F ( c ) F ( a ) ⊗ µ b,c µ a ⊗ b,c µ a,b ⊗ c F ( α ) Let a = { a , a } , b = { b , b } and c = { c , c } all be two-element sets, and let d a ∈ F ( a ), d b ∈ F ( b )and d c ∈ F ( c ) be given by the following graph structures: a a b b c c e a e b e c Then the graph ( d a × d b ) × d c ∈ ( F ( a ) ⊗ F ( b )) ⊗ F ( c ) = ( F ( a ) × F ( b )) × F ( c ) is an objectof the category given by the top left corner of the above hexagon. Starting from this top leftcorner and traversing the object ( d a × d b ) × d c through the hexagon right and then down to F ( a ⊗ ( b ⊗ c )) = F ( a + ( b + c )) results in a graph with vertex set a + ( b + c ) and edge set { e a } + ( { e b } + { e c } ), whereas traversing the hexagon down and then right results in a graph withthe same vertex set a + ( b + c ) but now edge set ( { e a } + { e b } ) + { e c } . These two graphs wouldvisually appear to be the same and indeed have the same sets of vertices, but their edge sets wouldonly be (naturally) isomorphic, causing the above hexagon to not commute on-the-nose as requiredby the definition of lax monoidal functor.One remedy to this as suggested by John Baez is to replace the finitely cocomplete category( FinSet , + ,
0) containing our graph structures with an equivalent strictified version courtesy of atheorem of Mac Lane:
Theorem 3.0.1 (Mac Lane [34]) . Given a (braided, symmetric) monoidal category C , there exists astrict (braided, symmetric) monoidal category C ′ and a (braided) monoidal equivalence F : C → C ′ . A monoidal equivalence F is a functor that is simultaneously a monoidal functor and an equiv-alence, and a strict (braided, symmetric) monoidal category is a (braided, symmetric) monoidalcategory in which the associator and left and right unitors are identity morphisms. By taking14ur graph structures from the strict monoidal category ( FinSet , + , a + ( b + c ) and edge sets { e a } + ( { e b } + { e c } ) and ( { e a } + { e b } ) + { e c } are nowidentified and thus the laxator hexagon commutes. A similar problem arises with two unitalitysquares which is also resolved by this strictification, and thus we obtain the lax monoidal functor F : ( FinSet , + , → ( Set , × ,
1) of Section 2.2.1 and are able to utilize Theorem 2.1.3. Unfortunately,we are unable to obtain the desired symmetric monoidal category of Fong’s original Theorem 2.1.2.Structured cospans will also serve as a remedy to this problem.
The main content of this chapter are foot-replaced double categories as introduced in a workwith Baez [3]. A special case of foot-replaced double categories are given by structured cospandouble categories. A cospan in any category is diagram of the form: ba a i o We call b the apex of the cospan, i and o the legs of the cospan, and a and a the feet of thecospan. In the framework of structured cospan double categories, given a functor L : A → X a structured cospan is a cospan in X of the form: xL ( a ) L ( a ) i o Formally, this is a cospan in X whose feet are objects of X , but from the perspective of structuredcospans, the feet of this cospan are the objects a and a in A . Here we are replacing the feet of thecospan in X with objects from another category A , hence the name ‘foot-replaced double category’. Theorem 3.1.1.
Given a double category X and a functor L : A → X , there is a unique doublecategory L X for which: • an object is an object of A , • a vertical 1-morphism is a morphism of A , • a horizontal 1-cell from a to a ′ is a horizontal 1-cell L ( a ) M −→ L ( a ′ ) of X , • a 2-morphism is a 2-morphism in X of the form: L ( a ) L ( b ) L ( a ′ ) L ( b ′ ) , ⇓ α ML ( f ) L ( g ) N composition of vertical 1-morphisms is composition in A , • composition of horizontal 1-morphisms are defined as in X , • vertical and horizontal composition of 2-morphisms is defined as in X , • the associator and unitors are defined as in X . The proof is a straightforward verification using the definition of a double category, which isDefinition A.2.5. Throughout this thesis we use ‘double category’ to mean ‘pseudo double category’:composition of horizontal 1-cells need not be strictly associative. However, if the double category X is strict, so is the foot-replaced double category L X .There is also a version of Theorem 3.1.1 for symmetric monoidal double categories. Theorem 3.1.2. If X is a symmetric monoidal double category, A is a symmetric monoidal categoryand L : A → X is a (strong) symmetric monoidal functor, then the double category L X becomessymmetric monoidal in a canonical way.Proof. As noted in Definition A.2.5, every double category D has not only a category of objects D , but also a category of arrows D with horizontal 1-cells of D as objects and 2-morphisms of D as morphisms. The definition of a symmetric monoidal double category, which is Definition A.2.13,can be expressed in terms of structure involving these categories.For the double category L X , the category of objects L X is just A . The category of arrows L X has horizontal 1-cells in X of this form: L ( a ) M −→ L ( b )as objects and diagrams in X of this form: L ( a ) L ( b ) L ( a ′ ) L ( b ′ ) ⇓ α ML ( f ) L ( g ) N as morphisms, which are composed vertically.As explained in Definition A.2.12, to make L X into a monoidal double category we need to dothe following:(1) We must choose a monoidal structure for L X = A and for L X . The category A is monoidalby hypothesis; we give L X a monoidal structure using the fact that X and the functor L arestrong monoidal, as follows. Given two objects of L X : L ( a ) M −→ L ( a ) L ( b ) N −→ L ( b )their tensor product is L ( a ⊗ b ) φ − a ,b −−−−→ L ( a ) ⊗ L ( b ) M ⊗ N −−−−→ L ( a ) ⊗ L ( b ) φ a ,b −−−−→ L ( a ⊗ b ) , φ a,b : L ( a ) ⊗ L ( b ) → L ( a ⊗ b ) for L . Note that φ is invertible because L is strong monoidal. Given two morphisms of L X : L ( a ) L ( a ) L ( b ) L ( b ) L ( a ′ ) L ( a ′ ) L ( b ′ ) L ( b ′ ) ⇓ α ⇓ β ML ( f ) L ( f ) M ′ NL ( g ) L ( g ) N ′ their tensor product is defined to be L ( a ⊗ b ) L ( a ⊗ b ) L ( a ′ ⊗ b ′ ) L ( a ′ ⊗ b ′ ). ⇓ α ⊗ β φ a ,b ( M ⊗ N ) φ − a ,b L ( f ⊗ g ) L ( f ⊗ g ) φ a ′ ,b ′ ( M ′ ⊗ N ′ ) φ − a ′ ,b ′ The monoidal unit for L X is L ( I ) ˆ U ( L ( I )) −−−−−→ L ( I ) (3.1)where I is the monoidal unit for A and ˆ U : X → X is the identity-assigning functor for X . Theassociator and unitors for L X are built from those in X . Explicitly, given three horizontal 1-cells M, N and P in L X : L ( a ) L ( a ′ ) L ( b ) L ( b ′ ) L ( c ) L ( c ′ ) M PN the associator α M,N,P : ( M ⊗ N ) ⊗ P ∼ −→ M ⊗ ( N ⊗ P ) in L X is given by: L (( a ⊗ b ) ⊗ c ) L (( a ′ ⊗ b ′ ) ⊗ c ′ ) L ( a ⊗ ( b ⊗ c )) L ( a ′ ⊗ ( b ′ ⊗ c ′ )) ⇓ α M,N,P φ a ′ ⊗ b ′ ,c ′ ( φ a ′ ,b ′ ⊗ L ( c ′ ) )(( M ⊗ N ) ⊗ P )( φ − a,b ⊗ L ( c ) ) φ − a ⊗ b,c L ( α a,b,c ) L ( α a ′ ,b ′ ,c ′ ) φ a ′ ,b ′ ⊗ c ′ (1 L ( a ′ ) ⊗ φ b ′ ,c ′ )( M ⊗ ( N ⊗ P ))(1 L ( a ) ⊗ φ − b,c ) φ − a,b ⊗ c (2) Any double category D has an identity-assigning functor U : D → D , and for D to bemonoidal we need U to preserve the monoidal unit. This is true for L X because U : A → L X mapsany object a ∈ A to L ( a ) ˆ U ( L ( a )) −−−−−→ L ( a ) , so U maps the monoidal unit I ∈ A to the monoidal unit for L X , given in Equation (3.1).(3) In a monoidal double category D the source and target functors S, T : D → D must bestrict monoidal. For L X this is easy to check, given the monoidal structures defined in item (1),because the source and target of an object L ( a ) M −→ L ( b )17f L X are a ∈ L X and b ∈ L X , respectively, and the source and target of a morphism L ( a ) L ( b ) L ( a ′ ) L ( b ′ ) ⇓ α ML ( f ) L ( g ) N in L X are the morphisms f : a → a ′ and g : b → b ′ in L X , respectively. We can choose the imagesof the source and target functors to ensure that they are strict symmetric monoidal, meaning thatfor two horizontal 1-cells M and N , S ( M ⊗ N ) = a ⊗ a ′ = S ( M ) ⊗ S ( N )and likewise for the target morphism T . The unit for the tensor product in L X is given in Equation(3.1), and applying S or T we obtain I ∈ L X .(4) A globular 2-morphism in a double category D is a morphism α in D such that Sα and T α are identity morphisms in D . In a monoidal double category D we must have invertibleglobular 2-morphisms χ : ( M ⊗ N ) ⊙ ( M ⊗ N ) ∼ −→ ( M ⊙ M ) ⊗ ( N ⊙ N )and µ : U A ⊗ B ∼ −→ ( U A ⊗ U B )expressing the compatiblity of the composition functor ⊙ : D × D D → D and identity-assigningfunctor U : D → D with the tensor product. These must make three diagrams commute, asdetailed in Definition A.2.12. In the case of L X this follows from the commutativity of the corre-sponding diagrams in X together with the natural isomorphisms given by the invertible laxators ofthe strong monoidal functor L : A → X . Explicitly, given composable horizontal 1-cells M , M , N and N in L X : L ( a ) L ( a ) L ( b ) L ( b ) L ( a ) L ( a ) L ( b ) L ( b ) M M N N the globular 2-morphism χ for L X is given by: L ( a ⊗ b ) L ( a ) ⊗ L ( b ) L ( a ) ⊗ L ( b ) L ( a ) ⊗ L ( b ) L ( a ) ⊗ L ( b ) L ( a ⊗ b ) L ( a ⊗ b ) L ( a ⊗ b ) ⇓ χ φ − a ,b φ a ,b φ − a ,b φ a ,b ( M ⊗ N ) ⊙ ( M ⊗ N )1 1( M ⊙ M ) ⊗ ( N ⊙ N ) where the middle χ in the above diagram in the corresponding globular 2-morphism for the sym-metric monoidal double category X and φ a i ,b i : L ( a i ) ⊗ L ( b i ) → L ( a i ⊗ b i ) is the natural isomorphism18f the strong monoidal functor L : A → X . Similarly, the other globular 2-morphism µ for L X isgiven by: L ( a ⊗ b ) L ( a ⊗ b ) L ( a ⊗ b ) L ( a ⊗ b ) ⇓ µ U ( L ( a ⊗ b )) φ a,b ( U ( L ( a )) ⊗ U ( L ( b ))) φ − a,b (5) In a monoidal double category, the associator and left and right unitors must be transforma-tions of double categories. This means that six diagrams must commute, as detailed in DefinitionA.2.12. In the case of L X this follows from the commuting of the corresponding diagrams in X together with the natural isomorphisms given by the invertible laxators of the strong monoidalfunctor L : A → X . For instance, one of the diagrams required to commute is given by: ( M ⊗ N ) ⊙ U a ⊗ b M ⊗ N ( M ⊗ N ) ⊙ ( U a ⊗ U b )( M ⊙ U a ) ⊗ ( N ⊙ U b ) ⊙ µρ ρ ⊗ ρ χ For the symmetric monoidal double category L X , this diagram may be seen as: ⇑ ⊙ µ ⇑ ρ ⇓ χ ⇓ ρ ⊗ ρ ( M ⊗ N ) ⊙ ( U a ⊗ U b ) L ( a ⊗ b ) L ( a ′ ⊗ b ′ ) ( M ⊗ N ) ⊙ U a ⊗ b L ( a ⊗ b ) L ( a ′ ⊗ b ′ ) M ⊗ NL ( a ⊗ b ) L ( a ′ ⊗ b ′ ) ( M ⊙ U a ) ⊗ ( N ⊙ U b ) L ( a ⊗ b ) L ( a ′ ⊗ b ′ ) M ⊗ NL ( a ⊗ b ) L ( a ′ ⊗ b ′ ) ρ ⊙ uχρ ⊗ ρφ a ′ ,b ′ ( M ⊗ N ) φ − a,b φ a,b ( U ( L ( a )) ⊗ U ( L ( b ))) φ − a,b φ a ′ ,b ′ ( M ⊗ N ) φ − a,b U ( L ( a ⊗ b ))1 1 φ a ′ ,b ′ ( M ⊗ N ) φ − a,b φ a ′ ,b ′ (( M ⊙ U ( L ( a ))) ⊗ ( N ⊙ U ( L ( b )))) φ − a,b φ a ′ ,b ′ ( M ⊗ N ) φ − a,b Here we have ‘unrolled’ the diagram to make it fit on the page; the reader should identify theobjects at the top of the diagram with those at the bottom.Similarly, a braided monoidal double category is a monoidal double category with the followingadditional structure.(6) D and D are braided monoidal categories.(7) The functors S and T are strict braided monoidal (i.e. they preserve the braidings).198) The following diagrams commute, expressing that the braiding is a transformation of doublecategories. ( M ⊙ M ) ⊗ ( N ⊙ N ) s / / χ (cid:15) (cid:15) ( N ⊙ N ) ⊗ ( M ⊙ M ) χ (cid:15) (cid:15) ( M ⊗ N ) ⊙ ( M ⊗ N ) s ⊙ s / / ( N ⊗ M ) ⊙ ( N ⊗ M ) U A ⊗ U B µ / / s (cid:15) (cid:15) U A ⊗ BU s (cid:15) (cid:15) U B ⊗ U A µ / / U B ⊗ A . These follow from the fact that X and X are braided monoidal categories and that the cor-responding functors S and T of X are strict braided monoidal and we can choose the source andtarget functors of L X to agree with the braidings of L X and L X , meaning that β ′ ( S ( M ⊗ N )) = β ′ ( S ( M ) ⊗ S ( N )) = β ′ ( a ⊗ a ′ ) = a ′ ⊗ a = S ( N ⊗ M ) = S ( β ( M ⊗ N ))and likewise for the target morphism T . The above diagrams commute in L X as the correspondingdiagrams commute in X and the laxators of the strong monoidal functor L are invertible.(9) D and D are symmetric monoidal categories.This follows from the fact that A , X and X are symmetric monoidal categories. Explicitly, thetriangle identity for L X is given by: ⇓ α M, L X ,N ⇑ r ⊗ N ⇓ M ⊗ ℓ M ⊗ (1 ⊗ N ) L ( a ⊗ (1 A ⊗ b )) L ( a ′ ⊗ (1 A ⊗ b ′ )) ( M ⊗ ⊗ NL (( a ⊗ A ) ⊗ b ) L (( a ′ ⊗ A ) ⊗ b ′ ) M ⊗ NL ( a ⊗ b ) L ( a ′ ⊗ b ′ ) M ⊗ NL ( a ⊗ b ) L ( a ′ ⊗ b ′ ) r ⊗ N α M, L X ,N M ⊗ ℓφ a ′ , A ⊗ b ′ ( M ⊗ ( φ A ,b ′ (1 L X ⊗ N ) φ − A ,b )) φ − a, A ⊗ b φ a ′ ⊗ A ,b ′ (( φ a ′ , A ( M ⊗ L X ) φ − a, A ) ⊗ N ) φ − a ⊗ A ,b L ( α a, A ,b ) L ( α a ′ , A ,b ′ ) φ a ′ ,b ′ ( M ⊗ N ) φ − a,b L ( r ′ ⊗ b ) L ( r ′ ⊗ b ′ ) φ a ′ ,b ′ ( M ⊗ N ) φ − a,b L (1 a ⊗ ℓ ′ ) L (1 a ′ ⊗ ℓ ′ ) Here we have again ‘unrolled’ the diagram to make it fit on the page; the objects at the top of thediagram should be identified with those at the bottom.Now for notation, let
M, N, P and Q be horizontal 1-cells in L X given by: L ( a ) L ( a ′ ) L ( b ) L ( b ′ ) L ( c ) L ( c ′ ) L ( d ) L ( d ′ ) MP NQ
20s horizontal 1-cells of the symmetric monoidal double category X together with the associator ˆ α of X , the following pentagon commutes: (( M ⊗ N ) ⊗ P ) ⊗ Q ( M ⊗ ( N ⊗ P )) ⊗ Q M ⊗ (( N ⊗ P ) ⊗ Q )( M ⊗ N ) ⊗ ( P ⊗ Q ) M ⊗ ( N ⊗ ( P ⊗ Q )) ˆ α M,N,P ⊗ Q ˆ α M,N ⊗ P,Q ˆ α M ⊗ N,P,Q ˆ α M,N,P ⊗ Q M ⊗ ˆ α N,P,Q
Unrolling the pentagon identity for L X , we obtain the following diagram: ⇓ α M,N,P ⊗ Q ⇑ α M ⊗ N,P,Q ⇓ α M,N ⊗ P,Q ⇓ M ⊗ α N,P,Q ⇑ α M,N,P ⊗ Q L (( a ⊗ ( b ⊗ c )) ⊗ d ) L (( a ′⊗ ( b ′⊗ c ′ )) ⊗ d ′ ) L ((( a ⊗ b ) ⊗ c ) ⊗ d ) L ((( a ′⊗ b ′ ) ⊗ c ′ ) ⊗ d ′ ) L (( a ⊗ b ) ⊗ ( c ⊗ d )) L (( a ′⊗ b ′ ) ⊗ ( c ′⊗ d ′ )) L ( a ⊗ (( b ⊗ c ) ⊗ d )) L ( a ′⊗ (( b ′⊗ c ′ ) ⊗ d ′ )) L ( a ⊗ ( b ⊗ ( c ⊗ d ))) L ( a ′⊗ ( b ′⊗ ( c ′⊗ d ′ ))) L ( a ′⊗ ( b ′⊗ ( c ′⊗ d ′ ))) L ( a ⊗ ( b ⊗ ( c ⊗ d ))) L ( α a,b,c ⊗ d ) L ( α a ′ ,b ′ ,c ′ ⊗ d ′ ) ( φa ′ ,b ′⊗ ( c ′⊗ d ′ ))(1 L ( a ′ ) ⊗ φb ′ ,c ′⊗ d ′ )(1 L ( a ′ ) ⊗ (1 L ( b ′ ) ⊗ φc ′⊗ d ′ ))( M ⊗ ( N ⊗ ( P ⊗ Q )))(1 L ( a ) ⊗ (1 L ( b ) ⊗ φ − c,d ))(1 L ( a ) ⊗ φ − b,c ⊗ d )( φ − a,b ⊗ ( c ⊗ d ))( φa ′⊗ ( b ′⊗ c ′ ) ,d ′ )( φa ′ ,b ′⊗ c ′⊗ L ( d ′ ))((1 L ( a ′ ) ⊗ φb ′ ,c ′ ) ⊗ L ( d ′ ))(( M ⊗ ( N ⊗ P )) ⊗ Q )((1 L ( a ) ⊗ φ − b,c ) ⊗ L ( d ))( φ − a,b ⊗ c ⊗ L ( d ))( φ − a ⊗ ( b ⊗ c ) ,d )( φ ( a ′⊗ b ′ ) ⊗ c ′ ,d ′ )( φa ′⊗ b ′ ,c ′⊗ L ( d ′ ))(( φa ′ ,b ′ ⊗ L ( c ′ )) ⊗ L ( d ′ ))((( M ⊗ N ) ⊗ P ) ⊗ Q )(( φ − a,b ⊗ L ( c )) ⊗ L ( d )) ⊗ ( φ − a ⊗ b,c ⊗ L ( d ))( φ − a ⊗ b ) ⊗ c,d ) L ( α a,b,c ⊗ d ) L ( α a ′ ,b ′ ,c ′ ⊗ d ′ ) ( φa ′⊗ b ′ ,c ′⊗ d ′ )( φa ′ ,b ′ ⊗ φc ′ ,d ′ )(( M ⊗ N ) ⊗ ( P ⊗ Q ))( φ − a,b ⊗ φ − c,d )( φ − a ⊗ b,c ⊗ d ) L ( α a ⊗ b,c,d ) L ( α a ′ ⊗ b ′ ,c ′ ,d ′ ) ( φa ′ , ( b ′⊗ c ′ ) ⊗ d ′ )(1 L ( a ′ ) ⊗ φb ′⊗ c ′ ,d ′ )(1 L ( a ′ ) ⊗ ( φb ′ ,c ′ ⊗ L ( d ′ )))( M ⊗ (( N ⊗ P ) ⊗ Q ))(1 L ( a ) ⊗ ( φ − b,c ⊗ L ( d )))(1 L ( a ) ⊗ φ − b ⊗ c,d )( φ − a, ( b ⊗ c ) ⊗ d ) L ( α a,b ⊗ c,d ) L ( α a ′ ,b ′ ⊗ c ′ ,d ′ ) ( φa ′ ,b ′⊗ ( c ′⊗ d ′ ))(1 L ( a ′ ) ⊗ φb ′ ,c ′⊗ d ′ )(1 L ( a ′ ) ⊗ (1 L ( b ′ ) ⊗ φc ′⊗ d ′ ))( M ⊗ ( N ⊗ ( P ⊗ Q )))(1 L ( a ) ⊗ (1 L ( b ) ⊗ φ − c,d ))(1 L ( a ) ⊗ φ − b,c ⊗ d )( φ − a,b ⊗ ( c ⊗ d )) L (1 a ⊗ α b,c,d ) L (1 a ′ ⊗ α b ′ ,c ′ ,d ′ ) in which the top and the bottom tensor products of horizontal 1-cells coincide. The red is tohighlight that the pentagon identity of X is nested within the pentagon identity of L X , andlikewise for the triangle identity on the previous page, although that one we have not colored. The most important example of a double category in this thesis is given by C sp ( X ) for somecategory X with pushouts. This double category has:(1) objects as those of X ,(2) vertical 1-morphisms as morphisms of X ,(3) horizontal 1-cells as cospans in X , and 214) 2-morphisms as maps of cospans in X given by commutative diagrams of the form: x yzx ′ y ′ z ′ of hgii ′ o ′ That C sp ( X ) is indeed a double category when X is a category with pushouts was shown byNiefield [32]; see also [18]. This also follows from Theorem 4.1.1 when the decorations are taken tobe trivial—see Corollary 4.1.2. Theorem 3.2.1.
Let L : A → X be a functor where X is a category with pushouts. Then thereexists a double category L C sp ( X ) for which: • an object is an object of A , • a vertical 1-morphism is a morphism of A , • a horizontal 1-cell from a to b is an L - structured cospan , meaning a cospan in X of theform: L ( a ) x L ( b ) i o • a 2-morphism is a map of L - structured cospans , meaning a commutative diagram in X of this form: L ( a ) L ( b ) xL ( a ′ ) L ( b ′ ) x ′ oL ( α ) L ( β ) fii ′ o ′ • composition of vertical 1-morphisms is morphism composition in A , • composition of horizontal 1-cells is done using chosen pushouts in X : L ( a ) x L ( b ) y L ( c ) x + L ( b ) y i o i o j x j y where j x and j y are the canonical morphisms from x and y into the pushout, the horizontal composite of two 2-morphisms: L ( a ) x L ( b ) L ( a ′ ) x ′ L ( b ′ ) L ( b ) y L ( c ) L ( b ′ ) y ′ L ( c ′ ) i i ′ o ′ o L ( α ) L ( β ) f i o L ( β ) i ′ o ′ L ( γ ) g is given by L ( a ) x + L ( b ) y L ( c ) L ( a ′ ) x ′ + L ( b ′ ) y ′ L ( c ′ ) . L ( α ) L ( γ ) f + L ( β ) gj x i j y o j x ′ i ′ j y ′ o ′ • The vertical composite of two 2-morphisms: L ( a ) y L ( b ) L ( a ′ ) y ′ L ( b ′ ) L ( α ) L ( β ) fi oi ′ o ′ L ( a ′ ) y ′ L ( b ′ ) L ( a ′′ ) y ′′ L ( b ′′ ) L ( α ′ ) L ( β ′ ) f ′ i ′ o ′ i ′′ o ′′ is given by L ( a ) y L ( b ) L ( a ′′ ) y ′′ L ( b ′′ ) . L ( α ′ α ) L ( β ′ β ) f ′ fi oi ′′ o ′′ • The associator and unitors are defined using the universal property of pushouts.Proof.
We apply Theorem 3.1.1 to the double category C sp ( X ).If the category X has not only pushouts but also finite colimits, meaning pushouts and an initialobject which will serve as the unit object for tensoring, then the aforementioned double category C sp ( X ) is in fact symmetric monoidal. Corollary 3.2.2.
Given a category X with finite colimits, the double category C sp ( X ) is symmetricmonoidal with the monoidal structure given by chosen coproducts in X . Thus: the tensor product of two objects x and x is x + x , • the tensor product of two vertical 1-morphisms is given by xy x ′ y ′ x + x ′ y + y ′ ⊗ = f f ′ f + f ′ • the tensor product of two horizontal 1-cells is given by x y z ⊗ x ′ y ′ z ′ = x + x ′ y + y ′ z + z ′ , i o i ′ o ′ i + i ′ o + o ′ • the tensor product of two 2-morphisms is given by x z y x z y x ′ z ′ y ′ x ′ z ′ y ′ ⊗ x + x ′ z + z ′ y + y ′ x + x ′ z + z ′ , y + y ′ = o f hgi i o o ′ f ′ h ′ g ′ i ′ i ′ o ′ o + o ′ f + f ′ h + h ′ g + g ′ i + i ′ i + i ′ o + o ′ • The unit for the tensor product is a chosen initial object of X , • The symmetry for any two objects x and y is defined using the canonical isomorphism x + y ∼ = y + x .Proof. This is just a special case of Theorem 4.1.3 where, as in Corollary 4.1.2, each F -decoratedcospan is once again equipped with the trivial decoration.We then have the following symmetric monoidal double category of structured cospans , theprimary result of the aforementioned work [3]. Theorem 3.2.3.
Let L : A → X be a functor preserving finite coproducts, where A has finitecoproducts and X has finite colimits. Then the double category L C sp ( X ) is symmetric monoidalwith the monoidal structure given by chosen coproducts in A and X . Thus:(1) the tensor product of two objects a and a is a + a ,
2) the tensor product of two vertical 1-morphisms is given by a b a b a ⊗ a b ⊗ b ⊗ = f f f + f (3) the tensor product of two horizontal 1-cells is given by L ( a ) x L ( b ) ⊗ L ( a ′ ) x ′ L ( b ′ ) = L ( a + a ′ ) x + x ′ L ( b + b ′ ) i o i ′ o ′ ( i + i ′ ) φ ( o + o ′ ) φ where the feet use the tensor product of A and the legs and apices use the tensor product of X and invertible laxators of L , and likewise(4) the tensor product of two 2-morphisms is given by: L ( a ) L ( b ) x L ( a ) L ( b ) x L ( a ′ ) L ( b ′ ) x ′ L ( a ′ ) L ( b ′ ) x ′ ⊗ L ( a + a ′ ) L ( b + b ′ ) x + x ′ L ( a + a ′ ) L ( b + b ′ ) x + x ′ = o L ( f ) L ( g ) αi i o o ′ L ( f ′ ) L ( g ′ ) α ′ i ′ i ′ o ′ ( o + o ′ ) φL ( f + f ′ ) L ( g + g ′ ) α + α ′ ( i + i ′ ) φ ( i + i ′ ) φ ( o + o ′ ) φ The unit for the tensor product is the initial object of X which is isomorphic to the image of theunit object of A under the functor L , and the symmetry for any two objects a and b is defined usingthe canonical isomorphism a + b ∼ = b + a . Theorem 3.2.3 is one of the main results on structured cospans in a joint work with Baez [3]. Themethod of proof used there however is different from the more direct approach taken here in thisthesis. The word ‘rex’ is a standard abbreviation of ‘right exact’, which means finitely cocontinuous,i.e., preserving finite colimits. Denoting by
Rex the 2-category of finitely cocomplete categories,finitely cocontinuous functors and natural transformations, it is shown that if A ∈ Rex , then C sp ( A ) is a ‘pseudocategory object’ in Rex —see Definition A.2.2. A morphism L : A → X thenyields the above symmetric monoidal double category L C sp ( X ) being realized as a pseudocategoryobject in Rex . Denoting by
SymMonCat the 2-category of symmetric monoidal categories,(strong) symmetric monoidal functors and monoidal natural transformations, there exists a 2-functor Φ :
Rex → SymMonCat which turns a finitely cocomplete category into a symmetric25onoidal category by prescription of chosen binary coproducts for every pair of objects to serveas their tensor product and a chosen initial object to serve as the monoidal unit. The rest of thesymmetric monoidal structure is then induced by these choices. This 2-functor Φ preserves thenecessary pullbacks and applying this 2-functor Φ to L C sp ( X ) then results in Φ( L C sp ( X )) as apseudocategory object in SymMonCat . A pseudocategory object in the 2-category
Cat is thesame as a double category. A pseudocategory object in
SymMonCat is almost the same as asymmetric monoidal double category, but not quite, because the source and target functors S and T are not required to be strict symmetric monoidal functors. Luckily, an easy verification showsthat that this is indeed the case for Φ( L C sp ( X )), so it is a symmetric monoidal double category.Analogous comments also apply for maps between structured cospan double categories, whichare given by weakly commuting squares in Rex : A XA ′ X ′ α ⇒ LF F L ′ Assuming L : A → X is a morphism in Rex is stronger than the hypothesis used in Theorem3.2.3, but this simplifies many proofs and also produces stronger results: not only can we tensorand compose structured cospans as we can in an ordinary symmetric monoidal double category ofstructured cospans, but we can even take finite colimits of structured cospans, themselves. This isnot the case for the symmetric monoidal double category L C sp ( X ) of Theorem 3.2.3 due to A onlybeing required to have finite coproducts and only requiring finite coproducts be preserved by L .A well-known result regarding adjoints is the following. Proposition 3.2.4.
Every left adjoint L : A → X preserves all colimits and every right adjoint R : X → A preserves all limits. The following is a particularly useful result on structured cospan double categories.
Corollary 3.2.5.
Let L : A → X be a left adjoint between two categories A and X with finite colimits.Then the double category L C sp ( X ) is symmetric monoidal with the monoidal structure given as inTheorem 3.2.3. The examples we present of structured cospan double categories, which are to be seen as im-provements of the corresponding examples of decorated cospans of the previous chapter, will beapplications of the above corollary. Another application may be found in the work of Cicala [15]who uses structured cospan double categories to study rewrite rules in a topos.
Definition 3.3.1.
Let
FinGraph be the category whose objects are finite graphs, which are diagramsin
FinSet of the form:
E N st f, g ) such that the following two squarescommute: EE ′ NN ′ EE ′ NN ′ f fss ′ g g tt ′ Define a functor L : FinSet → FinGraph where given a set N , L ( N ) is the discrete graph on N (with no edges) and given a function f : N → N ′ , L ( f ) : L ( N ) → L ( N ′ ) is the graph morphism thattakes vertices of L ( N ) to L ( N ′ ) as prescribed by the function f . This functor L preserves finitecoproducts as it is left adjoint to the forgetful functor R : FinGraph → FinSet that takes a graph(
E, N, s, t ) where N and E are finite to its underlying set of vertices N . The categories FinSet and
FinGraph both have finite colimits. By Corollary 3.2.5, we have the following.
Theorem 3.3.2.
Let L : FinSet → FinGraph be the left adjoint defined above. Then there exists asymmetric monoidal double category L C sp ( FinGraph ) which has:(1) finite sets as objects,(2) functions as vertical 1-morphisms,(3) open graphs which are cospans of graphs of the form L ( a ) x L ( b ) as horizontal 1-cells, and(4) maps of open graphs which are maps of cospans of graphs as 2-morphisms, as in thefollowing commutative diagram: L ( a ) L ( b ) xL ( a ′ ) L ( b ′ ) y L ( f ) L ( g ) h Recall from Section 2.2.2 that given a field k , a field with positive elements is a pair ( k, k + )where k + ⊂ k is a subset such that r ∈ k + for every nonzero r ∈ k and such that k + is closedunder addition, multiplication and division. A recent work of Baez and Fong [7] studies k -graphswhere a k -graph Γ is given by a diagram in Set of the form: k + E N r st E and N are finite sets. Here k is a field with positive elements and the finite sets E and N denote the sets of edges and nodes, respectively, of the k -graph Γ. An open k -graph is then givenby a cospan of finite sets: a N b i o where the apex N is decorated with a k -graph as above. Fong and Baez use the decoratedcospan machinery of Fong to construct a monoidal category F Cospan from a lax monoidal functor F : FinSet → Set . This functor F is defined on objects by: N
7→ { k + E N } r st and on morphisms by NN ′ k + E E NN ′ E NN ′ f f fr ss ′ r ′ tt ′ To fit the above construction into the framework of structured cospans, first we define a category
FinGraph k whose objects are given by finite k -graphs: k + E N r st and a morphism from this k -graph to another: k + E ′ N ′ r ′ s ′ t ′ consists of a pair of functions f : N → N ′ and g : E → E ′ such that the following diagrams commute: k + EE ′ EE ′ NN ′ EE ′ NN ′ f fr ss ′ r ′ g g g tt ′ Next, we define a left adjoint L : FinSet → FinGraph k which is defined on sets by: N k + ∅ N r st NN ′ k + ∅∅ NN ′ f fr str ′ ! s ′ t ′ Lemma 3.3.3.
The functor L : FinSet → FinGraph k defined above is left adjoint to the forgetfulfunctor R : FinGraph k → FinSet .Proof.
The functor L : FinSet → FinGraph k has a right adjoint given by the forgetful functor R : FinGraph k → FinSet which maps a finite k -graph k + E N r st to its underlying vertex set N . We then have a natural isomorphism hom FinGraph k ( L ( c ) , d ) ∼ =hom FinSet ( c, R ( d )). Lemma 3.3.4.
The category
FinGraph k has finite colimits.Proof. The category
FinGraph k has an initial object given by the empty k -graph as well as pushoutsgiven by taking the pushout of the underlying span of finite graphs which is done pointwise. Theorem 3.3.5.
Let L : FinSet → FinGraph k be the left adjoint as described above. Then thereexists a symmetric monoidal double category L C sp ( FinGraph k ) which has:(1) finite sets as objects,(2) functions as vertical 1-morphisms,(3) open k -graphs : that is, cospans of finite sets where the apex is equipped with a k -graph L ( a ) N L ( b ) i o k + E N r st as horizontal 1-cells, and(4) maps of cospans of finite sets equipped with a map of k -graphs L ( a ) N L ( b ) L ( a ′ ) N ′ L ( b ′ ) i oi ′ o ′ L ( h ) L ( h ) f + EE ′ EE ′ NN ′ EE ′ NN ′ f fr ss ′ r ′ g g g tt ′ as 2-morphisms.Proof. As FinGraph k has finite colimits, we get a symmetric monoidal double cate-gory C sp ( FinGraph k ) and hence a symmetric monoidal structured cospan double category L C sp ( FinGraph k ). For the last example, Baez and Pollard have constructed a black-boxing functor (cid:4) : Dynam → SemiAlgRel [10]. Here,
Dynam is a symmetric monoidal category of ‘open dynamical systems’ and
SemiAlgRel is a symmetric monoidal category of ‘semialgebraic relations’. A particular kind ofdynamical system is given by a Petri net with rates. Petri nets have also been studied extensivelyby Baez and Master [9] in the context of double categories and double functors.Recall that a Petri net consists of a set S of species , a set T of transitions and functions s, t : S × T → N . For a species σ ∈ S and a transition τ ∈ T , s ( σ, τ ) is the number of times thespecies σ appears as an input for the transition τ and t ( σ, τ ) is the number of times the species σ appears as an output for the transition τ . Definition 3.3.6. A Petri net with rates is a Petri net with finite sets of species and transitionstogether with a function r : T → [0 , ∞ ) where r ( τ ) is the rate of the transition τ .We can also say that a Petri net with rates is a diagram of the form: [0 , ∞ ) T N [ S ] r st where S and T are finite sets and N [ S ] is the free commutative monoid on S . An open Petri netwith rates is then given by a cospan of finite sets whose apex is equipped with a Petri net withrates. X S Y [0 , ∞ ) T N [ S ] r sti o A map of Petri nets with rates is given by a pair of functions f : S → S ′ and g : T → T ′ whichmake the following diagrams commute: [0 , ∞ ) TT ′ TT N [ S ] N [ S ′ ] TT ′ N [ S ] N [ S ′ ] N [ f ] N [ f ] r ss ′ r ′ g g g tt ′ X SS ′ Y ii ′ f oo ′ ∼ [0 , ∞ ) T T N [ S ] N [ S ′ ] T N [ S ] N [ S ′ ] N [ f ] N [ f ] r ss ′ r ′ tt ′ for some isomorphism f . Define a functor L : FinSet → Petri rates where for a finite set S , L ( S ) isthe discrete Petri net with rates with S as its set of species and no transitions. In other words, S [0 , ∞ ) ∅ N [ S ] r st Lemma 3.3.7.
The functor L : FinSet → Petri rates defined above is left adjoint to the forgetfulfunctor R : Petri rates → FinSet .Proof.
This is similar as to why the functors used in the previous two applications are also leftadjoints.
Lemma 3.3.8.
The category
Petri rates has finite colimits.Proof.
This is similar to the proof of Lemma 3.3.4 — the category
Petri rates has pushouts and aninitial object.
Theorem 3.3.9.
Let L : FinSet → Petri rates be the left adjoint described above. Then there exists asymmetric monoidal double category L C sp ( Petri rates ) which has:(1) finite sets as objects,(2) functions as vertical 1-morphisms,(3) cospans of sets whose apices are equipped with the stuff of a Petri net with rates as horizontal1-cells, and(4) maps of cospans as above as 2-morphisms, as in the following commutative diagrams. L ( a ) L ( b ) SL ( a ′ ) L ( b ′ ) S ′ oL ( h ) L ( h ) fii ′ o ′ , ∞ ) TT ′ TT N [ S ] N [ S ′ ] TT ′ N [ S ] N [ S ′ ] N [ f ] N [ f ] r ss ′ r ′ g g g tt ′ Proof.
This follows from Corollary 3.2.5, Theorem 3.3.7 and Lemma 3.3.8.
In this section we define maps between foot-replaced double categories. In Theorem 3.1.1 weshowed how to construct a foot-replaced double category L X starting from a pair( X , L : A → X )where X is a double category and L : A → X is a functor that maps the category A , which containsthe objects and morphisms of the foot-replaced double category L X , into the category of objects X of the double category X . Suppose that we have two foot-replaced double categories: L X obtainedfrom a pair ( X , L : A → X ) and L ′ X ′ obtained from a pair ( X ′ , L ′ : A ′ → X ′ ) . Then we can constructa map from L X to L ′ X ′ given a functor F : A → A ′ together with a double functor F : X → X ′ suchthat the following diagram commutes up to a specified isomorphism θ : A X A ′ X ′ ⇒ θ LF F L ′ In the case where L X and L ′ X ′ are symmetric monoidal and we wish to construct a symmetricmonoidal double functor between them, we will then require that both the functor F and doublefunctor F be symmetric monoidal, and that θ be monoidal as well. (For the definition of ‘symmetricmonoidal double functor’, see Definition A.2.14, and for the definition of ‘monoidal transformation’,see Definition A.1.11.) Theorem 3.4.1.
Let L X and L ′ X ′ be two foot-replaced double categories. Given a functor F : A → A ′ and a double functor F : X → X ′ such that the following diagram commutes up to isomorphism: A X A ′ X ′ ⇒ θ LF F L ′ the triple ( F, F , θ ) results in a double functor F F : L X → L ′ X ′ . This double functor F F maps objects,vertical 1-morphisms, horizontal 1-cells and 2-morphisms as follows:(1) Objects: a F ( a )32
2) Vertical 1-morphisms: a F ( a ) a ′ F ( a ′ ) f F ( f ) (3) Horizontal 1-cells: L ( a ) L ( b ) L ′ ( F ( a )) ∼ = F ( L ( a )) F ( L ( b )) ∼ = L ′ ( F ( b )) Mθ b F ( M ) θ − a (4) 2-morphisms: L ( a ) F ( L ( a )) L ′ ( F ( a )) L ′ ( F ( a ′ )) L ′ ( F ( b )) L ′ ( F ( b ′ )) F ( L ( b )) F ( L ( a ′ )) F ( L ( b ′ ))
7→ ⇓ F ( α ) L ( b ) L ( a ′ ) L ( b ′ ) ⇓ α L ′ ( F ( f )) L ′ ( F ( g )) θ − a θ − a ′ θ b θ b ′ ML ( f ) L ( g ) N F ( M ) F ( L ( f )) F ( L ( g )) F ( N ) Proof.
We will show that from the triple ( F, F , θ ) we can produce a double functor F F : L X → L ′ X ′ .This means that we must have F F = F : L X → L ′ X ′ and F F : L X → L ′ X ′ such that the following diagrams commute: L X L X L ′ X ′ L X L ′ X ′ L ′ X ′ L X L ′ X ′ F F S S ′ F F F T T ′ F where S, T and S ′ , T ′ are the source and target functors of the double categories L X and L ′ X ′ ,respectively, together with natural transformations F F ⊙ : F F ( M ) ⊙ F F ( N ) → F F ( M ⊙ N )for every pair of composable horizontal 1-cells M and N of L X and a natural transformation F F U : U ′ F ( a ) → F F ( U a )33or every object a ∈ L X that satisfy the standard coherence axioms of a monoidal category givenby the laxator hexagon and unitality squares.The functors F F = F and F F are defined as in the statement of the theorem. To see that theabove squares commute, if we focus on the left one, starting at the upper left corner, for an objectof L X which is given by a horizontal 1-cell, we have going right that: L ( a ) L ( b ) L ′ ( F ( a )) ∼ = F ( L ( a )) F ( L ( b )) ∼ = L ′ ( F ( b )) Mθ b F ( M ) θ − a and then going down yields source F ( a ). If we go down and then right, we get that the source ofthe top horizontal 1-cell is the object a which then maps to F ( a ) under the double functor F F . Amorphism in L X is given by a 2-morphism of the form L ( a ) L ( b ) L ( a ′ ) L ( b ′ ) ⇓ α ML ( f ) L ( g ) M ′ so, again focusing on the left square, going right gives L ′ ( F ( a )) L ′ ( F ( b )) L ′ ( F ( a ′ )) L ′ ( F ( b ′ )) ⇓ θ g F ( α ) θ − f θ b F ( M ) θ − a L ′ ( F ( f )) L ′ ( F ( g )) θ b ′ F ( N ) θ − a ′ and then going down yields source F ( f ). On the other hand, going down we get that the sourceof the original 2-morphism is f which then maps to F ( f ) under the double functor F F , and so theleft square commutes. The right square is analogous.That F F is functorial on vertical 1-morphisms is clear, as the pair F F acts as the functor F : A → A ′ on objects and vertical 1-morphisms. Given two vertically composable 2-morphisms in L X : L ( a ) L ( a ′ ) L ( b ′ ) L ( a ′′ ) L ( b ′′ ) ⇓ β L ( b ) L ( a ′ ) L ( b ′ ) ⇓ α ML ( f ) L ( g ) M ′ M ′ L ( f ′ ) L ( g ′ ) M ′′
34e wish to show that F F is functorial. If we first compose the above two 2-morphisms in L X , weget: L ( a ) L ( b ) L ( a ′′ ) L ( b ′′ ) ⇓ βα ML ( f ′ f ) L ( g ′ g ) M ′′ and then the image of this 2-morphism under F F is given by: L ′ ( F ( a )) L ′ ( F ( b )) L ′ ( F ( a ′′ )) L ′ ( F ( b ′′ )) ⇓ θ g ′ g F ( βα ) θ − f ′ f θ b F ( M ) θ − a L ′ ( F ( f ′ f )) L ′ ( F ( g ′ g )) θ b ′′ F ( M ′′ ) θ − a ′′ On the other hand, if we first map over the two 2-morphisms, we get L ′ ( F ( a )) L ′ ( F ( b )) L ′ ( F ( a ′ )) L ′ ( F ( b ′ )) ⇓ θ g F ( α ) θ − f θ b F ( M ) θ − a L ′ ( F ( f )) L ′ ( F ( g )) θ b ′ F ( M ′ ) θ − a ′ L ′ ( F ( a ′ )) L ′ ( F ( b ′ )) L ′ ( F ( a ′′ )) L ′ ( F ( b ′′ )) ⇓ θ g ′ F ( β ) θ − f ′ θ b ′ F ( M ′ ) θ − a ′ L ′ ( F ( f ′ )) L ′ ( F ( g ′ )) θ b ′′ F ( M ′′ ) θ − a ′′ and then composing these in L ′ X ′ yields L ′ ( F ( a )) L ′ ( F ( b )) L ′ ( F ( a ′′ )) L ′ ( F ( b ′′ )) ⇓ θ g ′ g F ( βα ) θ − f ′ f θ b F ( M ) θ − a L ′ ( F ( f ′ f )) L ′ ( F ( g ′ g )) θ b ′′ F ( M ′′ ) θ − a ′′ by the functoriality of F = F, F and L ′ .Now let M and N be two composable horizontal 1-cells in L X given by: L ( a ) L ( b ) L ( b ) L ( c ) M N
35e then have a natural transformation F F M,N : F F ( M ) ⊙ F F ( N ) → F F ( M ⊙ N )given by: L ′ ( F ( a )) L ′ ( F ( b )) L ′ ( F ( c )) L ′ ( F ( a )) L ′ ( F ( c )) ⇓ F F M,N := F M,N ◦ ⊙ X ′ θ b F ( M ) θ − a θ c F ( M ⊙ N ) θ − a θ c F ( N ) θ − b and for any object a , a natural transformation F F a : U ′ F F ( a ) → F F ( U a )given by: L ′ ( F ( a )) L ′ ( F ( a )) F ( L ( a )) F ( L ( a )) ⇓ F a L ′ ( F ( a )) L ′ ( F ( a )) L ′ ( F ( a )) L ′ ( F ( a )) θ a θ − a U ′ F ( a ) θ − a F ( U a ) θ − a The double functor F F is pseudo, lax or oplax depending on whether the double functor F is pseudo,lax or oplax, respectively.If both F : A → A ′ and F : X → X ′ are (strong) symmetric monoidal and θ : F L : L ′ F a monoidalnatural isomorphism, then F F : L X → L ′ X ′ is a (strong) symmetric monoidal double functor. Theorem 3.4.2.
Let L X and L ′ X ′ be symmetric monoidal foot-replaced double categories obtainedfrom pairs ( X , L : A → X ) and ( X ′ , L ′ : A ′ → X ′ ) , respectively, via Theorem 3.1.2. If F F : L X → L ′ X ′ is a foot-replaced double functor obtained from a square A X A ′ X ′ ⇒ θ LF F L ′ as in Theorem 3.4.1 with θ monoidal and F and F (strong) symmetric monoidal, then F F is a(strong) symmetric monoidal double functor of foot-replaced double categories.Proof. Since the functor F : A → A ′ is symmetric monoidal, for every pair of objects a and b of A ,we have a natural transformation µ a,b : F ( a ) ⊗ F ( b ) → F ( a ⊗ b )36ogether with a morphism ǫ : 1 L ′ X ′ → F (1 L X )where the unit object of L ′ X ′ is given by 1 L ′ X ′ = 1 A ′ ∼ = F (1 A ) and the unit object of L X is given by1 L X = 1 A . These together make the following diagrams commute for every triple of objects a, b, c of L X , which are just objects of A . Note that the object component of the double functor F F isjust F F = F . ( F ( a ) ⊗ F ( b )) ⊗ F ( c ) α ′ / / µ a,b ⊗ (cid:15) (cid:15) F ( a ) ⊗ ( F ( b ) ⊗ F ( c )) ⊗ µ b,c (cid:15) (cid:15) F ( a ⊗ b ) ⊗ F ( c ) µ a ⊗ b,c (cid:15) (cid:15) F ( a ) ⊗ F ( b ⊗ c ) µ a,b ⊗ c (cid:15) (cid:15) F (( a ⊗ b ) ⊗ c ) F α / / F ( a ⊗ ( b ⊗ c )) F ( a ) ⊗ L ′ X ′ F ( a ) F ( a ) ⊗ F (1 L X ) F ( a ⊗ L X ) 1 L ′ X ′ ⊗ F ( a ) F (1 L X ) ⊗ F ( a ) F ( a ) F (1 L X ⊗ a ) ⊗ ǫ F ( r a ) r F ( a ) µ a, L X ǫ ⊗ µ L X ,a ℓ F ( a ) F ( ℓ a ) Moreover, the following diagram commutes where by an abuse of notation, we denote the braidingsin both categories A and A ′ as β . F ( a ) ⊗ F ( b ) F ( b ) ⊗ F ( a ) F ( a ⊗ b ) F ( b ⊗ a ) µ a,b µ b,a β F ( a ) ,F ( b ) F ( β a,b ) The double functor F : X → X ′ is also symmetric monoidal, which means that for every pair ofhorizontal 1-cells M and N , we have a natural transformation F M,N : F ( M ) ⊗ F ( N ) → F ( M ⊗ N )and a morphism δ : U A ′ → F ( U A )which satisfy the usual axioms. From these, we can construct the corresponding transformationsfor F F . Given horizontal 1-cells M and M ′ in L X : L ( a ) L ( b ) L ( a ′ ) L ( b ′ ) M M ′ their images F F ( M ) and F F ( M ′ ) are given by: L ′ ( F ( a )) L ′ ( F ( b )) L ′ ( F ( a ′ )) L ′ ( F ( b ′ )) θ b F ( M ) θ − a θ b ′ F ( M ′ ) θ − a ′ F F ( M ) ⊗ F F ( M ′ ) is given by: L ′ ( F ( a ) ⊗ F ( a ′ )) L ′ ( F ( b ) ⊗ F ( b ′ )) σ F ( b ) ,F ( b ′ ) ( θ b F ( M ) θ − a ⊗ θ b ′ F ( M ′ ) θ − a ′ ) σ − F ( a ) ,F ( a ′ ) where σ F ( a ) ,F ( a ′ ) : L ′ ( F ( a )) ⊗ L ′ ( F ( a ′ )) → L ′ ( F ( a ) ⊗ F ( a ′ )) is the natural isomorphism comingfrom the (strong) symmetric monoidal functor L ′ : A ′ → X ′ , σ ′ a,a ′ : L ( a ) ⊗ L ( a ′ ) → L ( a ⊗ a ′ ) isthe natural isomorphism coming from the (strong) symmetric monoidal functor L : A → X , and µ ′ x,y : F ( x ) ⊗ F ( y ) → F ( x ⊗ y ) is the natural isomorphism coming from the (strong) symmetricmonoidal functor F : X → X ′ . On the other hand, M ⊗ M ′ is given by: L ( a ⊗ a ′ ) L ( b ⊗ b ′ ) µ b ⊗ b ′ ( M ⊗ M ′ ) µ − a ⊗ a ′ and the image F F ( M ⊗ M ′ ) is given by: L ′ ( F ( a ⊗ a ′ )) L ′ ( F ( b ⊗ b ′ )) θ b ⊗ b ′ F ( µ b ⊗ b ′ ) F ( M ⊗ M ′ ) F ( µ − a ⊗ a ′ ) θ − a ⊗ a ′ We then have a natural transformation ν ′ M,M ′ : F F ( M ) ⊗ F F ( M ′ ) → F F ( M ⊗ M ′ )given by the 2-isomorphism: L ′ ( F ( a ) ⊗ F ( a ′ )) L ′ ( F ( b ) ⊗ F ( b ′ )) L ′ ( F ( a ⊗ a ′ )) L ′ ( F ( b ⊗ b ′ )) ⇓ F F M,M ′ σ F ( b ) ,F ( b ′ ) ( θ b F ( M ) θ − a ⊗ θ b ′ F ( M ′ ) θ − a ′ ) σ − F ( a ) ,F ( a ′ ) θ b ⊗ b ′ F ( µ b ⊗ b ′ ) F ( M ⊗ M ′ ) F ( µ − a ⊗ a ′ ) θ − a ⊗ a ′ L ′ ( τ a,a ′ ) L ′ ( τ b,b ′ ) which we can rewrite as: L ′ ( F ( a ) ⊗ F ( a ′ )) L ′ ( F ( b ) ⊗ F ( b ′ )) L ′ ( F ( a ⊗ a ′ )) L ′ ( F ( b ⊗ b ′ )) ⇓ F F M,M ′ ( σ F ( b ) ,F ( b ′ ) ( θ b ⊗ θ b ′ ))( F ( M ) ⊗ F ( M ′ ))( σ F ( a ) ,F ( a ′ ) ( θ a ⊗ θ a ′ )) − ( θ b ⊗ b ′ F ( µ b ⊗ b ′ )) F ( M ⊗ M ′ )( θ a ⊗ a ′ F ( µ a ⊗ a ′ )) − L ′ ( τ a,a ′ ) L ′ ( τ b,b ′ ) For the unit constraint, the horizontal 1-cell unit of L X is given by U L (1 A ) : L (1 A ) L (1 A ) U L (1 A ) and the image F F ( U L (1 A ) ) is given by: L ′ ( F (1 A )) L ′ ( F (1 A )) θ A F ( U L (1 A ) ) θ − A
38n the other hand, the horizontal 1-cell unit of L ′ X ′ is given by U L ′ (1 A ′ ) : L ′ (1 A ′ ) L ′ (1 A ′ ) U L ′ (1 A ′ ) and we then get a natural transformation δ ′ : U L ′ (1 A ′ ) → F F ( U L (1 A ) ) given by: L ′ ( F (1 A )) L ′ ( F (1 A )) L ′ (1 A ′ ) L ′ (1 A ′ ) ⇓ F F U θ A F ( U L (1 A ) ) θ − A U L ′ (1 A ′ ) L ′ ( τ ) L ′ ( τ ) where τ : 1 A ′ → F (1 A ) comes from the (strong) symmetric monoidal functor F : A → A ′ .These transformations ν ′ and δ ′ together make the following diagrams commute for every tripleof horizontal 1-cells M, N, P of L X .( F F ( M ) ⊗ F F ( N )) ⊗ F F ( P ) α ′ / / ν ′ M,N ⊗ (cid:15) (cid:15) F F ( M ) ⊗ ( F F ( N ) ⊗ F F ( P )) ⊗ ν ′ N,P (cid:15) (cid:15) F F ( M ⊗ N ) ⊗ F F ( P ) ν ′ M ⊗ N,P (cid:15) (cid:15) F F ( M ) ⊗ F F ( N ⊗ P ) ν ′ M,N ⊗ P (cid:15) (cid:15) F F (( M ⊗ N ) ⊗ P ) F F ( α ) / / F F ( M ⊗ ( N ⊗ P )) F F ( M ) ⊗ U L ′ X ′ F F ( M ) F F ( M ) ⊗ F F ( U L X ) F F ( M ⊗ U L X ) U L ′ X ′ ⊗ F F ( M ) F F ( U L X ) ⊗ F F ( M ) F F ( M ) F F ( U L X ⊗ M ) ⊗ δ ′ F F ( r M ) r F F ( M ) ν ′ M,U L X δ ′ ⊗ ν ′ U L X ,M ℓ F F ( M ) F F ( ℓ M ) By another abuse of notation, the following diagram commutes where we denote the braiding inboth L X and L ′ X ′ by β . F F ( M ) ⊗ F F ( N ) F F ( M ⊗ N ) F F ( N ) ⊗ F F ( M ) F F ( N ⊗ M ) ν ′ M,N F F ( β M,N ) β F F ( M ) , F F ( N ) ν ′ N,M
Lastly, we have transformations Φ
M,N : ⊗ ◦ ( F F , F F ) ⇒ F F ◦ ⊗ and Φ U : I L ′ X ′ ⇒ F F ◦ I L X satisfyingthe axioms of a symmetric monoidal functor with respect to ⊗ which come from the correspondingtransformations Ψ M,N : ⊗ ◦ ( F , F ) ⇒ F ◦ ⊗ , Ψ U : I X ′ ⇒ F ◦ I X of the symmetric monoidal doublefunctor F , the natural isomorphisms µ a,b and µ of the symmetric (strong) monoidal functor F : A → A ′ , and the monoidal natural isomorphism θ : F L ⇒ L ′ F .39 .5 Transformations of foot-replaced double categories We can also consider double transformations between these foot-replaced double functors andsymmetric monoidal versions of such. By the previous section, we can produce a map between twofoot-replaced double categories L X = ( X , L : A → X ) and L ′ X ′ = ( X ′ , L ′ : A ′ → X ′ ) from a triple( F, F , θ ) as in the following diagram. A X A ′ X ′ ⇒ θ L F F L ′ This leads to a double functor F F : L X → L ′ X ′ by Theorem 3.4.1. Given another double functor G G : L X → L ′ X ′ coming from a triple ( G, G , ψ ), we can construct a foot-replaced double trans-formation from F F to G G from a pair ( φ, Φ) where φ : F ⇒ G is a natural transformation andΦ : F ⇒ G is a double transformation such that the following diagram commutes A X A ′ X ′ φ ⇐ Φ ⇐ ⇒ ψ ⇒ θ L G F G F L ′ meaning that the following composites are equal. A X A ′ X ′ A X A ′ X ′ φ ⇐ Φ ⇐ ⇒ θ ⇒ ψ = L L G F F G GF L ′ L ′ We will denote the double transformation that results from the pair ( φ, Φ) as φ Φ : F F ⇒ G G . Theorem 3.5.1.
Let F F : L X → L ′ X ′ and G G : L X → L ′ X ′ be double functors obtained from triples ( F, F , θ ) and ( G, G , ψ ) via Theorem 3.4.1, respectively. Given a double transformation Φ : F ⇒ G nd a transformation φ : F ⇒ G such that the diagrams above commute, then from the pair ( φ, Φ) we can construct a double transformation Ξ = φ Φ : F F ⇒ G G (see Definition A.2.9). The objectcomponent Ξ is given by the composite Ξ a = ψ − a L ′ ( φ a ) θ a : F F ( a ) → G G ( a ) and the arrow component Ξ is given by Φ , the arrow component of the double transformation Φ .Proof. Because Φ : F ⇒ G is a double transformation and the diagram on the previous page com-mutes, we have that the following equations hold. F F ( a ) F F ( b ) F F ( c ) G G ( a ) G G ( c ) F F ( a ) F F ( c ) = F F ( b ) G G ( a ) G G ( b ) G G ( c ) ⇓ F F ⊙ ⇓ Φ M ⊙ N F F ( a ) G G ( a ) F F ( c ) G G ( c ) ⇓ Φ M ⇓ Φ N ⇓ G G ⊙ a cF F ( M ) F F ( M ⊙ N ) F F ( N ) G G ( M ⊙ N ) Ξ a b G G ( N ) G G ( M ) G G ( M ⊙ N ) F F ( M ) F F ( N ) Ξ c F F ( a ) F F ( a ) G G ( a ) G G ( a ) F F ( a ) F F ( a ) = G G ( a ) G G ( a ) ⇓ F F U ⇓ Φ U a F F ( a ) G G ( a ) F F ( a ) G G ( a ) ⇓ U Ξ a ⇓ G G U a a U F F ( a ) F F ( U a ) G G ( U a ) Ξ a U G G ( a ) G G ( U a ) U F F ( a ) Ξ a Here we use the isomorphisms θ a : F ( L ( a )) ∼ −→ L ′ ( F ( a )) and ψ a : G ( L ( a )) ∼ −→ L ′ ( G ( a )) togetherwith the natural transformation φ : F ⇒ G to cook up the object component of the double naturaltransformation φ Φ : F F ⇒ G G . In detail, every object of L X is of the form L ( a ) for some a in A . Wethus have for every object L ( a ) in L X a map θ a : F ( L ( a )) ∼ −→ L ′ ( F ( a )). The natural transformation φ : F ⇒ G evaluated at a then gives a map φ a : F ( a ) → G ( a ) and applying the functor L ′ to the map φ a then gives a map L ′ ( φ a ) : L ′ ( F ( a )) → L ′ ( G ( a )). Then, we use the other natural isomorphism ψ a : G ( L ( a )) → L ′ ( G ( a )) to obtain a map ψ − a : L ′ ( G ( a )) ∼ −→ G ( L ( a )), and thusΞ a = ψ − a L ′ ( φ a ) θ a : F F ( a ) → G G ( a ) . Moreover, the map Ξ a for each object a will make the above equations hold for φ Φ : F F ⇒ G G asΞ a = ψ − a L ′ ( φ a ) θ a = ψ − a ψ a Φ L ( a ) = Φ L ( a ) and the corresponding equations utilizing the component Φ L ( a ) hold as Φ : X ⇒ X ′ is a doubletransformation.Finally, because Φ : F ⇒ G is a double transformation and by the commutativity of the diagramon the previous page, for a horizontal 1-cell M in L X we have that S (Φ M ) = Ξ S ( M ) and T (Φ M ) =Ξ T ( M ) . 41he double transformation φ Φ is a double natural isomorphism if and only if φ is a naturalisomorphism and Φ is a double natural isomorphism.As with functors of foot-replaced double categories, if both the transformation φ : F ⇒ G andthe double transformation Φ : F ⇒ G are symmetric monoidal, then φ Φ : F F ⇒ G G is a symmetricmonoidal double transformation of symmetric monoidal foot-replaced double functors. Theorem 3.5.2.
Let φ Φ : F F ⇒ G G be a foot-replaced double transformation between two sym-metric monoidal foot-replaced double functors F F : L X → L ′ X ′ and G G : L X → L ′ X ′ , where L X = ( X , L : A → X ) and L ′ X ′ = ( X ′ , L ′ : A ′ → X ′ ) . If φ : F ⇒ G is a monoidal transforma-tion and Φ : F ⇒ G is a monoidal double transformation, then φ Φ : F F ⇒ G G is a monoidal doubletransformation (see Definition A.2.15) of foot-replaced double functors.Proof. The double transformation φ Φ acts as Ξ (defined above) on objects and vertical 1-morphisms.This means that the following diagrams commute. F F ( a ) ⊗ F F ( b ) F F ( a ⊗ b ) G G ( a ) ⊗ G G ( b ) G G ( a ⊗ b ) µ a,b Ξ a ⊗ b Ξ a ⊗ Ξ b µ ′ a,b L ′ X ′ F F (1 L X ) G G (1 L X ) ǫ φ L X ǫ ′ Similarly, the double transformation φ Φ acts as Φ on horizontal 1-cells and 2-morphisms, whichmeans that the following diagrams commute. F F ( M ) ⊗ F F ( N ) G G ( M ) ⊗ G G ( N ) F F ( M ⊗ N ) G G ( M ⊗ N ) Φ M ⊗ Φ N µ ′ M,N µ M,N Φ M ⊗ N U L ′ X ′ F F ( U L X ) G G ( U L X ) δ Φ U L X δ ′ Hence both the object and arrow components are monoidal natural transformations and thus φ Φ : F F ⇒ G G is a symmetric monoidal double transformation.42 hapter 4 Decorated cospan double categories
In this chapter we present an improved version of Fong’s theory of decorated cospan categories[23] from the perspective of double categories. The main difference here is that, given a category A with finite colimits, we instead start with a pseudofunctor F : A → Cat rather than functor F : A → Set . The additional structure of
Cat viewed as a 2-category then allows us more flexibilityin defining what the isomorphism class of an F -decorated cospan consists of. This ultimately resultsin a second solution to the problems with the original incarnation of decorated cospans, structuredcospans being the first.Given a finitely cocomplete category A and a lax monoidal pseudofunctor F : A → Cat , the firstresult is the existence of a double category F C sp in which F -decorated cospans appear as horizontal1-cells, except now we can exploit the 2-categorical structure of Cat to define 2-morphisms. Thisis Theorem 4.1.1. In Theorem 4.1.3 we show that when this lax monoidal pseudofunctor F issymmetric, then the resulting double category F C sp is in fact symmetric monoidal. We then definemaps between decorated cospan double categories in Section 4.2. Finally, as both structured cospandouble categories and decorated cospan double categories are solutions to the problems with Fong’soriginal decorated cospans, in Section 4.3 we show that under certain conditions these approacheslead to equivalent symmetric monoidal double categories, the main result being Theorem 4.3.15. Theorem 4.1.1.
Let A be a category with finite colimits and F : ( A , + , → ( Cat , × , a laxmonoidal pseudofunctor. Then there exists a double category F C sp for which:(1) an object is an object of A ,(2) a vertical 1-morphism is a morphism of A ,(3) a horizontal 1-cell is an F -decorated cospan in A , which is a pair: a m b x ∈ F ( m ) i o
4) a 2-morphism is a map of F -decorated cospans in A , which is a pair consisting of acommutative diagram: aa ′ m bb ′ m ′ x ∈ F ( m ) x ′ ∈ F ( m ′ ) i of gi ′ o ′ h and a morphism ι : F ( h )( x ) → x ′ in F ( m ′ ) ,(5) composition of vertical 1-morphisms is composition in A ,(6) the composite of two horizontal 1-cells: a m bx ∈ F ( m ) b n cy ∈ F ( n ) i o i ′ o ′ is done using chosen pushouts in A : a m b n cm + nm + b n i o i ′ o ′ j j ′ ψψji ψj ′ o ′ where the decoration x ⊙ y on the apex is given by: λ − −−→ × x × y −−→ F ( m ) × F ( n ) φ m,n −−−→ F ( m + n ) F ( ψ ) −−−→ F ( m + b n ) (7) the vertical composite of two 2-morphisms: aa ′ m bb ′ m ′ x ∈ F ( m ) x ′ ∈ F ( m ′ ) ι α : F ( h )( x ) → x ′ a ′ a ′′ m ′ b ′ b ′′ m ′′ x ′ ∈ F ( m ′ ) x ′′ ∈ F ( m ′′ ) ι α ′ : F ( h ′ )( x ′ ) → x ′′ i of gi ′ o ′ hi ′ o ′ f ′ g ′ i ′′ o ′′ h ′ s given by: aa ′′ m bb ′′ m ′′ x ∈ F ( m ) x ′′ ∈ F ( m ′′ ) ι α ′ α : F ( h ′ h )( x ) → x ′′ i of ′ f g ′ gi ′′ o ′′ h ′ h where the morphism ι α ′ α comes from the pasting of the two diagrams representing the mor-phisms ι α and ι α ′ : F ( m ) F ( m ′′ ) ⇒ ι α ′ α = 1 F ( m ) F ( m ′ ) F ( m ′′ ) ⇒ ι α ⇒ ι α ′ F ( h ) F ( h ′ ) xx ′ x ′′ xx ′′ F ( h ′ h ) (8) the horizontal composite of two 2-morphisms: aa ′ m bb ′ m ′ x ∈ F ( m ) x ′ ∈ F ( m ′ ) b n cb ′ n ′ c ′ y ∈ F ( n ) y ′ ∈ F ( n ′ ) ι α : F ( h )( x ) → x ′ ι β : F ( h )( y ) → y ′ i o f gi ′ o ′ h i g h i ′ o ko ′ also uses chosen pushouts in A and is given by: aa ′ m + b n cc ′ m ′ + b ′ n ′ x ⊙ y ∈ F ( m + b n ) x ′ ⊙ y ′ ∈ F ( m ′ + b ′ n ′ ) ι α ⊙ β : F ( h + g h )( x ⊙ y ) → x ′ ⊙ y ′ jψ m i jψ n o f kjψ m ′ i ′ jψ n ′ o ′ h + g h where the morphism of decorations ι α ⊙ β is given by the diagram: ι α × ι β ⇒ λ − −−−→ × F ( m ) × F ( n ) F ( m ′ ) × F ( n ′ ) F ( m + n ) F ( m ′ + n ′ ) F ( m + b n ) F ( m ′ + b ′ n ′ ) φ m,n φ m ′ ,n ′ F ( j m,n ) F ( j m ′ ,n ′ ) F ( h + g h ) F ( h + h ) x × yx ′ × y ′ F ( h ) × F ( h ) roof. We begin by defining the functors U : F C sp → F C sp S, T : F C sp → F C sp and ⊙ : F C sp × F C sp F C sp → F C sp necessary to obtain a double category. The functor U : F C sp → F C sp is defined on objects as: a a a a ! a ∈ F ( a ) where ! a ∈ F ( a ) is the trivial decoration on a given by the composite of the unique map F (!) : F (0) → F ( a ) and the morphism φ : 1 → F (0) which comes from the structure of the laxmonoidal pseudofunctor F : A → Cat . For morphisms, the functor U is defined as: aa ′ aa ′ a aa ′ a ′ ! a ∈ F ( a )! a ′ ∈ F ( a ′ ) f f ff together with the morphism ι f = F ( f ) F (!) φ : 1 → F ( a ′ ). We also have source and target functors S, T : F C sp → F C sp where the source of the horizontal 1-cell a m b x ∈ F ( m ) i o is the object a in A and the source of the 2-morphism aa ′ m bb ′ m ′ x ∈ F ( m ) x ′ ∈ F ( m ′ ) ι : F ( h )( x ) → x ′ i of gi ′ o ′ h is the source of the underlying map of cospans in A , namely the morphism f in A ; the target functoris defined similarly. These functors satisfy the equations SU ( a ) = a = T U ( a )for all objects and morphisms of A .Given two composable horizontal 1-cells M and N : a m bx ∈ F ( m ) b n cy ∈ F ( n ) i o i ′ o ′ N ⊙ M is given by: a m b n cm + nm + b n i o i ′ o ′ j j ′ ψψji ψj ′ o ′ with the corresponding decoration of the apex x ⊙ y ∈ F ( m + b n ) being the element determinedby: 1 λ − −−→ × x × y −−→ F ( m ) × F ( n ) φ m,n −−−→ F ( m + n ) F ( ψ ) −−−→ F ( m + b n )where ψ : m + n → m + b n is the natural map from the coproduct to the pushout and φ m,n : F ( m ) × F ( n ) → F ( m + n ) is the natural transformation coming from the structure of the lax monoidalpseudofunctor F : A → Cat . The source and target functors satisfy the equations S ( N ⊙ M ) = S ( M ) and T ( N ⊙ M ) = T ( N ).Given three composable horizontal 1-cells M , M and M : a m bx ∈ F ( m ) b m cy ∈ F ( m ) c m dz ∈ F ( m ) i o i ′ o ′ i ′′ o ′′ we get a natural isomorphism a M ,M ,M : ( M ⊙ M ) ⊙ M → M ⊙ ( M ⊙ M ) which is the globular2-morphism given by a map of cospans (1 , σ, aa ( m + b m ) + c m ddm + b ( m + c m ) ( x ⊙ y ) ⊙ z ∈ F (( m + b m ) + c m ) x ⊙ ( y ⊙ z ) ∈ F ( m + b ( m + c m )) σ with the decorations on the cospan’s apices given by:( x ⊙ y ) ⊙ z := 1 ζ −→ F ( m + b m ) × F ( m ) φ m bm ,m −−−−−−−−→ F (( m + b m )+ m ) F ( j m bm ,m ) −−−−−−−−−−→ F (( m + b m )+ c m ) ζ = (1 × z ) ρ − F ( j m ,m ) φ m ,m ( x × y ) λ − and x ⊙ ( y ⊙ z ) := 1 ζ −→ F ( m ) × F ( m + c m ) φ m ,m cm −−−−−−−−→ F ( m +( m + c m )) F ( j m ,m cm ) −−−−−−−−−−→ F ( m + b ( m + c m )) ζ = ( x × λ − F ( j m ,m ) φ m ,m ( y × z ) ρ − ι σ : F ( σ )(( x ⊙ y ) ⊙ z ) → x ⊙ ( y ⊙ z ). The map σ : ( m + b m )+ c m → m + b ( m + c m ) is the universal map between two colimits of the same diagram. We can alsodefine left and right unitors as follows. Given a horizontal 1-cell M : a m b x ∈ F ( m ) i o if we, say, compose with the identity horizontal 1-cell of b on the right: a m bx ∈ F ( m ) b b b ! b ∈ F ( b ) i o where ! b = F (!) φ : 1 → F ( b ) is the trivial decoration on b , the result is: a m + b b b x ⊙ ! b ∈ F ( m + b b ) jψ m i jψ b where ψ m : m → m + b is the natural map into the coproduct and likewise for ψ b and j : m + b → m + b b is the natural map from the coproduct to the pushout. The decoration x ⊙ ! b : 1 → F ( m + b b )is given by: 1 λ − −−→ × x × ! b −−−→ F ( m ) × F ( b ) φ m,b −−−→ F ( m + b ) F ( j m,b ) −−−−−→ F ( m + b b ) . We then have that the right unitor R : M ⊙ b ∼ −→ M is given by the globular 2-morphism (1 , r, M : aa m + b b bbm x ⊙ ! b ∈ F ( m + b b ) x ∈ F ( m ) jψ m i jψ b i or where the isomorphism r : m + b b ∼ −→ m is a universal map together with the isomorphism ι r : F ( r )( x ⊙ ! b ) → x . The left unitor is similar. The source and target functor applied to theleft and right unitors and associators yield identities, and the left and right unitors together withthe associator satisfy the standard pentagon and triangle identities of a monoidal category or bi-category. Finally, for the interchange law, given four 2-morphisms α, β, α ′ and β ′ : aa ′ m bb ′ m ′ x ∈ F ( m ) x ′ ∈ F ( m ′ ) b n cb ′ n ′ c ′ y ∈ F ( n ) y ′ ∈ F ( n ′ ) ι α : F ( h )( x ) → x ′ ι β : F ( h )( y ) → y ′ a ′ a ′′ m ′ b ′ b ′′ m ′′ x ′ ∈ F ( m ′ ) x ′′ ∈ F ( m ′′ ) b ′ n ′ c ′ b ′′ n ′′ c ′′ y ′ ∈ F ( n ′ ) y ′′ ∈ F ( n ′′ ) ι α ′ : F ( h ′ )( x ′ ) → x ′′ ι β ′ : F ( h ′ )( y ′ ) → y ′′ i o f gi ′ o ′ h i g h i ′ o ko ′ i ′ o ′ f ′ g ′ i ′′ o ′′ h ′ i ′ g ′ h ′ i ′′ o ′ k ′ o ′′
48f we first compose horizontally we obtain: aa ′ m + b n cc ′ m ′ + b ′ n ′ x ⊙ y ∈ F ( m + b n ) x ′ ⊙ y ′ ∈ F ( m ′ + b ′ n ′ ) ι α ⊙ β : F ( h + g h )( x ⊙ y ) → x ′ ⊙ y ′ a ′ a ′′ m ′ + b ′ n ′ c ′ c ′′ m ′′ + b ′′ n ′′ x ′ ⊙ y ′ ∈ F ( m ′ + b ′ n ′ ) x ′′ ⊙ y ′′ ∈ F ( m ′′ + b ′′ n ′′ ) ι α ′ ⊙ β ′ : F ( h ′ + g ′ h ′ )( x ′ ⊙ y ′ ) → x ′′ ⊙ y ′′ . jψ m i jψ n o f kjψ m ′ i ′ jψ n ′ o ′ h + g h jψ m ′ i ′ jψ n ′ o ′ f ′ k ′ jψ m ′′ i ′′ jψ n ′′ o ′′ h ′ + g ′ h ′ To obtain the morphism of decorations for a horizontal composite, we have as initial data: ι α ⇒ F ( m ) F ( m ′ ) ι β ⇒ F ( n ) F ( n ′ ) xx ′ F ( h ) yy ′ F ( h ) These two 2-morphisms ι α and ι β are two 2-morphisms in the monoidal 2-category ( Cat , × ,
1) andso we can tensor them which results in: ι α × ι β ⇒ λ − −−−→ × F ( m ) × F ( n ) F ( m ′ ) × F ( n ′ ) F ( m + n ) F ( m ′ + n ′ ) F ( m + b n ) F ( m ′ + b ′ n ′ ) φ m,n φ m ′ ,n ′ F ( j m,n ) F ( j m ′ ,n ′ ) F ( h + g h ) F ( h + h ) x × yx ′ × y ′ F ( h ) × F ( h ) where the middle square commutes since F is a lax monoidal pseudofunctor and the right squarecommutes because we have taken a commutative square and applied the pseudofunctor F to it.The decorations x ⊙ y and x ′ ⊙ y ′ are given respectively by top and bottom composite of arrows andthe morphism of decorations ι α ⊙ β is given by composing ι α × ι β with the two commuting squares,which can equivalently be viewed as a morphism in F ( m ′ + b ′ n ′ ).Returning to the interchange law, composing the two horizontal compositions above verticallythen results in: aa ′′ m + b n cc ′′ m ′′ + b ′′ n ′′ x ⊙ y ∈ F ( m + b n ) x ′′ ⊙ y ′′ ∈ F ( m ′′ + b ′′ n ′′ ) ι ( a ′ ⊙ β ′ )( α ⊙ β ) : F (( h ′ + g ′ h ′ )( h + g h ))( x ⊙ y ) → x ′′ ⊙ y ′′ . jψ m i jψ n o f ′ f k ′ kjψ m ′′ i ′′ jψ n ′′ o ′′ ( h ′ + g ′ h ′ )( h + g h ) aa ′′ m bb ′′ m ′′ x ∈ F ( m ) x ′′ ∈ F ( m ′′ ) b n cb ′′ n ′′ c ′′ y ∈ F ( n ) y ′′ ∈ F ( n ′′ ) ι α ′ α : F ( h ′ h )( x ) → x ′′ ι β ′ β : F ( h ′ h )( y ) → y ′′ i o f ′ f g ′ gi ′′ o ′′ h ′ h i g ′ g h ′ h i ′′ o k ′ ko ′′ and then composing horizontally results in: aa ′′ m + b n cc ′′ m ′′ + b ′′ n ′′ x ⊙ y ∈ F ( m + b n ) x ′′ ⊙ y ′′ ∈ F ( m ′′ + b ′′ n ′′ ) ι ( α ′ α ) ⊙ ( β ′ β ) : F (( h ′ h ) + g ′ g ( h ′ h ))( x ⊙ y ) → x ′′ ⊙ y ′′ . jψ m i jψ n o f ′ f k ′ kjψ m ′′ i ′′ jψ n ′′ o ′′ ( h ′ h ) + g ′ g ( h ′ h ) As usual for the interchange law in double categories of this nature, only the ‘interior’ of the twocomposites appears different, but the two morphisms ( h ′ + g ′ h ′ )( h + g h ) : m + b n → m ′′ + b ′′ n ′′ and ( h ′ h ) + g ′ g ( h ′ h ) : m + b n → m ′′ + b ′′ n ′′ are the same universal map realized in two differentways. The two morphisms of decorations ι ( α ′ ⊙ β ′ )( α ⊙ β ) and ι ( α ′ α ) ⊙ ( β ′ β ) are obtained as two differentcompositions of four 2-morphisms in Cat , namely horizontally then vertically and vertically thenhorizontally. As
Cat is a 2-category, the interchange law for these 2-morphisms already holds, andas a result, the decoration morphisms ι ( α ′ ⊙ β ′ )( α ⊙ β ) : F (( h ′ + g ′ h ′ )( h + g h ))( x ⊙ y ) → x ′′ ⊙ y ′′ and ι ( α ′ α ) ⊙ ( β ′ β ) : F (( h ′ h ) + g ′ g ( h ′ h ))( x ⊙ y ) → x ′′ ⊙ y ′′ are also the same. Thus the interchange law for 2-morphisms holds and F C sp is a double category. Corollary 4.1.2.
Given a category A with pushouts, C sp ( A ) is a double category with the relevantstructure given as in Theorem 4.1.1.Proof. This is a special case of Theorem 4.1.1 where each F -decorated cospan is equipped with thetrivial decoration. Namely, given a cospan in A : a m b i o the trivial decoration on the apex m is given by the composite! m = F (!) φ : 1 → F ( m )50here φ : 1 → F (0) is the morphism between monoidal units coming from the structure of a laxmonoidal pseudofunctor and ! : 0 → m is the unique morphism from the initial object 0 of A to theobject m . By equipping each F -decorated cospan with the trivial decoration, all of the diagramsinvolving decorations commute trivially, and the proof of Theorem 4.1.1 reduces to a proof that C sp ( A ) is a double category.If the lax monoidal pseudofunctor F : ( A , + , → ( Cat , × ,
1) is symmetric lax monoidal, thenthe above double category F C sp is also symmetric monoidal. Theorem 4.1.3.
Let A be a category with finite colimits and F : ( A , + , → ( Cat , × , a symmet-ric lax monoidal pseudofunctor. Then the double category F C sp of Theorem 4.1.1 is symmetricmonoidal where:(1) the tensor product of two objects a and a is a chosen coproduct a + a ,(2) the tensor product of two vertical 1-morphisms is given by: a b a b a + a b + b ⊗ = f f f + f (3) the tensor product of two horizontal 1-cells: a m b x ∈ F ( m ) a m b x ∈ F ( m ) i o i o is given by: a + a m + m b + b x + x ∈ F ( m + m ) i + i o + o where the decoration on the apex is given by: x + x := 1 λ − −−→ × x × x −−−−→ F ( m ) × F ( m ) φ m ,m −−−−−→ F ( m + m ) where φ m ,m : F ( m ) × F ( m ) → F ( m + m ) is the laxator of the lax monoidal pseudofunctor F ,(4) the tensor product of two 2-morphisms: a a ′ m b b ′ m ′ x ∈ F ( m ) x ′ ∈ F ( m ′ ) a m b a ′ m ′ b ′ x ∈ F ( m ) x ′ ∈ F ( m ′ ) ι α : F ( h )( x ) → x ′ ι α : F ( h )( x ) → x ′ i o f g i ′ o ′ h i f h i ′ o g o ′ s given by: a + a a ′ + a ′ m + m b + b b ′ + b ′ m ′ + m ′ x + x ∈ F ( m + m ) x ′ + x ′ ∈ F ( m ′ + m ′ ) ι α + α : F ( h + h )( x + x ) → x ′ + x ′ i + i o + o f + f g + g i ′ + i ′ o ′ + o ′ h + h where ι α + α is given by the diagram: ι α × ι α ⇒ λ − −−−→ × F ( m ) × F ( m ) F ( m ′ ) × F ( m ′ ) F ( m + m ) F ( m ′ + m ′ ) φ m ,m φ m ′ ,m ′ F ( h + h ) x × x x ′ × x ′ F ( h ) × F ( h ) The unit for the tensor product is a chosen initial object of A and the symmetry for any two objects a and b is defined using the canonical isomorphism a + b ∼ = b + a .Proof. First we note that the category of objects F C sp = A is symmetric monoidal under binarycoproducts and the left and right unitors, associators and braidings are given as natural maps. Thecategory of arrows F C sp has:(1) objects as F -decorated cospans which are pairs: a m b x ∈ F ( m ) i o and(2) morphisms as maps of cospans in A aa ′ m bb ′ m ′ x ∈ F ( m ) x ′ ∈ F ( m ′ ) i of gi ′ o ′ h together with a morphism ι : F ( h )( x ) → x ′ .Given two objects M and M of F C sp : a m b x ∈ F ( m ) a m b x ∈ F ( m ) i o i o M ⊗ M is given by taking the coproducts of the cospans of A a + a m + m b + b x + x ∈ F ( m + m ) i + i o + o and where the decoration on the apex is obtained using the natural transformation of the symmetriclax monoidal pseudofunctor F : x + x := 1 λ − −−→ × x × x −−−−→ F ( m ) × F ( m ) φ m ,m −−−−−→ F ( m + m ) . The monoidal unit 0 is given by: ∈ F (0) ! ! where 0 is the monoidal unit of A and ! : 1 → F (0) is the morphism which is part of the structureof the symmetric lax monoidal pseudofunctor F : A → Cat . Tensoring an object with the monoidalunit, say, on the left: ∈ F (0) a m bx ∈ F ( m ) ⊗ ! ! i o results in: a m b ! + x ∈ F (0 + m ) ! + i ! + o where ! + x ∈ F (0 + m ) is given by1 λ − −−→ × ! × x −−−→ F (0) × F ( m ) φ ,m −−−→ F (0 + m ) . The left unitor is then an isomorphism in F C sp given by: aa m bbm ! + x ∈ F (0 + m ) x ∈ F ( m ) ! + i ! + oℓ ℓi oℓ where ℓ is the left unitor of ( A , + , ι λ : F ( ℓ )(! + x ) → x . Theright unitor is similar.Given three objects M , M and M in F C sp : a m b x ∈ F ( m ) a m b x ∈ F ( m ) a m b x ∈ F ( m ) i o i o i o tensoring the first two and then the third results in ( M ⊗ M ) ⊗ M : ( a + a ) + a ( m + m ) + m ( b + b ) + b ( x + x ) + x ∈ F (( m + m ) + m ) ( i + i ) + i ( o + o ) + o x + x ) + x : 1 → F (( m + m ) + m ) is given by:1 ( x × x ) × x −−−−−−−→ ( F ( m ) × F ( m )) × F ( m ) φ m ,m × −−−−−−→ F ( m + m ) × F ( m ) φ m m ,m −−−−−−−→ F (( m + m ) + m )whereas tensoring the last two and then the first results in M ⊗ ( M ⊗ M ): a + ( a + a ) m + ( m + m ) b + ( b + b ) x + ( x + x ) ∈ F ( m + ( m + m )) i + ( i + i ) o + ( o + o ) where x + ( x + x ) : 1 → F ( m + ( m + m )) is given by:1 x × ( x × x ) −−−−−−−→ F ( m ) × ( F ( m ) × F ( m )) × φ m ,m −−−−−−→ F ( m ) × F ( m + m ) φ m ,m m −−−−−−−→ F ( m + ( m + m )) . If we let a denote the associator of ( A , + , F C sp is then a map of cospans in A from ( M ⊗ M ) ⊗ M to M ⊗ ( M ⊗ M ) given by: ( a + a ) + a a + ( a + a ) ( m + m ) + m ( b + b ) + b b + ( b + b ) m + ( m + m ) ( x + x ) + x ∈ F (( m + m ) + m ) x + ( x + x ) ∈ F ( m + ( m + m )) ( i + i ) + i ( o + o ) + o a ai + ( i + i ) o + ( o + o ) a together with the isomorphism ι a : F ( a )(( x + x ) + x ) → x + ( x + x ). These associators andleft and right unitors together satisfy the pentagon and triangle identities of a monoidal category.If we denote the above associator simply as a and the left and right unitors as λ and ρ , respectively,then given four objects in F C sp , say M , M , M and M : a m b x ∈ F ( m ) a m b x ∈ F ( m ) a m b x ∈ F ( m ) a m b x ∈ F ( m ) i o i o i o i o the following pentagon of underlying cospans and maps of cospans commutes: (( M ⊗ M ) ⊗ M ) ⊗ M ( M ⊗ M ) ⊗ ( M ⊗ M ) M ⊗ ( M ⊗ ( M ⊗ M ))( M ⊗ ( M ⊗ M )) ⊗ M M ⊗ (( M ⊗ M ) ⊗ M ) a aa ⊗ a ⊗ a
54s well as the following pentagon of corresponding decorations in the category F ( m + ( m + ( m + m ))): F ( aa )((( x + x ) + x ) + x ) F ( a )(( x + x ) + ( x + x )) x + ( x + ( x + x )) F ((1 ⊗ a ) a )(( x + ( x + x )) + x ) F (1 ⊗ a )( x + (( x + x ) + x )) F ( a )( ι a ) ι a F ((1 ⊗ a ) a )( ι a ⊗ ) F (1 ⊗ a )( ι a ) ι ⊗ a since F ( aa )((( x + x ) + x ) + x ) = F ((1 ⊗ a ) a ( a ⊗ x + x ) + x ) + x )as the corresponding pentagon of cospan apices in the symmetric monoidal category ( A , + ,
0) com-mutes, and then appliying the pseudofunctor F to this commutative pentagon yields a commutativepentagon in Cat .Similarly, if we denote the left and right unitors as λ and ρ , respectively, then the followingtriangle of cospans and underlying maps of cospans commutes: ( M ⊗ ⊗ M M ⊗ M M ⊗ (0 ⊗ M ) ρ ⊗ ⊗ λa as well as the following triangle of corresponding decorations in the category F ( m + m ): F ( ρ ⊗ x + 0) + x ) x + x F (1 ⊗ λ )( x + (0 + x )) ι ρ ⊗ ι ⊗ λ F (1 ⊗ λ )( ι a ) since F ( ρ ⊗ x + 0) + x ) = F ((1 ⊗ λ ) a )(( x + 0) + x )as the corresponding triangle of cospan apices in the symmetric monoidal category ( A , + ,
0) com-mutes and applying the pseudofunctor F to this commutative triangle results in a commutativetriangle in Cat .For a tensor product of objects M ⊗ M in F C sp , the source and target functors S, T : F C sp → F C sp satisfy the following equations: S ( M ⊗ M ) = S ( M ) ⊗ S ( M ) T ( M ⊗ M ) = T ( M ) ⊗ T ( M ) . M and M in F C sp , we have a braiding β M ,M : M ⊗ M → M ⊗ M given by: a + a a + a m + m b + b b + b m + m x + x ∈ F ( m + m ) x + x ∈ F ( m + m ) i + i o + o β a ,a β b ,b i + i o + o β m ,m ι β M ,M : F ( β m ,m )( x + x ) ∼ −→ x + x where the vertical 1-morphisms are given by braidings in ( A , + , M ⊗ M M ⊗ M M ⊗ M β M ,M β M ,M as well as the following diagram of corresponding decorations in the category F ( m + m ): x + x x + x F ( β m ,m )( x + x ) ι β M ,M F ( β m ,m )( ι β M ,M ) since F ( β m ,m β m ,m )( x + x ) = x + x . Thus F C sp is also symmetric monoidal.Next we derive the globular isomorphisms required in the definition of a symmetric monoidaldouble category relating horizontal composition and the tensor product. Given four horizontal1-cells M , M , N and N respectively by: a m bx ∈ F ( m ) b m cx ∈ F ( m ) a ′ n b ′ y ∈ F ( n ) b ′ n c ′ y ∈ F ( n ) i o i o i ′ o ′ i ′ o ′ we have that ( M ⊗ N ) ⊙ ( M ⊗ N ) is given by: a + a ′ ( m + n ) + b + b ′ ( m + n ) c + c ′ ( x + y ) ⊙ ( x + y ) ∈ F (( m + n ) + b + b ′ ( m + n )) jψ ( i + i ′ ) jψ ( o + o ′ ) x + y ) ⊙ ( x + y ) ∈ F (( m + n ) + b + b ′ ( m + n )) is given by: × × × (1 × F ( m ) × F ( n )) × ( F ( m ) × F ( n )) F ( m + n ) × F ( m + n ) F (( m + n ) + ( m + n )) F (( m + n ) + b + b ′ ( m + n )) λ − λ − × λ − ( x × y ) × ( x × y ) φ m ,n × φ m ,n φ m + n ,m + n F ( j m + n ,m + n ) and ( M ⊙ M ) ⊗ ( N ⊙ N ) is given by: a + a ′ ( m + b m ) + ( n + b ′ n ) c + c ′ ( x ⊙ x ) + ( y ⊙ y ) ∈ F (( m + b m ) + ( n + b ′ n )) ( jψi ) + ( jψi ′ ) ( jψo ) + ( jψo ′ ) x ⊙ x ) + ( y ⊙ y ) ∈ F (( m + b m ) + ( n + b ′ n )) is given by: × × × (1 × F ( m ) × F ( m )) × ( F ( n ) × F ( n )) F ( m + m ) × F ( n + n ) F ( m + b m ) × F ( n + b ′ n ) F (( m + b m ) + ( n + b ′ n )) λ − λ − × λ − ( x × x ) × ( y × y ) φ m ,m × φ n ,n F ( j m ,m ) × F ( j n ,n ) φ m + b m ,n + b ′ n and where ψ and j are the natural maps into a coproduct and from a coproduct into a pushout,respectively. We then get a globular 2-morphism χ : ( M ⊗ N ) ⊙ ( M ⊗ N ) → ( M ⊙ M ) ⊗ ( N ⊙ N )given by: a + a ′ a + a ′ ( m + n ) + b + b ′ ( m + n ) c + c ′ c + c ′ ( m + b m ) + ( n + b ′ n )( x + y ) ⊙ ( x + y ) ∈ F (( m + n ) + b + b ′ ( m + n ))( x ⊙ x ) + ( y ⊙ y ) ∈ F (( m + b m ) + ( n + b ′ n )) jψ ( i + i ′ ) jψ ( o + o ′ )1 1( jψi ) + ( jψi ′ ) ( jψo ) + ( jψo ′ )ˆ χ ι ˆ χ : F ( ˆ χ )(( x + y ) ⊙ ( x + y )) → ( x ⊙ x ) + ( y ⊙ y )where ˆ χ is the universal map between two colimits of the same diagram. For two objects a, b ∈ A , U a + b is given by: a + b a + b a + b ! a + b ∈ F ( a + b ) a + b a + b where ! a + b : 1 φ −→ F (0) F (! a + b ) −−−−−→ F ( a + b ) . U a and U b given respectively by: a a a ! a ∈ F ( a ) b b b ! b ∈ F ( b ) a a b b and then U a + U b is given by: a + b a + b a + b ! a +! b ∈ F ( a + b ) a + 1 b a + 1 b where ! a +! b : 1 λ − −−→ × φ × φ −−−→ F (0) × F (0) F (! a ) × F (! b ) −−−−−−−→ F ( a ) × F ( b ) φ a,b −−→ F ( a + b ) . We then have the second globular isomorphism µ a,b : U a + b → U a + U b given by the identity 2-morphism: a + ba + b a + b a + ba + ba + b ! a + b ∈ F ( a + b )! a +! b ∈ F ( a + b ) a + b a + b a + 1 b a + 1 b ι a,b : ! a + b ∼ −→ ! a +! b where ! a + b and ! a +! b are both initial objects in F ( a + b ), hence isomorphic.There are many coherence laws to be checked, most of which are similar in flavor and make useof the two above globular isomorphisms. We check a few to give a sense of what these are like.For example, given horizontal 1-cells M i , N i , P i for i = 1 ,
2, the following commutative diagramexpresses the associativity isomorphism as a transformation of double categories. (( M ⊗ N ) ⊗ P ) ⊙ (( M ⊗ N ) ⊗ P ) ( M ⊗ ( N ⊗ P )) ⊙ ( M ⊗ ( N ⊗ P ))(( M ⊗ N ) ⊙ ( M ⊗ N )) ⊗ ( P ⊙ P ) ( M ⊙ M ) ⊗ (( N ⊗ P ) ⊙ ( N ⊗ P ))(( M ⊙ M ) ⊗ ( N ⊙ N )) ⊗ ( P ⊙ P ) ( M ⊙ M ) ⊗ (( N ⊙ N ) ⊗ ( P ⊙ P )) a ⊙ aχ χχ ⊗ ⊗ χa Here, a is the associator of F C sp and χ is the first globular isomorphism above. To see that thisdiagram does indeed commute, we first consider this diagram with respect to only the underlyingcospans of each horizontal 1-cell. For notation: a m bM = N = a ′ n b ′ P = a ′′ p b ′′ b m cM = N = b ′ n c ′ P = b ′′ p c ′′ x ∈ F ( m ) x ∈ F ( m ) y ∈ F ( n ) y ∈ F ( n ) z ∈ F ( p ) z ∈ F ( p ) ( a + a ′ ) + a ′′ (( m + n ) + p ) + (( b + b ′ )+ b ′′ ) (( m + n ) + p ) ( c + c ′ ) + c ′′ a + ( a ′ + a ′′ ) ( m + ( n + p )) + ( b +( b ′ + b ′′ )) ( m + ( n + p )) c + ( c ′ + c ′′ ) a + ( a ′ + a ′′ ) ( m + b m ) + (( n + p ) + ( b ′ + b ′′ ) ( n + p )) c + ( c ′ + c ′′ ) a + ( a ′ + a ′′ ) ( m + b m ) + (( n + b ′ n ) + ( p + b ′′ p )) c + ( c ′ + c ′′ )( a + a ′ ) + a ′′ (( m + n ) + ( b + b ′ ) ( m + n )) + ( p + b ′′ p ) ( c + c ′ ) + c ′′ ( a + a ′ ) + a ′′ (( m + b m ) + ( n + b ′ n )) + ( p + b ′′ p ) ( c + c ′ ) + c ′′ a + ( a ′ + a ′′ ) ( m + b m ) + (( n + b ′ n ) + ( p + b ′′ p )) c + ( c ′ + c ′′ ) a ⊙ a ι χ ι ⊗ χ ι χ ι χ ⊗ ι a ι which does indeed commute. Here, all of the vertical 1-morphisms on the left and right are associ-ators or identities, the middle vertical 1-morphisms labeled on the left are the 2-morphisms fromthe previous commutative diagram, and the cospan legs are natural maps into each colimit, all ofwhich are naturally isomorphic to each other as all the middle objects are colimits of the samediagram, namely the previous collection of cospans, taken in various ways. By identifying the topand bottom edges of the above diagram, it can be visualized as a hexagonal prism. Every face ofthis prism commutes. As for the morphisms of decorations, which are labeled on the right of theinterior vertical 1-morphisms, each isomorphism ι n goes from the domain under the image of thefunctor F applied to the natural isomorphism adjacent to it to the codomain as written, meaningthat, for example: ι : F ( a ⊙ a )((( x + y ) + z ) ⊙ (( x + y ) + z )) → ( x + ( y + z )) ⊙ ( x + ( y + z )) . The following diagram commutes in the category F (( m + b m ) + (( n + b ′ n ) + ( p + b ′′ p ))): F ( a ( χ ⊗ χ )((( x + y ) + z ) ⊙ (( x + y ) + z )) F ((1 ⊗ χ ) χ )(( x + ( y + z )) ⊙ ( x + ( y + z ))) F ( a ( χ ⊗ x + y ) ⊙ ( x + y )) + ( z ⊙ z )) F (1 ⊗ χ )(( x ⊙ x ) + (( y + z ) ⊙ ( y + z ))) F ( a )((( x ⊙ x ) + ( y ⊙ y )) + ( z ⊙ z )) ( x ⊙ x ) + (( y ⊙ y ) + ( z ⊙ z )) F ((1 ⊗ χ ) χ )( ι ) F ( a ( χ ⊗ ι ) F (1 ⊗ χ )( ι ) F ( a )( ι ) ι ι since F ( a ( χ ⊗ χ )((( x + y )+ z ) ⊙ (( x + y )+ z )) = F ((1 ⊗ χ ) χ ( a ⊙ a ))((( x + y )+ z ) ⊙ (( x + y )+ z ))as the hexagon formed by the morphisms between the cospan apices of the above underlying diagramof maps of cospans commutes and then applying the pseudofunctor F to this hexagon yields acommutative hexagon in Cat . 60nother requirement for a double category to be symmetric monoidal is that the braiding β ( − , − ) : F C sp × F C sp → F C sp × F C sp be a transformation of double categories, and one of the diagrams that is required to commute isthe following: ( M ⊙ M ) ⊗ ( N ⊙ N ) ( N ⊙ N ) ⊗ ( M ⊙ M )( M ⊗ N ) ⊙ ( M ⊗ N ) ( N ⊗ M ) ⊙ ( N ⊗ M ) β χχ β ⊙ β Using the same notation as the previous coherence diagram, the diagram for the underlying mapsof cospans becomes: a + a ′ ( m + b m ) + ( n + b ′ n ) c + c ′ a ′ + a ( n + b ′ n ) + ( m + b m ) c ′ + ca ′ + a ( n + m ) + ( b ′ + b ) ( n + m ) c ′ + ca + a ′ ( m + n ) + ( b + b ′ ) ( m + n ) c + c ′ a ′ + a ( n + m ) + ( b ′ + b ) ( n + m ) c ′ + c β ι χ ι χ ι β ⊙ β ι All the comments about the previous underlying coherence diagram of maps of cospans apply to thisone. As for the decorations, the following diagram commutes in the category F (( n + m ) + ( b ′ + b ) ( n + m )): F ( χβ )(( x ⊙ x ) + ( y ⊙ y )) F ( χ )(( y ⊙ y ) + ( x ⊙ x )) F ( β ⊙ β )(( x + y ) ⊙ ( x + y )) ( y + x ) ⊙ ( y + x ) F ( χ )( ι ) ι F ( β ⊙ β )( ι ) ι since F ( χβ )(( x ⊙ x ) + ( y ⊙ y )) = F (( β ⊙ β ) χ )(( x ⊙ x ) + ( y ⊙ y ))as the square formed by the morphisms between the cospan apices of the above underlying diagramof maps of cospans commutes and then applying the pseudofunctor F to this square yields acommutative square in Cat . The other diagrams are shown to commute similarly.
Given another symmetric lax monoidal pseudofunctor F ′ : A ′ → Cat , we can obtain anothersymmetric monoidal double category F ′ C sp . A map from F C sp to F ′ C sp will then be a doublefunctor H : F C sp → F ′ C sp whose object component is given by a finite colimit preserving functor61 = H : A → A ′ and whose arrow component is given by a functor H defined on horizontal 1-cellsby: a c bd ∈ F ( c ) H ( a ) H ( c ) H ( b ) θ c E ( d ) φ ∈ F ′ ( H ( c )) i o H ( i ) H ( o ) and on 2-morphisms by: aa ′ c bb ′ c ′ d ∈ F ( c ) d ′ ∈ F ( c ′ ) ι : F ( h )( d ) → d ′ H ( a ) H ( a ′ ) H ( c ) H ( b ) H ( b ′ ) H ( c ′ ) θ c E ( d ) φ ∈ F ′ ( H ( c )) θ c ′ E ( d ′ ) φ ∈ F ′ ( H ( c ′ )) E ( ι ) : F ′ ( H ( h ))( θ c E ( d ) φ ) → ( θ c ′ E ( d ′ ) φ ) f gh H ( f ) H ( g ) H ( h ) where E : Cat → Cat is a symmetric lax monoidal pseudofunctor such that the following diagramcommutes up to a monoidal natural isomorphism θ : EF ⇒ F ′ H : A Cat A ′ Cat ⇒ θ FH EF ′ We summarize this in the following theorem:
Theorem 4.2.1.
Given two finitely cocomplete categories A and A ′ , two symmetric lax monoidalpseudofunctors F : A → Cat and F ′ : A ′ → Cat , a finite colimit preserving functor H : A → A ′ ,a symmetric lax monoidal pseudofunctor E : Cat → Cat and a monoidal natural isomorphism θ : EF ⇒ F ′ H as in the following diagram, the triple ( H, E, θ ) induces a symmetric monoidaldouble functor H : F C sp → F ′ C sp as defined above. A Cat A ′ Cat ⇒ θ FH EF ′ Proof.
Recall that we can think of the object d ∈ F ( c ) as a morphism d : 1 → F ( c ) and themorphism ι : F ( h )( d ) → d ′ of F ( c ′ ) as a natural transformation in Cat : F ( c ) F ( c ′ ) ⇒ ι dd ′ F ( h ) E : Cat → Cat to this diagram yields: φ −→ E (1) E ( F ( c )) E ( F ( c ′ )) ⇒ E ( ι ) E ( d ) E ( d ′ ) E ( F ( h )) Then because the above square commutes up to the isomorphism θ : EF ⇒ F ′ H , we get: φ −→ E (1) E ( F ( c )) E ( F ( c ′ )) ⇒ E ( ι ) F ′ ( H ( c )) F ′ ( H ( c ′ )) θ c θ c ′ F ′ ( H ( h )) E ( d ) E ( d ′ ) E ( F (( h )) which results in a 2-morphism E ( ι ) : F ′ ( H ( h ))( θ c E ( d ) φ ) → ( θ c ′ E ( d ′ ) φ ) in F ′ ( H ( c ′ )). To checkthat the above recipe is functorial, suppose we are given two vertically composable 2-morphisms in F C sp : aa ′ c bb ′ c ′ d ∈ F ( c ) d ′ ∈ F ( c ′ ) ι : F ( h )( d ) → d ′ a ′ a ′′ c ′ b ′ b ′′ c ′′ d ′ ∈ F ( c ′ ) d ′′ ∈ F ( c ′′ ) ι ′ : F ( h ′ )( d ′ ) → d ′′ f ghf ′ g ′ h ′ If we first compose these, the result is: aa ′′ c bb ′′ c ′′ d ∈ F ( c ) d ′′ ∈ F ( c ′′ ) ι ′ ι : F ( h ′ h )( d ) → d ′′ f ′ f g ′ gh ′ h and then the image of this 2-morphism under the double functor H is given by: H ( a ) H ( a ′′ ) H ( c ) H ( b ) H ( b ′′ ) H ( c ′′ ) θ c E ( d ) φ ∈ F ′ ( H ( c )) θ c ′′ E ( d ′′ ) φ ∈ F ′ ( H ( c ′′ )) E ( ι ′ ι ) : F ′ ( H ( h ′ h ))( θ c E ( d ) φ ) → ( θ c ′′ E ( d ′′ ) φ ) . H ( f ′ f ) H ( g ′ g ) H ( h ′ h )
63n the other hand, applying the double functor H first gives: H ( a ) H ( a ′ ) H ( c ) H ( b ) H ( b ′ ) H ( c ′ ) θ c E ( d ) φ ∈ F ′ ( H ( c )) θ c ′ E ( d ′ ) φ ∈ F ′ ( H ( c ′ )) E ( ι ) : F ′ ( H ( h ))( θ c E ( d ) φ ) → ( θ c ′ E ( d ′ ) φ ) H ( a ′ ) H ( a ′′ ) H ( c ′ ) H ( b ′ ) H ( b ′′ ) H ( c ′′ ) θ c ′ E ( d ′ ) φ ∈ F ′ ( H ( c ′ )) θ c ′′ E ( d ′′ ) φ ∈ F ′ ( H ( c ′′ )) E ( ι ′ ) : F ′ ( H ( h ′ ))( θ c ′ E ( d ′ ) φ ) → ( θ c ′′ E ( d ′′ ) φ ) H ( f ) H ( g ) H ( h ) H ( f ′ ) H ( g ′ ) H ( h ′ ) and then composing these gives: H ( a ) H ( a ′′ ) H ( c ) H ( b ) H ( b ′′ ) H ( c ′′ ) θ c E ( d ) φ ∈ F ′ ( H ( c )) θ c ′′ E ( d ′′ ) φ ∈ F ′ ( H ( c ′′ )) E ( ι ′ ι ) : F ′ ( H ( h ′ h ))( θ c E ( d ) φ ) → ( θ c ′′ E ( d ′′ ) φ ) . H ( f ′ f ) H ( g ′ g ) H ( h ′ h ) Thus H is functorial on 2-morphisms, and it is evident that H satisfies the equations S H = HS and T H = HT .Given two composable horizontal 1-cells M and N in F C sp : a c bd M ∈ F ( c ) b c a d N ∈ F ( c ) i o i o composing first gives M ⊙ N : a c + b c a d M ⊙ N ∈ F ( c + b c ) ψj c i ψj c o where d : 1 λ − −−→ × d × d −−−−→ F ( c ) × F ( c ) φ c ,c −−−−→ F ( c + c ) F ( j ) −−−→ F ( c + b c ) . The image of this horizontal 1-cell is then given by H ( M ⊙ N ): H ( a ) H ( c + b c ) H ( a ) d H ( M ⊙ N ) = θ c + b c E ( d M ⊙ N ) φ ∈ F ′ ( H ( c + b c )) H ( ψj c i ) H ( ψj c o ) d H ( M ⊙ N ) = θ c + b c E ( d M ⊙ N ) φ : 1 φ −→ E (1) E ( d M ⊙ N ) −−−−−−→ E ( F ( c + b c )) θ c bc −−−−−→ F ′ ( H ( c + b c )) . On the other hand, the image of each horizontal 1-cell under the double functor H is given respec-tively by H ( M ) and H ( N ): H ( a ) H ( c ) H ( b ) θ c E ( d M ) φ ∈ F ′ ( H ( c )) H ( b ) H ( c ) H ( a ) θ c E ( d N ) φ ∈ F ′ ( H ( c )) H ( i ) H ( o ) H ( i ) H ( o ) Composing these then gives H ( M ) ⊙ H ( N ): H ( a ) H ( c ) + H ( b ) H ( c ) H ( a ) d H ( M ) ⊙ H ( N ) ∈ F ′ ( H ( c ) + H ( b ) H ( c )) Ψ j H ( c ) H ( i ) Ψ j H ( c ) H ( o ) where d H ( M ) ⊙ H ( N ) : 1 ( θc × θc E ( dM ) × E ( dN )) φ −−−−−−−−−−−−−−−−−→ F ′ ( H ( c )) × F ′ ( H ( c )) Φ H ( c ,H ( c −−−−−−−−→ F ′ ( H ( c )+ H ( c )) F ′ ( J ) −−−→ F ′ ( H ( c )+ H ( b ) H ( c )) . We then have a comparison constraint: H M,N : H ( M ) ⊙ H ( N ) ∼ −→ H ( M ⊙ N )given by the globular 2-isomorphism: H ( a ) H ( a ) H ( c ) + H ( b ) H ( c ) H ( a ) H ( a ) H ( c + b c ) d H ( M ) ⊙ H ( N ) ∈ F ′ ( H ( c ) + H ( b ) H ( c )) d H ( M ⊙ N ) ∈ F ′ ( H ( c + b c )) ι κ − : F ′ ( κ − )( d H ( M ) ⊙ H ( N ) ) → d H ( M ⊙ N ) . Ψ j H ( c ) H ( i ) Ψ j H ( c ) H ( o )1 1 H ( ψj c i ) H ( ψj c o ) κ − where κ is the natural isomorphism κ : H ( c + b c ) ∼ −→ H ( c ) + H ( b ) H ( c )which comes from the finite colimit preserving functor H : A → A ′ . The above diagram commutesby a similar argument to the one used in Theorem 4.3.15. Similarly, for every object c ∈ A , wehave a unit comparison constraint H U : U H ( c ) → H ( U c )given by the globular 2-isomorphism: H ( c ) H ( c ) H ( c ) H ( c ) H ( c ) H ( c ) ! H ( c ) ∈ F ′ ( H ( c )) θ c E (! c ) φ ∈ F ′ ( H ( c )) ι : ! H ( c ) → ( θ c E (! c ) φ ) in F ′ ( H ( c )). Thesecomparison constrains satisfy the coherence axioms of a monoidal category, namely that thesediagrams commute: ( H ( M ) ⊙ H ( N )) ⊙ H ( P ) H ( M ⊙ N ) ⊙ H ( P ) H (( M ⊙ N ) ⊙ P ) H ( M ) ⊙ ( H ( N ) ⊙ H ( P )) H ( M ) ⊙ H ( N ⊙ P ) H ( M ⊙ ( N ⊙ P )) H M,N ⊙ H M ⊙ N,P a H ( a ′ ) H M,N ⊙ P ⊙ H N,P U H ( a ) ⊙ H ( M ) H ( U a ) ⊙ H ( M ) H ( M ) H ( U a ⊙ M ) H ( M ) ⊙ U H ( b ) H ( M ) ⊙ H ( U b ) H ( M ) H ( M ⊙ U b ) H U ⊙ λ H U a ,M H ( λ ′ ) 1 ⊙ H U ρ H M,U b H ( ρ ′ ) The diagrams involving the morphisms of decorations are similar to those in Theorem 4.1.3. Thisshows that H = ( H, E, θ ) is a double functor. Next we show that this double functor is symmetricmonoidal. First, that the object component H = H is symmetric monoidal is clear as H : A → A ′ preserves finite colimits. As for the arrow component H , given two horizontal 1-cells M and M in F C sp : a c b d M ∈ F ( c ) a c b d M ∈ F ( c ) i o i o their tensor product M ⊗ M in F C sp is given by: a + a c + c b + b d M ⊗ M ∈ F ( c + c ) i + i o + o d M ⊗ M : 1 d × d −−−−→ F ( c ) × F ( c ) φ c ,c −−−−→ F ( c + c )and the image of this horizontal 1-cell under the double functor H is H ( M ⊗ M ) given by: H ( a + a ) H ( c + c ) H ( b + b ) d H ( M ⊗ M ) = θ c + c E ( d M ⊗ M ) φ ∈ F ′ ( H ( c + c )) . H ( i + i ) H ( o + o ) On the other hand, the image of M and M is given by H ( M ) and H ( M ): H ( a ) H ( c ) H ( b ) d H ( M ) = θ c E ( d M ) φ ∈ F ′ ( H ( c )) H ( a ) H ( c ) H ( b ) d H ( M ) = θ c E ( d M ) φ ∈ F ′ ( H ( c )) H ( i ) H ( o ) H ( i ) H ( o ) H ( M ) ⊗ H ( M ) is given by: H ( a ) + H ( a ) H ( c ) + H ( c ) H ( b ) + H ( b ) d H ( M ) ⊗ H ( M ) ∈ F ′ ( H ( c ) + H ( c )) H ( i ) + H ( i ) H ( o ) + H ( o ) d H ( M ⊗ H ( M : 1 ( φ × φ )( λ − × λ − −−−−−−−−−−−→ E (1) × E (1) ( θc × θc E ( dM × E ( dM −−−−−−−−−−−−−−−−−−→ F ′ ( H ( c )) × F ′ ( H ( c )) Φ H ( c ,H ( c −−−−−−−−→ F ′ ( H ( c )+ H ( c )) . We then have a natural 2-isomorphism µ M ,M : H ( M ) ⊗ H ( M ) → H ( M ⊗ M ) in F ′ C sp givenby: H ( a ) + H ( a ) H ( a + a ) H ( c ) + H ( c ) H ( b ) + H ( b ) H ( b + b ) H ( c + c ) d H ( M ) ⊗ H ( M ) ∈ F ′ ( H ( c ) + H ( c )) d H ( M ⊗ M ) ∈ F ′ ( H ( c + c )) H ( i ) + H ( i ) H ( o ) + H ( o ) κ κH ( i + i ) H ( o + o ) κ ι κ : F ′ ( κ )( d H ( M ) ⊗ H ( M ) ) → d H ( M ⊗ M ) where κ denotes the natural isomorphism arising from H preserving finite colimits. This natural2-isomorphism together with the associators of F C sp and F ′ C sp , respectively α and α ′ , make thefollowing diagram commute: ( H ( M ) ⊗ H ( M )) ⊗ H ( M ) H ( M ⊗ M ) ⊗ H ( M ) H (( M ⊗ M ) ⊗ M ) H ( M ) ⊗ ( H ( M ) ⊗ H ( M )) H ( M ) ⊗ H ( M ⊗ M ) H ( M ⊗ ( M ⊗ M )) µ M ,M ⊗ µ M ⊗ M ,M α ′ H ( α ) µ M ,M ⊗ M ⊗ µ M ⊗ M with the corresponding diagram of decorations in F ′ ( H ( c + ( c + c ))): F ′ ( ακ ( κ + 1))( d ( H ( M ) ⊗ H ( M )) ⊗ H ( M ) ) F ′ ( ακ )( d H ( M ⊗ M ) ⊗ H ( M ) ) F ′ ( α )( d H (( M ⊗ M ) ⊗ M ) ) F ′ ( κ (1 + κ ))( d H ( M ) ⊗ ( H ( M ) ⊗ H ( M )) ) F ′ ( κ )( d H ( M ) ⊗ H ( M ⊗ M ) ) d H ( M ⊗ ( M ⊗ M )) F ′ ( ακ )( ι κ + 1) F ′ ( α )( ι κ ) F ′ ( κ (1 + κ ))( ι α ′ ) ι α ι κ F ′ ( κ )(1 + ι κ ) where F ′ ( ακ ( κ + 1))( d ( H ( M ) ⊗ H ( M )) ⊗ H ( M ) ) = F ′ ( κ ( κ + 1) α ′ )( d ( H ( M ) ⊗ H ( M )) ⊗ H ( M ) )67s the corresponding hexagon for the finite colimit preserving functor H : A → A ′ commutes. Themap µ M ,M is also compatible with the braidings β and β ′ of F C sp and F ′ C sp , respectively, andmake the necessary square commute as a consequence of the corresponding commutative squareinvolving braidings from the finite colimit preserving functor H : A → A ′ .The monoidal unit of F C sp is given by: A A A ! A ∈ F (1 A ) where 1 A is the monoidal unit of the finitely cocomplete category A . The image of this horizontal1-cell under H is given by: H (1 A ) H (1 A ) H (1 A ) θ A E (! A ) φ ∈ F ′ ( H (1 A )) as H preserves finite colimits. We then have a 2-isomorphism in F ′ C sp given by: µ : 1 F ′ C sp → H (1 F C sp ) A ′ H (1 A ) 1 A ′ A ′ H (1 A ) H (1 A ) ! A ′ ∈ F ′ (1 A ′ ) θ A E (! A ) φ ∈ F ′ ( H (1 A )) κ κ κ together with the morphism ι µ : F ′ ( κ )(! A ′ ) → ( θ A E (! A ) φ ) in F ′ ( H (1 A )). The following squarethen commutes for any horizontal 1-cell M of F C sp : A ′ ⊗ H ( M ) H (1 A ) ⊗ H ( M ) H ( M ) H (1 A ⊗ M ) µ ⊗ ℓ µ A ,M H ( ℓ ′ ) where we have abbreviated the monoidal units of F C sp and F ′ C sp as 1 A and 1 A ′ , respectively.The diagram of corresponding decorations is given by: F ′ ( ℓ )( d ! A ′ ⊗ d H ( M ) ) d H ( M ) F ′ ( H ( ℓ ′ ) κ )( d ! H (1 A ) ⊗ d H ( M ) ) F ′ ( H ( ℓ ′ ))( d H (1 A ⊗ M ) ) ι ℓ F ′ ( H ( ℓ ′ ) κ )( ι µ ⊗ ) F ′ ( H ( ℓ ′ ))( ι κ ) ι H ( ℓ ′ ) where F ′ ( ℓ )( d ! A ′ ⊗ d H ( M ) ) = F ′ ( H ( ℓ ′ ) κ ( µ ⊗ d ! A ′ ⊗ d H ( M ) )68ince the corresponding square involving left unitors for the finite colimit preserving functor H : A → A ′ commutes. The other square involving the right unitors r and r ′ is similar. The comparisonand unit constraints H M,N and H U are monoidal transformations and this suffices for a functor ofsymmetric monoidal double categories which are isofibrant, which F C sp and F ′ C sp are by Lemma5.2.1. Note that because the comparison constraints µ and µ ( − , − ) are both isomorphisms, thesymmetric monoidal double functor H is strong. In this section we compare the double categories obtained via structured cospans and decoratedcospans. Under conditions discovered by Christina Vasilakopoulou, the two frameworks will beshown to be equivalent as double categories. This is Theorem 4.3.15 and the main content of thissection. But first, we make precise what it meant by an ‘equivalence of double categories’.We define an equivalence of double categories following Shulman [38]. Given a double category A , we write f A g ( M, N ) for the set of 2-morphisms in A of the form: A | M / / f (cid:15) (cid:15) ⇓ a B g (cid:15) (cid:15) C | N / / D We call M and N the horizontal source and target of the 2-morphism a , respectively, andlikewise we call f and g the vertical source and target of the 2-morphism a , respectively. Thus f A g ( M, N ) denotes the set of 2-morphisms in A with horizontal source and target M and N andvertical source and target f and g . Definition 4.3.1.
A (possibly lax or oplax) double functor F : A → X is full (respectively, faithful )if F : A → X is full (respectively, faithful) and each map F : f A g ( M, N ) → F ( f ) X F ( g ) ( F ( M ) , F ( N ))is surjective (respectively, injective). Definition 4.3.2.
A (possibly lax or oplax) double functor F : A → X is essentially surjective if we can simultaneously make the following choices:(1) For each object x ∈ X , we can find an object a ∈ A together with a vertical 1-isomorphism α x : F ( a ) → x , and(2) For each horizontal 1-cell N : x → x of X , we can find a horizontal 1-cell M : a → a of A and a 2-isomorphism a N of X as in the following diagram: F ( a ) | F ( M ) / / α x (cid:15) (cid:15) ⇓ a N F ( a ) α x (cid:15) (cid:15) x | N / / x efinition 4.3.3. A double functor F : A → X is strong if the comparison and unit constraintsare globular isomorphisms, meaning that for each composable pair of horizontal 1-cells M and N we have a natural isomorphism F M,N : F ( M ) ⊙ F ( N ) ∼ −→ F ( M ⊙ N )and for each object a ∈ A a natural isomorphism F a : ˆ U F ( a ) ∼ −→ F ( U a ) . Shulman [38, Theorem 7.8] proved that a strong double functor is part of a ‘double equivalence’if and only if it is full, faithful and essentially surjective in the sense of a double functor as givenabove. We will take this theorem and use it as the definition of a double equivalence.
Definition 4.3.4.
Given a strong double functor F : A → X , F is part of a double equivalence if and only if F is full, faithful and essentially surjective. We say that F : A → X is a doubleequivalence and that A and X are equivalent as double categories. Definition 4.3.5.
Given a double equivalence F : A → X , if F , A and X are all symmetric monoidal,then F is a symmetric monoidal double equivalence , and A and X are equivalent as symmetricmonoidal double categories.Given a symmetric lax monoidal pseudofunctor F : ( A , + , → ( Cat , × , R : R F → A by the Grothendieck construction, as explained in Definition 4.3.9. Moreover,if the pseudofunctor F : A → Cat factors through
Rex → Cat as an ordinary pseudofunctor,the category R F will have finite colimits and this functor R will preserve finite colimits and beright adjoint to a fully faithful left adjoint L : A → R F between two categories with finite colimitswhich then allows for the construction of a structured cospan double category. The bridge whichallows us to obtain a left adjoint L : ( A , + , → ( R F, + ,
0) from a lax monoidal pseudofunctor F : ( A , + , → ( Cat , × ,
1) is established in Lemma 4.3.11, Corollary 4.3.12 and Proposition 4.3.13.In this case, the resulting decorated cospan double category F C sp and structured cospan doublecategory L C sp ( R F ) are equivalent as symmetric monoidal double categories.First we find conditions under which an opfibration has a left adjoint. This bridge between thenotions of opfibration and left adjoint is due to Christina Vasilakopoulou, who together with Baezand the author have investigated this situation and its consequences in more detail [4].The definitions of 2-category and pseudofunctor are given in Definitions A.3.2 and A.3.4, re-spectively, of the Appendix. Definition 4.3.6.
Let
Rex denote the 2-category of categories with finite colimits and finite colimitpreserving functors.
Definition 4.3.7.
A functor R : X → A is a Grothendieck opfibration if for any object a ∈ A and every object x ∈ X such that R ( x ) = a , for any morphism f : a → b there exists a cocartesianlifting of f . This means that there exists a morphism β in X whose domain is x which satisfies thefollowing universal property: for any morphism g : b → b ′ in A and morphism γ : x → y ′ in X such70hat R ( γ ) = g ◦ f , there exists a unique morphism δ : y → y ′ such that γ = δ ◦ β and R ( δ ) = g . y ′ (cid:20) (cid:20) x β / / (cid:22) (cid:22) γ ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ y (cid:21) (cid:21) ∃ ! δ ❧❧❧❧❧ in X b ′ a f = R ( β ) / / g ◦ f = R ( γ ) ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ b g ❦❦❦❦❦❦❦❦❦❦ in A We call X the total category and A the base category of the opfibration R : X → A .For any object a ∈ A , the fiber category X a consists of all objects x ∈ X such that R ( x ) = a and all morphisms γ : x → x ′ such that R ( f ) = 1 a . The axiom of choice allows us to select acocartesian lifting for any f : a → b which we denote byCocart( f, x ) : x → f ! ( x ) . This choice also induces reindexing functors f ! : X a → X b between any two fiber categories X a and X b . Note that by the universal property of a cocartesianlifting, we have natural isomorphisms (1 a ) ! ∼ = 1 X a and for any composable morphisms f and g in A ,( f ◦ g ) ! ∼ = f ! ◦ g ! . If these natural isomorphisms are equalities, we say that R is a split opfibration . Definition 4.3.8.
Let
OpFib ( A ) be the 2-subcategory of the slice 2-category of Cat / A of opfi-brations over A , cocartesian lifting preserving functors and natural transformations with verticalcomponents.There is a 2-equivalence between opfibrations and pseudofunctors which is given by the wellknown ‘Grothendieck construction’. Definition 4.3.9.
Given a pseudofunctor F : A → Cat where A is a category with trivial 2-morphisms, the Grothendieck category R F has:(1) objects as pairs ( a, x ∈ F ( a )) and(2) a morphism from a pair ( a, x ∈ F ( a )) to another pair ( b, y ∈ F ( b )) is given by a pair( f : a → b, ι : F ( f )( x ) → y ) in A × F ( b ). Note that a morphism can be viewed as a morphismtogether with a 2-morphism: ab F ( a ) F ( b ) ι ⇒ f xy F ( f ) There is an opfibration R : R F → A where the fiber categories are given by ( R F ) a = F ( a )and the associated reindexing functors are given by f ! = F ( f ). We call the entirety of this the Grothendieck construction of the pseudofunctor F .71he Grothendieck construction provides one direction of a well known equivalence. Theorem 4.3.10. (1) Every opfibration R : X → A gives rise to a pseudofunctor F R : A → Cat .(2) Every pseudofunctor F : A → Cat gives rise to an opfibration R F : R F → A .(3) The above two correspondences give rise to an equivalence of 2-categories [ A , Cat ] ps ≃ OpFib ( A ) such that F R F ∼ = F and R F R ∼ = R . Moeller and Vasilakopoulou [35] have generalized the Grothendieck construction to the monoidalsituation, meaning that lax monoidal pseudofunctors F : A → Cat correspond bijectively tomonoidal structures on the total category R F such that the corresponding opfibration R F : R F → A is a strict monoidal functor and the tensor product ⊗ R F preserves cocartesian liftings. If A iscocartesian monoidal, there is a further correspondence given by:lax monoidal pseudofunctors F : ( A , + , → ( Cat , × , ≃ monoidal opfibrations R : ( X , ⊗ , I ) → ( A , + ,
0) (4.1) ≃ pseudofunctors F : A → MonCat
The second equivalence is due to Shulman [38]. In detail, given a lax monoidal structure ( φ, φ )on a pseudofunctor F , each fiber category inherits a monoidal structure via: ⊗ a : F ( a ) × F ( a ) φ a,a −−→ F ( a + a ) F ( ∇ ) −−−→ F ( a ) (4.2) I x : φ −→ F (0) F (!) −−→ F ( a ) . These correspondences further restrict when the Grothendieck category R F is cocartesianmonoidal itself. In this case, the monoidal opfibration clauses for R : ( X , + , → ( A , + ,
0) results ina functor (strictly) preserving coproducts and the initial object, and these bijectively correspond topseudofunctors F : A → cocartCat where cocartCat is the 2-category of cocartesian categories,coproduct preserving functors and natural transformations. The following statement, which relatesthe existence of any class of colimits in the total category of an opfibration to their existence inthe fibers, then brings pushouts into the picture by addressing when opfibrations preserve all finitecolimits. For more details, see the work of Hermida [29]. Lemma 4.3.11.
Let J be a small category and R : X → A an opfibration. If the base category A has J -colimits, then the following are equivalent:(1) All the fiber categories have J -colimits and all reindexing functors preserve them.(2) The total category X has J -colimits and R preserves them. F : A → Cat landing in the sub-2-category
Rex of finitely cocomplete categories and finite colimit preserving functors. The second part regardsthe existence of colimits globally in the total category R F . These two combine to result in: Corollary 4.3.12.
Let A be a category with finite colimts and F : ( A , + , → ( Cat , × , a laxmonoidal pseudofunctor. If the pseudofunctor A → MonCat via the correspondence in Equation4.1 factors through
Rex , meaning that each F ( a ) is finitely cocomplete and that the associatedreindexing functors are finitely cocontinuous, then the Grothendieck category R F has all finitecolimits and the corresponding opfibration R F : R F → A preserves them. For applications to structured cospans, we want a left adjoint L F to the induced monoidalopfibration R F : R F → A of the Grothendieck construction of F . Gray found sufficient conditionsfor the existence of such a left adjoint. Proposition 4.3.13 ([27, Prop. 4.4]) . Let R : X → A be an opfibration. Then R is a right adjoint left inverse , meaning that the unit η : 1 A → RL is an identity, if and only if its fibers have initialobjects which are preserved by the reindexing functors.Proof. The left adjoint L : A → X maps an object a to the initial object in its fiber which we denoteby ⊥ a or ! a in other sections of this thesis. By construction, we have that R ( L ( a )) = R ( ⊥ a ) = a .For a morphism f : a → a ′ , L ( f ) is given by: ⊥ a Cocart( f, ⊥ a ) −−−−−−−−→ f ! ( ⊥ a ) −→ ⊥ a ′ where the second arrow is the unique isomorphism between initial objects in the fiber above a ′ as f ! preserves them. For more details, see Gray [27, Proposition 4.4].Notice that under Lemma 4.3.11, if A has an initial object 0 A , then the above conditions areequivalent to X having an initial object 0 X above 0 A . Then ⊥ a is the cocartesian lifting of theunique map ! a : 0 A → a in the base category A :0 X (cid:15) (cid:15) Cocart(! a , X ) / / (! a ) ! (0 X ) =: ⊥ a (cid:15) (cid:15) in X A ! a / / a in A Furthermore, if R = R F for a pseudofunctor F : A → Cat as in Theorem 4.3.10, the reindexingfunctors (! a ) ! of the opfibration are given by F (! a ) and therefore ⊥ a = ( a, F (! a )(0 X )). Lastly, if thepseudofunctor ( F, φ, φ ) : ( A , + , → ( Cat , × ,
1) is lax monoidal to begin with, the Grothendieckconstruction in the cocartesian case expresses ⊥ a as the image of the composite1 φ −→ F (0 A ) F (! a ) −−−→ F ( a ) . Regarding the opposite direction, which is not needed in the proof of the main result of thischapter below, we have the following result. For a discussion on the ‘strict cocontinuity’ condition,we refer to the work of Cicala and Vasilakopoulou [16].73 roposition 4.3.14.
Suppose that R : X → A is a right adjoint and left inverse. If X and A bothhave chosen pushouts and initial objects and R strictly preserves them, then R is an opfibration. Before presenting the main proof, we outline a sketch. Given a lax monoidal pseudofunctor F : ( A , + , A ) → ( Cat , × , F C sp has A as its categoryof objects, horizontal 1-cells as F -decorated cospans given by pairs ( a → m ← b, x ∈ F ( m )) and2-morphisms as maps of cospans k : m → m ′ together with a morphism F ( k )( x ) → x ′ as in Theorem4.1.3.When the pseudofunctor F factors through Rex , by Corollary 4.3.12, the Grothendieck construc-tion yields a finitely cocomplete Grothendieck category R F such that the corresponding opfibration R F : ( R F, + , → ( A , + ,
0) preserves all finite colimits. In particular, the initial object is preservedand so Lemma 4.3.11 and Corollary 4.3.13 apply to obtain a left adjoint L F : A → R F which isright inverse to R F . This left adjoint is explicitly defined on objects by L ( a ) = ( a, ⊥ a ) where ⊥ a is initial in the finitely cocomplete category F ( a ). We can also express ⊥ a as ⊥ a = F (! a ) φ .Diagrammatically, this process can be expressed as: F : A → Cat ∫ F A R F A ∫ F L F ⊥ R F From this left adjoint L F : A → R F which goes between finitely cocomplete categories andpreserves finite colimits, we can obtain a double category of structured cospans L F C sp ( R F ). Thisdouble category will also have A as its category of objects, but now horizontal 1-cells are given bycospans of the form L F ( a ) → x ← L F ( b ) in the Grothendieck category R F . Explicitly, horizontal1-cells are given by:( a, ⊥ a ) ( i : a → m in A !: F ( i )( ⊥ a ) → x in F ( m ) −−−−−−−−−−−−−−−−→ ( m, x ) ( o : b → m in A !: F ( o )( ⊥ b ) → x in F ( m ) ←−−−−−−−−−−−−−−−− ( b, ⊥ b ) (4.3)where x ∈ F ( m ), as in Definition 4.3.9. A 2-morphism is given explicitly by: ( a, ⊥ a ) ( m, x ) ( b, ⊥ b )( a ′ , ⊥ a ′ ) ( m ′ , x ′ ) ( b ′ , ⊥ b ′ ) i : a → m in A !: F ( i )( ⊥ a ) → x in F ( m ) f : a → a ′ in A χ a : F ( f )( ⊥ a ) ∼ = ⊥ a ′ in F ( a ′ ) k : m → m ′ in A ι : F ( k )( x ) → x ′ in F ( m ′ ) o : b → m in A !: F ( o )( ⊥ b ) → x in F ( m ) g : b → b ′ in A χ b : F ( g )( ⊥ b ) ∼ = ⊥ b ′ in F ( b ′ ) i ′ : a ′ → m ′ in A !: F ( i ′ )( ⊥ a ′ ) → x ′ in F ( m ′ ) o ′ : b ′ → m ′ in A !: F ( o ′ )( ⊥ b ′ ) → x ′ in F ( m ′ ) L F applied to vertical 1-morphismsin F C sp , which are just morphisms of A . Each of the above squares commutes which says that ki = i ′ f and ko = o ′ g in A . Then in the Grothendieck category, we have: F ( k ◦ i )( ⊥ a ) ∼ = −→ F k ( F i ( ⊥ a )) F k (!) −−−→
F k ( x ) ι −→ x ′ = (4.4) F ( i ′ ◦ f )( ⊥ a ) ∼ = −→ F i ′ ( F f ( ⊥ a )) F i ′ ( χ a ) −−−−→ F i ′ ( ⊥ a ′ ) ! −→ x ′ in F ( m ′ ). Note that all the maps in the above equality are unique and originate from initialobjects, which are preserved by reindexing functors. Thus no extra conditions are imposed onthese morphisms, and likewise for the square involving o and o ′ .We define a double functor E : L F C sp ( R F ) → F C sp whose object component is the identity onthe category A . Given a horizontal 1-cell:( a, ⊥ a ) ( i : a → m in A !: F ( i )( ⊥ a ) → x in F ( m ) −−−−−−−−−−−−−−−−→ ( m, x ) ( o : b → m in A !: F ( o )( ⊥ b ) → x in F ( m ) ←−−−−−−−−−−−−−−−− ( b, ⊥ b ) (4.5)the image is given by a i −→ m o ←− b together with the decoration x ∈ F ( m ) . Note that this is actually a bijective correspondence as the unique maps from the initial objectsin the fibers provides no extra information. Given a 2-morphism of L F -structured cospans as inEquation (4.3), the image is given by the following map of cospans in A : a m ba ′ m ′ b ′ if k o gi ′ o ′ together with the morphism ι : F ( k )( x ) → x ′ as in Equation (4.3). This is again a bijectivecorrespondence and commutativity of Equation (4.4) holds by initiality of the domain.The double functor E = ( E , E ) is in fact strong. We have natural isomorphisms: E ( M ) ⊙ E ( N ) ∼ −→ E ( M ⊙ N )ˆ U E ( m ) ∼ −→ E ( U m )for any composable horizontal 1-cells: M = ( a, ⊥ a ) i −→ ( m, x ) o ←− ( b, ⊥ b )and N = ( b, ⊥ b ) i ′ −→ ( n, y ) o ′ ←− ( c, ⊥ c )and any object m ∈ L F C sp ( R F ). The horizontal composite E ( M ) ⊙ E ( N ) is given as in Theorem4.1.1 via a pushout and decoration: m + b na c, j m ◦ i j n ◦ o ′ F ( m ) × F ( n ) F ( m + n ) F ( m + b n ) x × y φ m,n F ( j ) j m : m → m + b n and j n : n → m + b n are the canonical maps into a pushout. If wefirst compose M and N in the structured cospan double category L F C sp ( R F ) by using fiberwisepushouts constructed using Lemma 4.3.11, we obtain: ( m + b n, F ( j m ) x + ⊥ m + bn F ( j n ) y )( m, x ) ( n, y )( a, ⊥ a ) ( b, ⊥ b ) ( c, ⊥ c ) and the image of this composite is given by the cospan a −→ m + b n ←− c together with the samedecoration as the following diagram commutes: F ( m ) × F ( n ) F ( m + n ) F ( m + b n ) × F ( m + b n ) F (( m + b n ) + ( m + b n )) F ( m + b n ) φF ( j m ) × F ( j n ) F ( j ) φ F ( ∇ ) as the pushout is over an initial object and hence really a coproduct. The fiberwise coproduct in F ( m + b n ) is given as in Equation (4.2).Lastly, for the identity morphisms, we have that U m is given by:( m, ⊥ m ) −→ ( m, ⊥ m ) ←− ( m, ⊥ m )with 1 m as the A -component of the cospan legs together with isomorphisms between initial objectsin the fibers. Hence E ( U m ) is the identity cospan on m in A together with the ‘initial decoration’or ‘trivial decoration’ ⊥ m ∈ F ( m ). On the other hand, U E ( m ) is the same cospan and decoration.This concludes the outline that E is a strong double functor.Finally, here is the main result relating structured and decorated cospans [4]. Theorem 4.3.15.
Let A be a category with finite colimits and F : A → Cat a symmetric laxmonoidal pseudofunctor such that F factors through Rex as above. Then the symmetric monoidaldouble category L C sp ( R F ) built using structured cospans and the symmetric monoidal double cat-egory F C sp built using decorated cospans are equivalent as symmetric monoidal double categories. We will sometimes denote a decoration x ∈ F ( m ) as d E ( M ) ∈ F ( R ( x )) where M is a horizontal1-cell of L C sp ( X ) = L F C sp ( Z F ) , and given an object a ∈ L C sp ( X ), the initial decoration or trivial decoration will be denoted as ⊥ a ∈ F ( a ) or ! a ∈ F ( a ). Note, that as mentioned above, ⊥ a is determined by the unique map! a : 0 A → a . The object d E ( M ) is not to be mistaken for an object of A which we will denote by a, b and c , or m and n with various primes and subscripts. Proof of Theorem 4.3.15.
As each F : A → Cat factors through
Rex , there exists a fully faithfulleft adjoint L : A → R F of the Grothendieck construction R : R F → A of F , R F is finitelycocomplete and R preserves finite colimits. 76ext we define a double functor E , prove it is a double equivalence, and show it is symmetricmonoidal. For notation, let R F = X . We define a double functor E : L C sp ( X ) → F C sp as follows:the object component of the double functor E is given by E = 1 A as both double categories L C sp (X)and F C sp have objects and morphisms of A as objects and vertical 1-morphisms, respectively. Thefunctor E is trivially an equivalence of categories.Given a horizontal 1-cell M of L C sp ( X ), which is a cospan in X of the form: L ( c ) x L ( c ′ ) i o the image E ( M ) is given by the pair: c R ( x ) c ′ x ∈ F ( R ( x )) R ( i ) η c R ( o ) η c ′ where R : X → A is the right adjoint to the functor L : A → X and η : 1 A → RL is the unit ofthe adjunction L ⊣ R which is an isomorphism since L is fully faithful. Similarly, the image of a2-morphism α : M → N in L C sp ( X ): L ( c ) x L ( c ) L ( c ′ ) x ′ L ( c ′ ) i oi ′ o ′ L ( f ) α L ( g ) is given by the 2-morphism E ( α ) : E ( M ) → E ( N ) in F C sp given by: c R ( x ) c c ′ R ( x ′ ) c ′ x ∈ F ( R ( x )) x ′ ∈ F ( R ( x ′ )) R ( i ) η c R ( o ) η c R ( i ′ ) η c ′ R ( o ′ ) η c ′ f R ( α ) g together with a morphism ι : F ( R ( α ))( x ) → x ′ in F ( R ( x ′ )) which comes from the Grothendieckconstruction of the pseudofunctor F : A → Cat . That E is a functor is clear. For E , given twovertically composable 2-morphisms M and M ′ in L C sp ( X ), L ( c ) x L ( c ) L ( c ′ ) x ′ L ( c ′ ) L ( c ′ ) x ′ L ( c ′ ) L ( c ′′ ) x ′′ L ( c ′′ ) i oi ′ o ′ L ( f ) α L ( g ) i ′ o ′ i ′′ o ′′ L ( f ′ ) α ′ L ( g ′ ) M ′ M is given by: L ( c ) x L ( c ) L ( c ′′ ) x ′′ L ( c ′′ ) i oi ′′ o ′′ L ( f ′ f ) α ′ α L ( g ′ g ) and the image of this 2-morphism E ( M ′ M ) is given by: c R ( x ) c c ′′ R ( x ′′ ) c ′′ x ∈ F ( R ( x )) x ′′ ∈ F ( R ( x ′′ )) R ( i ) η c R ( o ) η c R ( i ′′ ) η c ′′ R ( o ′′ ) η c ′′ f ′ f R ( α ′ α ) g ′ g together with a morphism ι M ′ M : F ( R ( α ′ α ))( x ) → x ′′ in F ( R ( x ′′ )). On the other hand, the indi-vidual images E ( M ) and E ( M ′ ) are given by: c R ( x ) c c ′ R ( x ′ ) c ′ x ∈ F ( R ( x )) x ′ ∈ F ( R ( x ′ )) c ′ R ( x ′ ) c ′ c ′′ R ( x ′′ ) c ′′ x ′ ∈ F ( R ( x ′ )) x ′′ ∈ F ( R ( x ′′ )) R ( i ) η c R ( o ) η c R ( i ′ ) η c ′ R ( o ′ ) η c ′ f R ( α ) gR ( i ′ ) η c ′ R ( o ′ ) η c ′ R ( i ′′ ) η c ′′ R ( o ′′ ) η c ′′ f ′ R ( α ′ ) g ′ together with morphisms ι M : F ( R ( α ))( x ) → x ′ in F ( R ( x ′ )) and ι M ′ : F ( R ( α ′ )( x ′ ) → x ′′ in F ( R ( x ′′ )), respectively. The vertical composite E ( M ′ ) E ( M ) of the above two 2-morphisms isgiven by E ( M ′ M ) as R is a functor and ι M ′ M = ι M ′ ι M . The functors E and E satisfy theequations E S = S E and E T = T E .To show that E is part of a double equivalence, we need to show it is essentially surjective, full,faithful and strong. To show it is essentially surjective, given a horizontal 1-cell in F C sp : c c c x ∈ F ( c ) i o we can find a 2-isomorphism in F C sp whose codomain is the above horizontal 1-cell and whosedomain is the image of the following horizontal 1-cell in L C sp ( X ): L ( c ) x L ( c ) i ′ o ′ F C sp given by: c R ( x ) c c c c x ∈ F ( R ( x )) x ∈ F ( c ) R ( i ′ ) η c R ( o ′ ) η c i o R ( e ) η c ) − ι : F (( R ( e ) η c ) − )( x ) → x where e : L ( c ) → x is given by the unique map from the trivial decoration on c to x ∈ F ( c ). Theobject and arrow components E and E satisfy the equations S E = E S and T E = E T .To show that the double functor E is full and faithful, we need to show that the map E : f L C sp ( X ) g ( M, N ) → E ( f ) F C sp E ( g ) ( E ( M ) , E ( N ))is bijective for arbitrary vertical 1-morphisms f and g and horizontal 1-cells M and N of L C sp ( X ).Consider a 2-morphism in L C sp ( X ) with horizontal source and target M and N , respectively andvertical source and target f and g , respectively: L ( c ) x L ( c ) L ( c ′ ) x ′ L ( c ′ ) Mf N g i oi ′ o ′ L ( f ) α L ( g ) Thus the set f L C sp( X ) g ( M, N )consists of triples ( f, α, g )rendering the above diagram commutative where f and g are morphisms of A and α is a morphismof X . The image of the above 2-morphism under the double functor E has horizontal source andtarget given by E ( M ) and E ( N ), respectively, and vertical source and target given by E ( f ) and E ( g ), respectively: c R ( x ) c c ′ R ( x ′ ) c ′ x ∈ F ( R ( x )) x ′ ∈ F ( R ( x ′ )) E ( M ) E ( f ) E ( N ) E ( g ) R ( i ) η c R ( o ) η c R ( i ′ ) η c ′ R ( o ′ ) η c ′ f R ( α ) g ι : F ( R ( α ))( x ) → x ′ of F ( R ( x ′ )). Thus the set E ( f ) F C sp E ( g ) ( E ( M ) , E ( N ))consists of 4-tuples ( f, R ( α ) , g, ι )rendering the above diagram commutative and where f, g and R ( α ) are morphisms of A and ι isa morphism in F ( R ( x ′ )). The morphisms R ( α ) : R ( x ) → R ( x ′ ) and ι : F ( R ( α ))( x ) → x ′ togetherdetermine the morphism α : x → x ′ in X and conversely: given two objects x = ( c, x ∈ F ( c )) and x ′ = ( c ′ , x ′ ∈ F ( c ′ )) of X = R F , a morphism from α : x → x ′ is a pair( h : c → c ′ , ι : F ( h )( x ) → x ′ )where h : c → c ′ is given by R ( α ) : R ( x ) → R ( x ′ ). This shows that E is fully faithful.Next we show that the double functor E is strong by exhibiting a natural isomorphism E M,N : E ( M ) ⊙ E ( N ) ∼ −→ E ( M ⊙ N )for every pair of composable horizontal 1-cells M and N of L C sp ( X ) and for every object c ∈ L C sp ( X ) a natural isomorphism E c : ˆ U E ( c ) ∼ −→ E ( U c )where U and ˆ U are the unit functors of L C sp ( X ) and F C sp , respectively. For any object c , thehorizontal 1-cell ˆ U E ( c ) is given by ˆ U c which is given by the pair: c c c ! c ∈ F ( c ) The horizontal 1-cell U c is given by L ( c ) L ( c ) L ( c ) and so E ( U c ) is given by the pair: c R ( L ( c )) c ! c ∈ F ( R ( L ( c ))) η c η c We can then obtain the natural isomorphism E c : ˆ U E ( c ) ∼ −→ E ( U c ) as the 2-morphism c c cc R ( L ( c )) c ! c ∈ F ( c )! R ( L ( c )) ∈ F ( R ( L ( c ))) η c η c η c ι : F ( η c )(! c ) ! −→ ! R ( L ( c )) in F C sp . 80ext, given composable horizontal 1-cells M and N in L C sp ( X ): L ( c ) x L ( c ) L ( c ) x ′ L ( c ) i o i ′ o ′ their images E ( M ) and E ( N ) are given by: c R ( x ) c c R ( x ′ ) c x ∈ F ( R ( x )) x ′ ∈ F ( R ( x ′ )) R ( i ) η c R ( o ) η c R ( i ′ ) η c R ( o ′ ) η c and so E ( M ) ⊙ E ( N ) is given by: c R ( x ) + c R ( x ′ ) c d E ( M ) ⊙ E ( N ) ∈ F ( R ( x ) + c R ( x ′ )) jψR ( i ) η c jψR ( o ′ ) η c d E ( M ) ⊙ E ( N ) : 1 λ − −−→ × x × x ′ −−−→ F ( R ( x )) × F ( R ( x ′ )) φR ( x ) ,R ( x ′ ) −−−−−−−→ F ( R ( x )+ R ( x ′ )) F ( jR ( x ) ,R ( x ′ )) −−−−−−−−−→ F ( R ( x )+ c R ( x ′ )) where ψ denotes each natural map into the coproduct and j denotes the natural map from thecoproduct to the pushout. On the other hand, M ⊙ N is given by L ( c ) x + L ( c ) x ′ L ( c ) Jζi Jζo ′ where ζ is a natural map into a coproduct and J is the natural map from the coproduct to thepushout. Then E ( M ⊙ N ) is given by c R ( x + L ( c ) x ′ ) c d E ( M ⊙ N ) = x + L ( c ) x ′ ∈ F ( R ( x + L ( c ) x ′ )) R ( Jζi ) η c R ( Jζo ′ ) η c and so E M,N : E ( M ) ⊙ E ( N ) ∼ −→ E ( M ⊙ N ) is given by the 2-morphism: c R ( x ) + c R ( x ′ ) c c R ( x + L ( c ) x ′ ) c d E ( M ) ⊙ E ( N ) ∈ F ( R ( x ) + c R ( x ′ )) d E ( M ⊙ N ) ∈ F ( R ( x + L ( c ) x ′ )) jψR ( i ) η c jψR ( o ′ ) η c R ( Jζi ) η c R ( Jζo ′ ) η c σ First, the right adjoint R also preserves finite colimits and so we have a natural isomorphism κ : R ( x ) + R ( L ( c )) R ( x ′ ) → R ( x + L ( c ) x ′ ) . Also, since the left adjoint L : A → X is fully faithful, the unit of the adjunction L ⊣ R at the object c gives a natural isomorphism η c : c → R ( L ( c )) which results in a natural isomorphism j η c : R ( x ) + c R ( x ′ ) → R ( x ) + R ( L ( c )) R ( x ′ ) . σ : = κj η c : R ( x ) + c R ( x ′ ) → R ( x + L ( c ) x ′ ) . Next, to see that the above diagram commutes, it suffices to show that for the object c ∈ A , R ( J ) R ( ζ ) R ( i ) η c ( c ) = R ( J ζi ) η c ( c ) ! = σjψR ( i ) η c ( c ) = κj η c ψR ( i ) η c ( c ) . This follows as R ( i ) η c : c → R ( x ) and the following diagram commutes: R ( x ) R ( x ) + R ( x ′ ) R ( x ) + c R ( x ′ ) R ( x + L ( c ) x ′ ) R ( x + x ′ ) R ( x ) + R ( L ( c )) R ( x ′ ) jψ R ( J ) R ( ζ ) j η c σκ Lastly, this map of cospans comes with an isomorphism ι : F ( σ )( d E ( M ) ⊙ E ( N ) ) → d E ( M ⊙ N ) in F ( R ( x + L ( c ) x ′ )). This shows that E is strong, and so E : L C sp ( X ) ∼ −→ F C sp is part of a doubleequivalence by a Theorem of Shulman [38, Theorem 7.8].Next we will show that this equivalence of double categories E : L C sp ( X ) → F C sp is symmetricmonoidal. First, note that we have a natural isomorphism ǫ : 1 F C sp → E (1 L C sp ( X ) ) and naturalisomorphisms µ c ,c : E ( c ) ⊗ E ( c ) → E ( c ⊗ c ) for every pair of objects c , c ∈ L C sp ( X ) bothof which are given by identities since both double categories L C sp ( X ) and F C sp have A as theircategory of objects and E = 1 A . The diagrams utilizing these maps that are required to commutedo so trivially.For the arrow component E , we have a natural isomorphism δ : U F C sp → E ( U L C sp ( X ) ) wherethe horizontal 1-cell U F C sp is given by: A A A ! A ∈ F (1 A ) where ! A = φ : 1 → F (1 A ) is the trivial decoration which comes from the structure of the symmetriclax monoidal pseudofunctor F : A → Cat . The horizontal 1-cell U L C sp ( X ) is given by: L (1 A ) L (1 A ) L (1 A ) where here we make use of the fact that the left adjoint L : ( A , + , A ) → ( X , + , X ) preserves allcolimits and thus L (1 A ) ∼ = 1 X . The horizontal 1-cell E ( U L C sp ( X ) ) is then given by the pair: A R ( L (1 A )) 1 A ! R ( L (1 A )) ∈ F ( R ( L (1 A ))) η A η A δ is then given by the 2-morphism: A A A A R ( L (1 A )) 1 A ! A ∈ F (1 A )! R ( L (1 A )) ∈ F ( R ( L (1 A ))) η A η A η A ι η A : F ( η A )(! A ) → ! R ( L (1 A )) of F C sp . This is just the natural isomorphism E A from earlier.Given two horizontal 1-cells M and N of L C sp ( X ): L ( c ) x L ( c ) L ( c ′ ) x ′ L ( c ′ ) i o i ′ o ′ their images E ( M ) and E ( N ) are given by: c R ( x ) c c ′ R ( x ′ ) c ′ x ∈ F ( R ( x )) x ′ ∈ F ( R ( x ′ )) R ( i ) η c R ( o ) η c R ( i ′ ) η c ′ R ( o ′ ) η c ′ and so E ( M ) ⊗ E ( N ) is given by: c + c ′ R ( x ) + R ( x ′ ) c + c ′ d E ( M ) ⊗ E ( N ) ∈ F ( R ( x ) + R ( x ′ )) R ( i ) η c + R ( i ′ ) η c ′ R ( o ) η c + R ( o ) η c ′ where d E ( M ) ⊗ E ( N ) : 1 λ − −−→ × x × x ′ −−−→ F ( R ( x )) × F ( R ( x ′ )) φ R ( x ) ,R ( x ′ ) −−−−−−−→ F ( R ( x ) + R ( x ′ )) . On the other hand, M ⊗ N is given by L ( c + c ′ ) x + x ′ L ( c + c ′ ) ( i + i ′ ) φ − c ,c ′ ( o + o ′ ) φ − c ,c ′ and E ( M ⊗ N ) is given by: c + c ′ R ( x + x ′ ) c + c ′ d E ( M ⊗ N ) = x + x ′ ∈ F ( R ( x + x ′ )) . R (( i + i ′ ) φ − c ,c ′ ) η c + c ′ R (( o + o ′ ) φ − c ,c ′ ) η c + c ′ We then have a natural 2-isomorphism µ M,N : E ( M ) ⊗ E ( N ) ∼ −→ E ( M ⊗ N ) in F C sp given by: c + c ′ R ( x ) + R ( x ′ ) c + c ′ c + c ′ R ( x + x ′ ) c + c ′ d E ( M ) ⊗ E ( N ) ∈ F ( R ( x ) + R ( x ′ )) d E ( M ⊗ N ) ∈ F ( R ( x + x ′ )) ι µ : F ( κ )( d E ( M ) ⊗ E ( N ) ) → d E ( M ⊗ N ) R ( i ) η c + R ( i ′ ) η c ′ R ( o ) η c + R ( o ′ ) η c ′ R (( i + i ′ ) φ − c ,c ′ ) η c + c ′ R (( o + o ′ ) φ − c ,c ′ ) η c + c ′ κ κ is the isomorphism which comes from R : X → A preserving finite colimits.The natural isomorphisms δ and µ satisfy the left and right unitality squares, associativityhexagon and braiding square. To see this, let M , M and M be horizontal 1-cells in L C sp ( X )given by: L ( c ) x L ( c ′ ) L ( c ) x L ( c ′ ) L ( c ) x L ( c ′ ) i o i o i o The left unitality square: F C sp ⊗ E ( M ) E (1 L C sp ( X ) ) ⊗ E ( M ) E ( M ) E (1 L C sp ( X ) ⊗ M ) δ ⊗ µ ,M λ ′ E ( λ ) has an underlying diagram of maps of cospans given by: E (1 L C sp ( X ) ) ⊗ E ( M )1 A + c R ( L (1 A )) + R ( x ) 1 A + c ′ F C sp ⊗ E ( M )1 A + c A + R ( x ) 1 A + c ′ E ( M ) c R ( x ) c ′ E (1 L C sp ( X ) ⊗ M )1 A + c ′ R ( L (1 A ) + x ) 1 A + c ′ E ( M ) c R ( x ) c ′ λδ ⊗ µ ,M E ( λ ) η A + R ( i ) η c η A + R ( o ) η c ′ R ( i ) η c R ( o ) η c ′ η A + 1 ι R ( i ) η c R ( o ) η c ′ λ A λ A ι λ A ( µ L (1 A ) ,d )( η A + R ( i ) η c ) ( µ L (1 A ) ,d )( η A + R ( o ) η c ′ )1 µ L (1 A ) ,x ι R ( i ) η c R ( o ) η c ′ λ A R ( λ X ) ι λ A with the corresponding maps of decorations amounting to the following commutative diagram in F ( R ( x )): F ( λ A )(! A + x ) F ( R ( λ X )( µ L (1 A ) ,x ))(! R ( L (1 A )) + x ) x F ( R ( λ X ))( x !+1 ) F ( R ( λ X )( µ L (1 A ) ,x ))( ι ) ι ι F ( R ( λ X ))( ι ) where x !+1 is the decoration x on the object R ( L (1 A ) + x ) ∈ A . The above square commutesbecause F ( λ A )(! A + x ) = F ( R ( λ X )( µ L (1 A ) ,x )( η A + 1))(! A + x )as the corresponding left unitality square for the finite colimit preserving functor R : ( X , X , +) → ( A , A , +) commutes. The right unitality square is similar. The associator hexagon: ( E ( M ) ⊗ E ( M )) ⊗ E ( M ) E ( M ⊗ M ) ⊗ E ( M ) E (( M ⊗ M ) ⊗ M ) E ( M ) ⊗ ( E ( M ) ⊗ E ( M )) E ( M ) ⊗ E ( M ⊗ M ) E ( M ⊗ ( M ⊗ M )) µ M ,M ⊗ µ M ⊗ M ,M ⊗ µ M ,M µ M ,M ⊗ M a ′ E ( a ) ( E ( M ) ⊗ E ( M )) ⊗ E ( M ) E ( M ⊗ M ) ⊗ E ( M ) E (( M ⊗ M ) ⊗ M ) E ( M ⊗ ( M ⊗ M )) E ( M ) ⊗ ( E ( M ) ⊗ E ( M )) E ( M ) ⊗ E ( M ⊗ M ) E ( M ⊗ ( M ⊗ M ))( c + c ) + c ( R ( x ) + R ( x )) + R ( x ) ( c ′ + c ′ ) + c ′ ( c + c ) + c R ( x + x ) + R ( x ) ( c ′ + c ′ ) + c ′ ( c + c ) + c R (( x + x ) + x ) ( c ′ + c ′ ) + c ′ c + ( c + c ) R ( x + ( x + x )) c ′ + ( c ′ + c ′ ) c + ( c + c ) R ( x ) + ( R ( x ) + R ( x )) c ′ + ( c ′ + c ′ ) c + ( c + c ) R ( x ) + R ( x + x ) c ′ + ( c ′ + c ′ ) c + ( c + c ) R ( x + ( x + x )) c ′ + ( c ′ + c ′ ) µ M ,M ⊗ µ M ⊗ M ,M E ( a ) a ′ ⊗ µ M ,M µ M ,M ⊗ M ( R ( i ) η c + R ( i ) η c ) + R ( i ) η c ( R ( o ) η c ′ + R ( o ) η c ′ ) + R ( o ) η c ′ R ( i + i ) η c + c + R ( i ) η c R ( o + o ) η c ′ + c ′ + R ( o ) η c ′ κ + 1 ι R (( i + i ) + i ) η ( c + c )+ c R (( o + o ) + o ) η ( c ′ + c ′ )+ c ′ κ ι R ( i + ( i + i )) η c +( c + c ) R ( o + ( o + o )) η c ′ +( c ′ + c ′ ) a A R ( a X ) ι a A R ( i ) η c + ( R ( i ) η c + R ( i ) η c ) R ( o ) η c ′ + ( R ( o ) η c ′ + R ( o ) η c ′ ) a A a A ι a A R ( i ) η c + R ( i + i ) η c + c R ( o ) η c ′ + R ( o + o ) η c ′ + c ′ κ ι R ( i + ( i + i )) η c +( c + c ) R ( o + ( o + o )) η c ′ +( c ′ + c ′ ) κ ι Here, due to limited space, we have omitted the natural isomorphisms φ c i ,c j : L ( c i )+ L ( c j ) → L ( c i + c j ) on the inward pointing morphisms which make up the legs of each cospan. The correspondingmaps of decorations amount to the following commutative diagram in F ( R ( x + ( x + x ))): F (( κ )(1 + κ )( a A ))(( x + x ) + x ) F (( R ( a X ))( κ ))(( x + x ) + x ) F ( R ( a X ))(( x + x ) + x ) x + ( x + x ) F (( κ )(1 + κ ))(( x + x ) + x ) F ( κ )( x + ( x + x )) F (( R ( a X ))( κ ))( ι ) F ( R ( a X ))( ι ) ι F (( κ )(1 + κ ))( ι ) F ( κ )( ι ) ι The above hexagon commutes because F (( κ )(1 + κ )( a A ))(( x + x ) + x ) = F (( R ( a X ))( κ )( κ + 1))(( x + x ) + x )85s the corresponding associator hexagon for the finite colimit preserving functor R : ( X , X , +) → ( A , A , +) commutes. Lastly, the braiding square: E ( M ) ⊗ E ( M ) E ( M ) ⊗ E ( M ) E ( M ⊗ M ) E ( M ⊗ M ) β ′ µ M ,M µ M ,M E ( β ) has underlying map of cospans given by: E ( M ) ⊗ E ( M ) E ( M ) ⊗ E ( M ) E ( M ⊗ M ) E ( M ⊗ M ) E ( M ⊗ M ) c + c R ( x ) + R ( x ) c ′ + c ′ c + c R ( x ) + R ( x ) c ′ + c ′ c + c R ( x + x ) c ′ + c ′ c + c R ( x + x ) c ′ + c ′ c + c R ( x + x ) c ′ + c ′ β ′ µ M ,M µ M ,M E ( β ) R ( i ) η c + R ( i ) η c R ( o ) η c ′ + R ( o ) η c ′ R ( i ) η c + R ( i ) η c R ( o ) η c ′ + R ( o ) η c ′ β A β A ι β A R ( i + i ) η c + c R ( o + o ) η c ′ + c ′ κ ι R ( i + i ) η c + c R ( o + o ) η c ′ + c ′ κ ι R ( i + i ) η c + c R ( o + o ) η c ′ + c ′ β A R ( β X ) ι β A Again, we have omitted the natural isomorphisms φ c i ,c j on the inward pointing morphisms oneach cospan leg due to space restrictions. The corresponding maps of decorations amount to thefollowing commutative diagram in F ( R ( x + x )): F (( κ )( β A ))( x + x ) F ( κ )( x + x ) F ( R ( β X ))( x + x ) x + x F ( κ )( ι ) F ( R ( β X ))( ι ) ι ι The above square commutes because F (( κ )( β A ))( x + x ) = F (( R ( β X ))( κ ))( x + x )as the corresponding braiding square for the finite colimit preserving functor R : ( X , X , +) → ( A , A , +) commutes. The comparison and unit constraints E M,N and E c are monoidal naturaltransformations, and as both L C sp ( X ) and F C sp are isofibrant by Lemmas 5.1.2 and 5.2.1, respec-tively, the double functor E : L C sp ( X ) → F C sp is symmetric monoidal. In this section we present the three examples that were illustrated with the original decoratedcospans as well as structured cospans. The first example regarding graphs was mentioned in theintroduction and is the easiest example to keep in mind. The next two examples, taking on more86f an applied flavor, consist of electrical circuits and Petri nets. Each of these has been studiedextensively in work on ‘black-boxing’ [5, 7, 8, 9, 10]. Black-boxing is a way of describing the behaviorof an open system, that is, a system with prescribed inputs and outputs such as the terminals of anelectrical circuit, by observing the activity at the inputs and the outputs, typically while the systemis in a ‘steady state’. The relation between the activity at inputs and outputs can be seen as amorphism in some category of relations. A black-boxing functor sending open electrical circuits toLagrangian linear relations was first constructed using Fong’s theory of decorated cospans [7], andlater via the theory of props [5]. A black-boxing double functor sending open Petri nets to relationswas constructed using structured cospans [9]. A black-boxing functor for a special class of Markovprocesses was constructed using decorated cospans [8]; later it was generalized and enhanced to adouble functor [2], as explained in Chapter 6.
As a first example, let L : FinSet → FinGraph be the functor that assigns to a set N the discretegraph L ( N ) which is the edgeless graph with N as its set of vertices. Both FinSet and
FinGraph have finite colimits and the functor L : FinSet → FinGraph is left adjoint to the forgetful functor R : FinGraph → FinSet which assigns to a finite graph G its underlying finite set of vertices, R ( G ).Using structured cospans and appealing to Theorem 3.2.3, we get a symmetric monoidal doublecategory L C sp ( FinGraph ) which has:(1) finite sets as objects,(2) functions as vertical 1-morphisms,(3) open graphs , or cospans of graphs of the form L ( N ) G L ( M ) I O as horizontal 1-cells, where L ( N ) and L ( M ) are discrete graphs on the sets N and M , re-spectively, G is a graph and I and O are graph morphisms, and(4) maps of cospans of graphs of the form L ( N ) G L ( M ) L ( N ) G L ( M ) I O I O L ( f ) α L ( g ) as 2-morphisms, where L ( f ) and L ( g ) are maps of discrete graphs induced by the underlyingfunctions f and g , respectively, and α : G → G is a graph morphism.This is precisely Theorem 3.3.2. We can obtain a similar symmetric monoidal double categoryusing decorated cospans. Let F : FinSet → Cat be the symmetric lax monoidal pseudofunctor thatassigns to a finite set N the category of all graph structures whose underlying set of vertices is N .Thus, F ( N ) is the category where: 871) objects are given by graphs each having N as their set of vertices E N st and(2) morphisms are given by maps of edges f : E → E ′ making the following two triangles com-mute: EE ′ N EE ′ N ss ′ f f tt ′ The laxator µ N ,N : F ( N ) × F ( N ) → F ( N + N )for this symmetric lax monoidal pseudofunctor F is analogous to the laxator for the monoidalfunctor F of Section 2.2.1. By Theorem 4.1.3, we have the following: Theorem 4.4.1.
Let F : FinSet → Cat be the symmetric lax monoidal pseudofunctor which assignsto a finite set N the category of all graph structures whose underlying set of vertices is N . Thenthere exists a symmetric monoidal double category F C sp which has:(1) finite sets as objects,(2) functions as vertical 1-morphisms,(3) horizontal 1-cells as pairs: N P M G ∈ F ( P ) i o which can also be thought of as open graphs, and(4) 2-morphisms as maps of cospans of finite sets N N M P M P G ∈ F ( P ) G ∈ F ( P ) i o f go i h together with a graph morphism ι : F ( h )( G ) → G in F ( P ) .Proof. This follows immediately from Theorem 4.1.3.88e thus have two symmetric monoidal double categories: L C sp ( FinGraph ) obtained from struc-tured cospans and F C sp obtained from decorated cospans. Both of these double categories have FinSet as their categories of objects, open graphs as horizontal 1-cells and maps of open graphsas 2-morphisms, and by Theorem 4.3.15, we have an equivalence of symmetric monoidal doublecategories L C sp ( FinGraph ) ≃ F C sp . Corollary 4.4.2.
The symmetric monoidal double category L C sp ( FinGraph ) of Theorem 3.3.2 andthe symmetric monoidal double category F C sp of Theorem 4.4.1 are equivalent.Proof. This follows immediately from Theorem 4.3.15.
In a previous work [10], Baez and Fong attempted to use decorated cospans to construct asymmetric monoidal category of open k -graphs. Now we can fix the problems in this construction.Recall from Definition 2.2.3 that given a field k with positive elements, a k -graph is given by adiagram in Set of the form:
E Vk + r st Here the finite sets E and V are the sets of edges and vertices, respectively, and if we take the field k = R , the function r : E → R + assigns to each edge e ∈ E a positive real number r ( e ) ∈ R + whichcan be interpreted as the resistance at the edge e . We restrict to finite sets to avoid convergenceissues with certain summations. An open k -graph is then given by a cospan of finite sets V YX i o where the apex V is equipped with the structure of a k -graph. See the original paper for moredetails [10].Let FinGraph k be the category whose objects are given by k -graphs and morphisms by morphismsof k -graphs, where a morphism of k -graphs is given by a pair of functions f : E → E ′ and g : V → V ′ between the edge sets and vertex sets, respectively, of two k -graphs that respect the source andtarget functions of each, and such that the resistances of each edge are preserved. In the originalwork introducing structured cospans, it is shown that the category FinGraph k has finite colimits[3]. We can then obtain a double category of open k -graphs by defining a left adjoint L : FinSet → FinGraph k that assigns to a finite set V the discrete k -graph L ( V ) given by the k -graph with V as itsset of vertices and no edges. The resulting symmetric monoidal double category L C sp ( FinGraph k )has:(1) finite sets as objects,(2) functions as vertical 1-morphisms,(3) open k -graphs as horizontal 1-cells V YX k + E V r sti o k -graphs between the apices. X X Y V Y V i o h h ′ o i g k + E E E E V V E E V V g gr s s r f f f t t We can also obtain a similar double category using decorated cospans: define a pseudofunctor F : FinSet → Cat that assigns to a finite set V the category of all k -graph structures on the set V and to a function f : V → V ′ the corresponding functor F ( f ) : F ( V ) → F ( V ′ ) between decora-tion categories. Both categories FinSet and
Cat are symmetric monoidal and the pseudofunctor F : FinSet → Cat is symmetric lax monoidal, as given a k -graph structure on a finite set V de-noted by an element K ∈ F ( V ) and a k -graph structure on a finite set V denoted by an element K ∈ F ( V ), there is a natural k -graph structure φ V ,V ( K , K ) on V + V . Thus we get a naturaltransformation φ V ,V : F ( V ) × F ( V ) → F ( V + V )as well as a morphism φ : 1 → F ( ∅ ) which together satisfy the coherence conditions of a monoidalfunctor. The braiding is also clear as the following diagram commutes: F ( V ) × F ( V ) F ( V ) × F ( V ) F ( V + V ) F ( V + V ) φ V ,V φ V ,V β ′ V ,V F ( β V ,V ) Thus the pseudofunctor F is symmetric lax monoidal and so by Theorem 4.1.3 we have the following: Theorem 4.4.3.
Let F : FinSet → Cat be the symmetric lax monoidal pseudofunctor which assignsto a finite set N the category of all k -graph structures whose underlying set of vertices is N . Thenthere exists a symmetric monoidal double category F C sp which has:(1) objects as finite sets,(2) vertical 1-morphisms as functions,(3) horizontal 1-cells as cospans of sets together with the structure of a k -graph given by anelement of the image of the apex under the pseudofunctor F : U V W K ∈ F ( V ) i o which are open k -graphs, and
4) 2-morphisms as maps of cospans of finite sets U W V U W V K ∈ F ( V ) K ∈ F ( V ) o f ghi i o together with a morphism of k -graphs ι : F ( h )( K ) → K in F ( V ) . Corollary 4.4.4.
The symmetric monoidal double category L C sp ( FinGraph k ) of Theorem 3.3.5 andthe symmetric monoidal double category F C sp of Theorem 4.4.3 are equivalent.Proof. This follows immediately from Theorem 4.3.15.
In a previous work, Baez and Master used the framework of structured cospans to obtain asymmetric monoidal double category of ‘open Petri nets’ [9]. Recall from Definition 2.2.5 that aPetri net is given by a diagram in
Set of the form: T N [ S ] . st Here, T is the finite set of transitions and S is the finite set of species , and N [ S ] is the freecommutative monoid on the set S . Each transition then has a formal linear combination of speciesgiven by an element of N [ S ] as its source and target as prescribed by the functions s and t ,respectively. An example of a Petri net is given by: HO α H O This Petri net has a single transition α with 2H + O as its source and H O as its target. See theoriginal paper for more details on Petri nets [9].Each set of species S gives rise to a discrete Petri net L ( S ) with S as its set of species and notransitions. Baez and Master note the existence of a left adjoint L : Set → Petri where
Petri is thecategory whose objects are Petri nets and whose morphisms are ‘morphisms of Petri nets’. Theyalso show that
Petri has finite colimits and thus using Theorem 3.2.3 obtain a symmetric monoidaldouble category O pen ( Petri ) of open Petri nets which has:(1) objects given by sets,(2) vertical 1-morphisms given by functions, 913) horizontal 1-cells as open Petri nets which are given by cospans in
Petri of the form: L ( X ) P L ( Y ) I O and(4) 2-morphisms as maps of cospans in
Petri of the form: L ( X ) L ( Y ) P L ( X ) L ( Y ) P O L ( f ) L ( g ) αI I O We can also obtain a similar double category using decorated cospans: define a pseudofunctor F : Set → Cat where given a set s , F ( s ) is the category of all Petri net structures with s as its setof species. This pseudofunctor F is symmetric lax monoidal as both ( Set , + , ∅ ) and ( Cat , × ,
1) aresymmetric monoidal and given Petri nets P ∈ F ( s ) and P ′ ∈ F ( s ′ ), we can place them side by sideand consider them together as a single Petri net P + P ′ ∈ F ( s + s ′ ) with set of species s + s ′ , andthus we have natural transformations φ s,s ′ : F ( s ) × F ( s ′ ) → F ( s + s ′ ) for any two sets s and s ′ . Theother structure morphism between monoidal units φ : 1 → F ( ∅ ) is defined by the unique morphismfrom the terminal category to the empty Petri net with the empty set for its set of species, whichis the only possible Petri net on the empty set. All of the diagrams that are required to commuteare straightforward. Appealing to Theorem 4.1.3, we have the following: Theorem 4.4.5.
Let F : Set → Cat be the symmetric lax monoidal pseudofunctor which assignsto a set S the category of all Petri nets whose set of species is S . Then there exists a symmetricmonoidal double category F C sp which has:(1) objects given by sets,(2) vertical 1-morphisms given by functions,(3) horizontal 1-cells given by open Petri nets presented as pairs: X Z Y P ∈ F ( Z ) i o and(4) 2-morphisms as maps of cospans in Set : X Y Z X Y Z P ∈ F ( Z ) P ∈ F ( Z ) o f ghi i o together with a morphism of Petri nets ι : F ( h )( P ) → P in F ( Z ) . O pen ( Petri ) of open Petri nets obtainedfrom structured cospans and a symmetric monoidal double category F C sp of open Petri nets obtainfrom decorated cospans, and these two symmetric monoidal double categories are equivalent. Corollary 4.4.6.
The symmetric monoidal double category O pen ( Petri ) constructed by Baez andMaster [9] utilizing structures cospans and the symmetric monoidal double category F C sp of The-orem 4.4.5 are equivalent.Proof. This follows immediately from Theorem 4.3.15.We may also construct a symmetric monoidal double category of open Petri nets with ratesusing decorated cospans, and this is equivalent to the symmetric monoidal double category L C sp ( Petri rates ) of Theorem 3.3.9. 93 hapter 5
A brief digression to bicategories
If one prefers bicategories to double categories, one will be happy to learn that all of the mainresults in this thesis on double categories have bicategorical analogues thanks to a result of MikeShulman [37]. Bicategories are defined in Section A.3 of the Appendix. First we discuss the re-lationship between 2-categories and double categories. As we are mainly interested in symmetricmonoidal double categories, we are similarly primarily interested in ‘symmetric monoidal bicate-gories’. We will not define monoidal, braided monoidal, ‘sylleptic’ monoidal or symmetric monoidalbicategories here. These definitions can be found in a work of Mike Stay [39].The first thing we point out is that 2-categories are just a special case of strict double categoriesand that every strict double category has at least two canonical underlying 2-categories. Given astrict double category C , there exists:(1) a 2-category H ( C ) called the horizontal 2-category of C which has:(a) objects as objects of C ,(b) morphisms as horizontal 1-cells of C , and(c) 2-morphisms as 2-morphisms of C with identity vertical 1-morphisms, also known as globular 2-morphisms of C .(2) a 2-category V ( C ) called the vertical 2-category of C which has:(a) objects as objects of C ,(b) morphisms as vertical 1-morphisms of C , and(c) 2-morphisms as 2-morphisms of C with identity horizontal 1-cells, where now composi-tion of 2-morphisms is given by horizontal composition of 2-morphisms in C .Every pseudo double category C has an underlying bicategory H ( C ) given by as above. Using ourconventions, there is no underlying vertical bicategory V ( C ) as restricting the horizontal source andtarget of 2-morphisms, namely the horizontal 1-cells, to be identities does not force the horizontalsource and target of the composite 2-morphisms in C to also be identities, due to the compositionof horizontal 1-cells in a pseudo double category being neither strictly unital nor associative.Sometimes when the pseudo double category C is symmetric monoidal, the symmetric monoidalstructure can be lifted to the horizontal bicategory H ( C ). This is due to the following result ofShulman [37]. The definitions of ‘isofibrant’ and ‘symmetric monoidal double category’ are givenin Definitions A.2.7 and A.2.12, respectively. 94 heorem 5.0.1 ([37, Thm. 1.2]) . Let X be an isofibrant symmetric monoidal pseudo double cate-gory. Then the horizontal bicategory H ( X ) of X is a symmetric monoidal bicategory which has:(1) objects as those of X ,(2) morphisms as horizontal 1-cells of X , and(3) 2-morphisms as globular 2-morphisms of X . The property of being isofibrant, meaning fibrant on vertical 1-isomorphisms, is precisely whatallows the horizontal bicategory H ( X ) to inherit the portion of the symmetric monoidal structurethat resides in the category of objects of X , namely, the associators, left and right unitors andbraidings.In the previous chapters we constructed various symmetric monoidal double categories whichare in fact isofibrant, and thus have underlying symmetric monoidal bicategories. Every foot-replaced double category L X has an underlying foot-replaced bicategory H ( L X ) givenby taking the 2-morphisms of H ( L X ) to be globular 2-morphisms of L X . Lemma 5.1.1.
Given a double category X , a category A and a functor L : A → X , there is abicategory H ( L X ) for which: • objects are objects of A , • morphisms from a ∈ A to a ′ ∈ A are horizontal 1-cells M : L ( a ) → L ( a ′ ) of L X , • L X , • composition of morphisms is horizontal composition of horizontal 1-cells in L X , • vertical and horizontal composition of 2-morphisms is vertical and horizontal composition of2-cells in L X . If the double category X is isofibrant symmetric monoidal and we have a strong symmetricmonoidal functor L : A → X , then Shulman’s Theorem 5.0.1 allows us to lift the monoidal structureof the foot-replaced double category L X to obtain a symmetric monoidal foot-replaced bicategory H ( L X ). Lemma 5.1.2. If X is an isofibrant symmetric monoidal double category, A is a symmetricmonoidal category and L : A → X is a (strong) symmetric monoidal functor, then the bicategory H ( L X ) becomes symmetric monoidal in a canonical way. Lemma 5.1.3. If X is a category with finite colimits, then the symmetric monoidal double category C sp ( X ) is isofibrant. roof. A vertical 1-isomorphism in C sp ( X ) is a isomorphism f : x → y in X . We take its companionˆ f to be the cospan x y y. f The unit horizontal 1-cells U x and U y are given respectively by x x x and y y y and the accompanying 2-morphisms are given by x yyy yy and x xxx yy f f ff f respectively. An easy calculation verifies Equation (A.1). Theorem 5.1.4.
Let L : A → X be a functor where X is a category with pushouts. Then there is abicategory H ( L C sp ( X )) for which:(1) an object is an object of A ,(2) a morphism from a to b is given by a cospan in X of the form: L ( a ) x L ( b ) with composition the same as composition of horizontal 1-cells in Theorem 3.2.1 and(3) 2-morphisms are given by maps of cospans which are commutative diagrams of the form: L ( a ) L ( b ) xx ′ α with horizontal and vertical composition of 2-morphisms given by horizontal and vertical com-position of globular 2-morphisms in Theorem 3.2.1. Theorem 5.1.5.
Let L : A → X be a functor preserving finite coproducts, where A has finitecoproducts and X has finite colimits. Then the bicategory of Theorem 5.1.4 is symmetric monoidalwith the monoidal structure given by:(1) the tensor product of two objects a and a is a + a ,(2) the tensor product of two morphisms is given by the tensor product of two horizontal 1-cellsin Theorem 3.2.3 and
3) the tensor product of two 2-morphisms is given by: L ( a ) x x ′ x ′ x ′ + x ′ L ( a ′ ) ⊗ L ( a ) x L ( a ′ ) = L ( a + a ) x + x L ( a ′ + a ′ ) i o i ′ o ′ i o i ′ o ′ ( i + i ) φ − ( o + o ) φ − ( i ′ + i ′ ) φ − ( o ′ + o ′ ) φ − α α α + α where φ is the natural isomorphism φ a ,a : L ( a ) ⊗ L ( a ) → L ( a + a ) of the strong symmetricmonoidal functor L . The unit for the tensor product is the initial object of A , and the symmetryfor any two objects a and b is defined using the canonical isomorphism a + b ∼ = b + a . In Section 3.3.2, we constructed a symmetric monoidal double category L C sp ( FinGraph ) of opengraphs. This double category is isofibrant by Lemma 5.1.3, and so we may extract from it asymmetric monoidal bicategory in which open graphs appear as morphisms.
Theorem 5.1.6.
There exists a symmetric monoidal bicategory
OpenFinGraph = H ( L C sp ( FinGraph )) which has:(1) finite sets as objects,(2) open graphs: that is, cospans of graphs of the form L ( a ) x L ( b ) as morphisms, and(3) maps of cospans of graphs as 2-morphisms, as in the following commutative diagram: L ( a ) L ( b ) xy h We can then decategorify this symmetric monoidal bicategory
OpenFinGraph to obtain asymmetric monoidal category D ( OpenFinGraph ) which has:(1) finite sets as objects, and(2) isomorphism classes of open graphs L ( a ) x L ( b )
97s morphisms, where two open graphs are isomorphic if the following diagram commutes: L ( a ) L ( b ) xy h ∼ Here, the graph isomorphism h : x → y is really a pair of bijections f : N → N ′ and g : E → E ′ between the vertex and edge sets of the graphs x and y that make the following diagram commute: xy EE ′ NN ′ EE ′ NN ′ h f fss ′ g g tt ′ In Section 3.3, we constructed a symmetric monoidal double category of open k -graphs. Thissymmetric monoidal double category is in fact isofibrant by Lemma 5.1.3, so we can apply Theorem5.0.1 to obtain a symmetric monoidal bicategory: Theorem 5.1.7.
There exists a symmetric monoidal bicategory
OpenFinGraph k = H ( L C sp ( FinGraph k )) where:(1) objects are finite sets,(2) morphisms are open k -graphs : L ( a ) N L ( b ) i o k + E N r st which are open graphs where the apex of the cospan representing the open graph is equippedwith the structure of a k -graph, and(3) 2-morphisms are maps of open k -graphs , which are maps of cospans such that the followingdiagrams commute L ( a ) N L ( b ) N ′ i oi ′ o ′ f + EE ′ EE ′ NN ′ EE ′ NN ′ f fr ss ′ r ′ g g g tt ′ for some morphisms f and g .Proof. We obtain
OpenFinGraph k = H ( L C sp ( FinGraph k )) by applying Theorems 5.1.4 and 5.1.5to the functor L : FinSet → FinGraph k of Theorem 3.3.5.We can then decategorify this symmetric monoidal bicategory OpenFinGraph k to obtain asymmetric monoidal category D ( OpenFinGraph k ) where:(1) objects are finite sets, and(2) morphisms are isomorphism classes of open k -graphs, where two open k -graphs are in thesame isomorphism class if the following diagrams commute: L ( a ) N L ( b ) N ′ i oi ′ o ′ f ∼ k + EE ′ EE ′ NN ′ EE ′ NN ′ f fr ss ′ r ′ g g g tt ′ for some isomorphisms f and g .To make contact with Baez and Fong’s original work on black-boxing electrical circuits [7], recallthat we have a monoidal category F Cospan obtained from the original incarnation of decoratedcospans. The monoidal category D ( OpenFinGraph k ) constructed in this section is not onlysymmetric, but also contains more isomorphisms. For example, consider the following two open k -graphs: a N b a N b i o i o k + E N k + E ′ N r st r ′ s ′ t ′ where E = E ′ but there exists a bijection g : E ∼ −→ E ′ such that s = s ′ ◦ g and t = t ′ ◦ g ; thisjust says that the two networks look the same but have different edge labels. Then these two open k -graphs give different morphisms in F Cospan , but the same morphism in D ( OpenFinGraph k ).99e can define a functor G : F Cospan → D ( OpenFinGraph k ) that is the identity on objectsand that identifies open graphs that are isomorphic in the sense of (2) above. Then we can considerthe following diagram: D ( OpenFinGraph k ) F Cospan LagRel k G (cid:4)(cid:4) Here the top functor (cid:4) : F Cospan → LagRel k is the original black-boxing functor constructed byBaez and Fong [7]. While we shall not prove it here, one can extend this functor to a new one, alsocalled (cid:4) , defined on D ( OpenFinGraph k ). This also promotes the original black-box functor froma mere monoidal functor to a symmetric monoidal functor (cid:4) : D ( OpenFinGraph k ) → LagRel k . In Section 3.3.3, we constructed a symmetric monoidal double category of open Petri nets withrates. This symmetric monoidal double category is also isofibrant by Lemma 5.1.3, so we can applyTheorem 5.0.1 to obtain a symmetric monoidal bicategory:
Theorem 5.1.8.
There exists a symmetric monoidal bicategory
Petri rates = H ( L C sp ( Petri rates )) where:(1) objects are finite sets,(2) morphisms are open Petri nets with rates : L ( a ) S L ( b ) i o T N [ S ][0 , ∞ ) r st which are cospans of Petri nets whose apices are equipped with a function r : T → [0 , ∞ ) assigning a rate r ( t ) to every transition t ∈ T , and(3) 2-morphisms are maps of open Petri nets with rates , which are maps of open Petri netssuch that the following diagrams commute: L ( a ) SS ′ L ( b ) ii ′ f oo ′ , ∞ ) TT ′ TT ′ N [ S ] N [ S ] TT ′ N [ S ] N [ S ] N [ f ] N [ f ] r ss ′ r ′ g g g tt ′ for some morphisms f and g .Proof. We obtain
Petri rates = H ( L C sp ( Petri rates )) by applying Theorems 5.1.4 and 5.1.5 to thefunctor L : FinSet → Petri rates of Theorem 3.3.9.Once again, we can then decategorify this bicategory
Petri rates to obtain a symmetric monoidalcategory D ( Petri rates ) where:(1) objects are finite sets, and(2) morphisms are isomorphism classes of open Petri nets with rates, where two open Petri netswith rates are in the same isomorphism class if the following diagrams commute: L ( a ) SS ′ L ( b ) ii ′ f oo ′ [0 , ∞ ) TT ′ TT ′ N [ S ] N [ S ] TT ′ N [ S ] N [ S ] N [ f ] N [ f ] r ss ′ r ′ g g g tt ′ for some isomorphisms f and g .We can define a functor G : Petri rates → D ( Petri rates ) that is the identity on objects and identifiesmorphisms in
Petri rates , if they are in the same isomorphism class in the sense of (2) above. Wecan then consider the following diagram:
Petri rates D ( Petri rates ) SemiAlgRel G (cid:4)(cid:4) Here the top functor (cid:4) : Petri rates → SemiAlgRel was constructed by Baez and Pollard [10]. Whilewe shall not prove it here, one can extend this functor to a new one, also called (cid:4) , defined on D ( Petri rates ). 101 .1.4 Maps of foot-replaced bicategories
A result of Hansen and Shulman [28] not only allows us to lift symmetric monoidal doublecategories to their underlying symmetric monoidal horizontal-edge bicategories, but also mapsbetween such.
Corollary 5.1.9.
Given two symmetric monoidal foot-replaced double categories L X and L ′ X ′ anda symmetric monoidal double functor F F : L X → L ′ X ′ between the two, the symmetric monoidaldouble functor F F induces a functor of symmetric monoidal bicategories between the underlyinghorizontal-edge bicategories of the foot-replaced double categories L X and L ′ X ′ . H ( F F ) : H ( L X ) → H ( L ′ X ′ ) Proof.
This follows immediately from the work of Hansen and Shulman [28].
Lemma 5.2.1.
The double category F C sp constructed in Theorem 4.1.1 is fibrant.Proof. Let f : c → c ′ be a vertical 1-morphism in F C sp . We can lift f to the companion horizontal1-cell ˆ f : c c ′ c ′ ! c ′ ∈ F ( c ′ ) f and then obtain the following two 2-morphisms: cc ′ c ′ c ′ c ′ c ′ ! c ′ ∈ F ( c ′ )! c ′ ∈ F ( c ′ ) c c cc c ′ c ′ ! c ∈ F ( c )! c ′ ∈ F ( c ′ ) ι c ′ = 1 ! c ′ ι f : F ( f )(! c ) → ! c ′ f f
11 11 11 ff f which satisfy the equations: c c c ! c ∈ F ( c ) cc ′ c ′ c ′ c ′ c ′ ! c ′ ∈ F ( c ′ )! c ′ ∈ F ( c ′ ) c c cc ′ c ′ c ′ ! c ∈ F ( c )! c ′ ∈ F ( c ′ ) ι f : F ( f )(! c ) → ! c ′ ι c ′ = 1 ! c ′ ι f : F ( f )(! c ) → ! c ′ = f ff f
11 11 1 f f f c ′ cc cc ′ c ′ c ′ c ′ c ′ ! c ′ ∈ F ( c ′ )! c ′ ∈ F ( c ′ ) ι c ′ = 1 ! c ′ ! c ∈ F ( c )! c ′ ∈ F ( c ′ ) ι f : F ( f )(! c ) → ! c ′ ∼ = cc c ′ c ′ c ′ c ′ ! c ′ ∈ F ( c ′ )! c ′ ∈ F ( c ′ ) ι c ′ = 1 ! c ′ ff f f
11 111 f f The right hand sides of the above two equations are given respectively by the 2-morphisms U f and 1 ˆ f . The conjoint of f is given by the F -decorated cospan ˇ f which is just the opposite of thecompanion above: c ′ c ′ c ! c ′ ∈ F ( c ′ ) f Corollary 5.2.2.
Let ( C , + , be a category with finite colimts and F : C → Cat a symmetric laxmonoidal pseudofunctor. Then there exists a symmetric monoidal bicategory F Csp : = H ( F C sp ) which has:(1) objects as those of A ,(2) morphisms as F -decorated cospans: a c b d ∈ F ( c ) i o and(3) 2-morphisms as maps of cospans in A of the form: a c bc ′ d ∈ F ( c ) d ′ ∈ F ( c ′ ) i oi ′ h o ′ together with a morphism ι : F ( h )( d ) → d ′ in F ( c ′ ) .Proof. This follows immediately from Shulman’s Theorem 5.0.1 above applied to the fibrant sym-metric monoidal double category F C sp .This symmetric monoidal bicategory F Csp is a superior version of the symmetric monoidal bi-category F Cospan ( A ) constructed earlier in a previous work [18], in that there is greater flexibilityin what 2-morphisms are allowed. 103 .2.1 Maps of decorated cospan bicategories Just as a result of Hansen and Shulman [28] allows us to lift maps of symmetric monoidal foot-replace double categories to maps between their underlying horizontal-edge bicategories, we canalso lift maps between symmetric monoidal decorated cospan double categories to maps betweentheir underlying horizontal-edge bicategories.
Corollary 5.2.3.
Given two symmetric monoidal decorated cospan double categories F C sp and F ′ C sp and a symmetric monoidal double functor H : F C sp → F ′ C sp between the two, the sym-metric monoidal double functor H induces a functor of symmetric monoidal bicategories between theunderlying horizontal-edge bicategories of the decorated cospan double categories F C sp and F ′ C sp . H ( H ) : H ( F C sp ) → H ( F ′ C sp ) Proof.
This follows immediately from the work of Hansen and Shulman [28].
We can then decategorify the symmetric monoidal bicategory F Csp to obtain a symmetricmonoidal category similar to the monoidal one obtained using Fong’s result, but symmetric andwith larger isomorphism classes of morphisms:
Corollary 5.2.4.
Given a symmetric lax monoidal pseudofunctor F : A → Cat where A is acategory with finite colimits whose monoidal structure is given by binary coproducts, there exists asymmetric monoidal category F Csp : = D ( F Csp ) where:(1) objects are those of A and(2) morphisms are isomorphism classes of F -decorated cospans of A , where an F -decorated cospanis given by a pair: a c b d ∈ F ( c ) i o and given another F -decorated cospan: a c ′ b d ′ ∈ F ( c ′ ) i ′ o ′ these two F -decorated cospans are in the same isomorphism class if there exists an isomor-phism f : c → c ′ such that following diagram commutes: a cc ′ b i ′ o ′ i of and there exists an isomorphism ι : F ( f )( d ) → d ′ in F ( c ′ ) . In this symmetric monoidal category, isomorphism classes are as they should morally be, andthe instance of two graphs having different but isomorphic edge sets does not prevent them frombeing in the same isomorphism class. 104 .3 A biequivalence of compositional frameworks
In Chapter 4, it is mentioned that given a symmetric monoidal pseudofunctor F : ( A , + , → ( Cat , × ,
1) such that F factors as an ordinary pseudofunctor F → Rex ֒ → Cat , where
Rex is the 2-category of finitely cocomplete categories, finite coproduct preserving functors and nat-ural transformations, we can obtain a fully faithful left adjoint L : ( A , + , → ( X , + ,
0) where( X , + ,
0) := ( R F, + , R : X → A preserves finite colimits. Fromthe pseudofunctor F : A → Cat , we can obtain a symmetric monoidal double category of decoratedcospans by Theorem 4.1.3. From the left adjoint L : A → X , we can obtain a symmetric monoidaldouble category of structured cospans by Theorem 3.2.3. By Theorem 4.3.15, we have an equiva-lence of symmetric monoidal double categories F C sp ≃ L C sp ( X ). In the previous sections of thepresent chapter, we proved that each of these symmetric monoidal double categories are fibrant andgive rise to underlying symmetric monoidal bicategories F Csp and H ( L C sp ( X )), respectively, byTheorem 5.0.1 due to Shulman. We can use another result due to Shulman [38] to lift the doubleequivalence of double categories to a biequivalence of bicategories. Proposition 5.3.1 ([38, Prop. B.3]) . An equivalence of fibrant double categories induces a biequiv-alence of horizontal bicategories.
Corollary 5.3.2.
The bicategories F Csp and H ( L C sp ( X )) are biequivalent. Both the double equivalence and biequivalence are in fact isomorphisms. See [4] for more details.105 hapter 6
Coarse-graining open Markovprocesses
A ‘Markov process’ is a stochastic model describing a sequence of transitions between states inwhich the probability of a transition depends only on the current state. The only Markov processeswe consider here are continuous-time Markov chains with a finite set of states. Such a Markovprocess can be drawn as a labeled graph: ab c d / In this example the set of states is X = { a, b, c, d } . The numbers labeling edges are transitionrates, so the probability π i ( t ) of being in state i ∈ X at time t ∈ R evolves according to a lineardifferential equation ddt π i ( t ) = X j ∈ X H ij π j ( t )called the ‘master equation’, where the matrix H can be read off from the diagram: H = − / − / − − . If there is an edge from a state j to a distinct state i , the matrix entry H ij is the number labelingthat edge, while if there is no such edge, H ij = 0. The diagonal entries H ii are determined by the106equirement that the sum of each column is zero. This requirement says that the rate at whichprobability leaves a state equals the rate at which it goes to other states. As a consequence, thetotal probability is conserved: ddt X i ∈ X π i ( t ) = 0and is typically set equal to 1.However, while this sum over all states is conserved, the same need not be true for the sum of π i ( t ) over i in a subset Y ⊂ X . This poses a challenge to studying a Markov process as built fromsmaller parts: the parts are not themselves Markov processes. The solution is to describe them as‘open’ Markov processes. These are a generalization in which probability can enter or leave fromcertain states designated as inputs and outputs: ab c d inputs outputs / In an open Markov process, probabilities change with time according to the ‘open master equation’,a generalization of the master equation that includes inflows and outflows. In the above example,the open master equation is ddt π a ( t ) π b ( t ) π c ( t ) π d ( t ) = − / − / − − π a ( t ) π b ( t ) π c ( t ) π d ( t ) + I a ( t ) I b ( t )00 − O d ( t ) . To the master equation we have added a term describing inflows at the states a and b and subtracteda term describing outflows at the state d . The functions I a , I b and O d are not part of the data ofthe open Markov process. Rather, they are arbitrary smooth real-valued functions of time. Wethink of these as provided from outside—for example, though not necessarily, from the rest of alarger Markov process of which the given open Markov process is part.Open Markov processes can be seen as morphisms in a category, since we can compose two openMarkov processes by identifying the outputs of the first with the inputs of the second. Compositionlets us build a Markov process from smaller open parts—or conversely, analyze the behavior of aMarkov process in terms of its parts. Categories of this sort have been studied in a number of papers[7, 8, 24, 36], but here we go further and construct a double category to describe coarse-graining.‘Coarse-graining’ is a widely used method of simplifying a Markov process by mapping its setof states X onto some smaller set X ′ in a manner that respects, or at least approximately respects,the dynamics [1, 14]. Here we introduce coarse-graining for open Markov processes. We show howto extend this notion to the case of maps p : X → X ′ that are not surjective, obtaining a generalconcept of morphism between open Markov processes.Since open Markov processes are already morphisms in a category, it is natural to treat mor-phisms between them as 2-morphisms. To this end, we construct a double category M ark with:1071) finite sets as objects,(2) functions as vertical 1-morphisms,(3) open Markov processes as horizontal 1-cells,(4) morphisms between open Markov processes as 2-morphisms.Composition of open Markov processes is only weakly associative, so this double category is notstrict. In fact, M ark is symmetric monoidal: this captures the fact that we can not only composeopen Markov processes but also ‘tensor’ them by setting them side by side. For example, if wecompose this open Markov process: inputs outputs with the one shown before: inputs outputs / we obtain this open Markov process: inputs outputs / inputs outputs / If we fix constant probabilities at the inputs and outputs, there typically exist solutions of theopen master equation with these boundary conditions that are constant as a function of time.These are called ‘steady states’. Often these are nonequilibrium steady states, meaning that thereis a nonzero net flow of probabilities at the inputs and outputs. For example, probability can flowthrough an open Markov process at a constant rate in a nonequilibrium steady state.In previous work, Baez, Fong and Pollard studied the relation between probabilities and flowsat the inputs and outputs that holds in steady state [8, 10]. They called the process of extractingthis relation from an open Markov process ‘black-boxing’, since it gives a way to forget the internalworkings of an open system and remember only its externally observable behavior. They provedthat black-boxing is compatible with composition and tensoring. This result can be summarizedby saying that black-boxing is a symmetric monoidal functor.For the main result [2], we show that black-boxing is compatible with morphisms between openMarkov processes. To make this idea precise, we prove that black-boxing gives a map from thedouble category M ark to another double category, called L inRel , which has:(1) finite-dimensional real vector spaces U, V, W, . . . as objects,(2) linear maps f : V → W as vertical 1-morphisms from V to W ,(3) linear relations R ⊆ V ⊕ W as horizontal 1-cells from V to W ,(4) squares V V W W R ⊆ V ⊕ V gf S ⊆ W ⊕ W obeying ( f ⊕ g ) R ⊆ S as 2-morphisms.Here a ‘linear relation’ from a vector space V to a vector space W is a linear subspace R ⊆ V ⊕ W .Linear relations can be composed in the same way as relations [6]. The double category L inRel M ark it is strict:that is, composition of linear relations is associative.The main result, Theorem 6.6.3, says that black-boxing gives a symmetric monoidal doublefunctor (cid:4) : M ark → L inRel . The hardest part is to show that black-boxing preserves composition of horizontal 1-cells: thatis, black-boxing a composite of open Markov processes gives the composite of their black-boxings.Luckily, for this we can adapt a previous argument [10] due to Baez and Pollard. Thus, the newcontent of this result concerns the vertical 1-morphisms and especially the 2-morphisms, whichdescribe coarse-grainings.An alternative approach to studying morphisms between open Markov processes would usebicategories rather than double categories. The symmetric monoidal double categories M ark and L inrel can be converted into symmetric monoidal bicategories using Shulman’s technique [37]. In[2], Baez and the author conjectured that the black-boxing double functor would determine a functorbetween these symmetric monoidal bicategories, and Hansen and Shulman [28] consequently provedthis conjecture: see Theorem 6.6.4. However, double categories seem to be a simpler framework forcoarse-graining open Markov processes.It is worth comparing some related work. Baez, Fong and Pollard constructed a symmetricmonoidal category where the morphisms are open Markov processes [8, 10]. As in this chapter, theyonly consider Markov processes where time is continuous and the set of states is finite. However,they formalized such Markov processes in a slightly different way than is done here: they defined aMarkov process to be a directed multigraph where each edge is assigned a positive number calledits ‘rate constant’. In other words, they defined it to be a diagram(0 , ∞ ) E r o o t / / s / / X where X is a finite set of vertices or ‘states’, E is a finite set of edges or ‘transitions’ between states,the functions s, t : E → X give the source and target of each edge, and r : E → (0 , ∞ ) gives therate constant of each edge. They explained how from this data one can extract a matrix of realnumbers ( H ij ) i,j ∈ X called the ‘Hamiltonian’ of the Markov process, with two familiar properties:(1) H ij ≥ i = j ,(2) P i ∈ X H ij = 0 for all j ∈ X .A matrix with these properties is called ‘infinitesimal stochastic’, since these conditions are equiv-alent to exp( tH ) being stochastic for all t ≥ X together with an infinitesimal stochastic matrix ( H ij ) i,j ∈ X .This allows us to work more directly with the Hamiltonian and the all-important master equation ddt π ( t ) = Hπ ( t )which describes the evolution of a time-dependent probability distribution π ( t ) : X → R .Clerc, Humphrey and Panangaden have constructed a bicategory [17] with finite sets as objects,‘open discrete labeled Markov processes’ as morphisms, and ‘simulations’ as 2-morphisms. In their110ramework, ‘open’ has a similar meaning as it does in the works listed above. These open discretelabeled Markov processes are also equipped with a set of ‘actions’ which represent interactionsbetween the Markov process and the environment, such as an outside entity acting on a stochasticsystem. A ‘simulation’ is then a function between the state spaces that map the inputs, outputsand set of actions of one open discrete labeled Markov process to the inputs, outputs and set ofactions of another.Another compositional framework for Markov processes is given by de Francesco Albasini, Saba-dini and Walters [25] in which they construct an algebra of ‘Markov automata’. A Markov automa-ton is a family of matrices with nonnegative real coefficients that is indexed by elements of a binaryproduct of sets, where one set represents a set of ‘signals on the left interface’ of the Markovautomata and the other set analogously for the right interface. Before explaining open Markov processes we should recall a bit about Markov processes. Asmentioned in the Introduction, we use ‘Markov process’ as a short term for ‘continuous-time Markovchain with a finite set of states’, and we identify any such Markov process with the infinitesimalstochastic matrix appearing in its master equation. We make this precise with a bit of terminologythat is useful throughout the chapter.Given a finite set X , we call a function v : X → R a ‘vector’ and call its values at points x ∈ X its ‘components’ v x . We define a ‘probability distribution’ on X to be a vector π : X → R whosecomponents are nonnegative and sum to 1. As usual, we use R X to denote the vector space offunctions v : X → R . Given a linear operator T : R X → R Y we have ( T v ) i = P j ∈ X T ij v j for some‘matrix’ T : Y × X → R with entries T ij . Definition 6.2.1.
Given a finite set X , a linear operator H : R X → R X is infinitesimal stochas-tic if(1) H ij ≥ i = j and(2) P i ∈ X H ij = 0 for each j ∈ X .The reason for being interested in such operators is that when exponentiated they give stochasticoperators. Definition 6.2.2.
Given finite sets X and Y , a linear operator T : R X → R Y is stochastic if forany probability distribution π on X , T π is a probability distribution on Y .Equivalently, T is stochastic if and only if(1) T ij ≥ i ∈ Y , j ∈ X and(2) P i ∈ Y T ij = 1 for each j ∈ X .If we think of T ij as the probability for j ∈ X to be mapped to i ∈ Y , these conditions makeintuitive sense. Since stochastic operators are those that preserve probability distributions, thecomposite of stochastic operators is stochastic. 111n Lemma 6.3.7 we recall that a linear operator H : R X → R X is infinitesimal stochastic if andonly if its exponential exp( tH ) = ∞ X n =0 ( tH ) n n !is stochastic for all t ≥
0. Thus, given an infinitesimal stochastic operator H , for any time t ≥ tH ) : R X → R X to any probability distribution π ∈ R X and get aprobability distribution π ( t ) = exp( tH ) π. These probability distributions π ( t ) obey the master equation ddt π ( t ) = Hπ ( t ) . Moreover, any solution of the master equation arises this way.All the material so far is standard [33, Sec. 2.1]. We now turn to open Markov processes.
Definition 6.2.3.
We define a
Markov process to be a pair (
X, H ) where X is a finite set and H : R X → R X is an infinitesimal stochastic operator. We also call H a Markov process on X . Definition 6.2.4.
We define an open Markov process to consist of finite sets X , S and T andinjections S X T i o together with a Markov process (
X, H ). We call S the set of inputs and T the set of outputs .Thus, an open Markov process is a cospan in FinSet with injections as legs and a Markov processon its apex. We do not require that the injections have disjoint images. We often abbreviate anopen Markov process as S ( X, H ) T i o or simply S i ( X, H ) o T .Given an open Markov process we can write down an ‘open’ version of the master equation,where probability can also flow in or out of the inputs and outputs. To work with the open masterequation we need two well-known concepts: Definition 6.2.5.
Let f : A → B be a map between finite sets. The linear map f ∗ : R B → R A sends any vector v ∈ R B to its pullback along f , given by f ∗ ( v ) = v ◦ f. The linear map f ∗ : R A → R B sends any vector v ∈ R A to its pushforward along f , given by( f ∗ ( v ))( b ) = X { a : f ( a )= b } v ( a ) . f ∗ and f ∗ as matrices with respect to the standard bases of R A and R B , they are simplytransposes of one another.Now, suppose we are given an open Markov process S ( X, H ) T i o together with inflows I : R → R S and outflows O : R → R T , arbitrary smooth functions of time.We write the value of the inflow at s ∈ S at time t as I s ( t ), and similarly for the outflows and otherfunctions of time. We say that a function v : R → R X obeys the open master equation if dv ( t ) dt = Hv ( t ) + i ∗ ( I ( t )) − o ∗ ( O ( t )) . This says that for any state j ∈ X the time derivative of v j ( t ) takes into account not only the usualterm from the master equation, but also those of the inflows and outflows.If the inflows and outflows are constant in time, a solution v of the open master equation thatis also constant in time is called a steady state . More formally: Definition 6.2.6.
Given an open Markov process S i ( X, H ) o T together with I ∈ R S and O ∈ R T , a steady state with inflows I and outflows O is an element v ∈ R X such that Hv + i ∗ ( I ) − o ∗ ( O ) = 0 . Given v ∈ R X , we call i ∗ ( v ) ∈ R S and o ∗ ( v ) ∈ R T the input probabilities and output proba-bilities , respectively. Definition 6.2.7.
Given an open Markov process S i ( X, H ) o T , we define its black-boxing to be the set (cid:4) (cid:0) S i ( X, H ) o T (cid:1) ⊆ R S ⊕ R S ⊕ R T ⊕ R T consisting of all 4-tuples ( i ∗ ( v ) , I, o ∗ ( v ) , O ) where v ∈ R X is some steady state with inflows I ∈ R S and outflows O ∈ R T .Thus, black-boxing records the relation between input probabilities, inflows, output probabilitiesand outflows that holds in steady state. This is the ‘externally observable steady state behavior’of the open Markov process. It has already been shown [8, 10] that black-boxing can be seen asa functor between categories. Here we go further and describe it as a double functor betweendouble categories, in order to study the effect of black-boxing on morphisms between open Markovprocesses. There are various ways to approximate a Markov process by another Markov process on a smallerset, all of which can be considered forms of coarse-graining [14]. A common approach is to take aMarkov process H on a finite set X and a surjection p : X → X ′ and create a Markov process on X ′ . In general this requires a choice of ‘stochastic section’ for p , defined as follows:113 efinition 6.3.1. Given a function p : X → X ′ between finite sets, a stochastic section for p isa stochastic operator s : R X ′ → R X such that p ∗ s = 1 X ′ .It is easy to check that a stochastic section for p exists if and only if p is a surjection. In Lemma6.3.9 we show that given a Markov process H on X and a surjection p : X → X ′ , any stochasticsection s : R X ′ → R X gives a Markov process on X ′ , namely H ′ = p ∗ Hs.
Experts call the matrix corresponding to p ∗ the collector matrix , and they call s the distributormatrix [14]. The names help clarify what is going on. The collector matrix, coming from thesurjection p : X → X ′ , typically maps many states of X to each state of X ′ . The distributor matrix,the stochastic section s : R X ′ → R X , typically maps each state in X ′ to a linear combination ofmany states in X . Thus, H ′ = p ∗ Hs distributes each state of X ′ , applies H , and then collects theresults.In general H ′ depends on the choice of s , but sometimes it does not: Definition 6.3.2.
We say a Markov process H on X is lumpable with respect to a surjection p : X → X ′ if the operator p ∗ Hs is independent of the choice of stochastic section s : R X ′ → R X .This concept is not new [14]. In Theorem 6.3.10 we show that it is equivalent to anothertraditional formulation, and also to an even simpler one: H is lumpable with respect to p if andonly if p ∗ H = H ′ p ∗ . This equation has the advantage of making sense even when p is not asurjection. Thus, we can use it to define a more general concept of morphism between Markovprocesses: Definition 6.3.3.
Given Markov processes (
X, H ) and ( X ′ , H ′ ), a morphism of Markov pro-cesses p : ( X, H ) → ( X ′ , H ′ ) is a map p : X → X ′ such that p ∗ H = H ′ p ∗ .There is a category Mark with Markov processes as objects and the morphisms as defined above,where composition is the usual composition of functions. But what is the meaning of such amorphism? Using Lemma 6.3.7 one can check that for any Markov processes (
X, H ) and ( X ′ , H ′ ),and any map p : X → X ′ , we have p ∗ H = H ′ p ∗ ⇐⇒ p ∗ exp( tH ) = exp( tH ′ ) p ∗ for all t ≥ . Thus, p is a morphism of Markov processes if evolving a probability distribution on X via exp( tH )and then pushing it forward along p is the same as pushing it forward and then evolving it viaexp( tH ′ ).We can also define morphisms between open Markov processes: Definition 6.3.4. A morphism of open Markov processes from the open Markov process S i ( X, H ) o T to the open Markov process S ′ i ′ ( X ′ , H ′ ) o ′ T ′ is a triple of functions f : S → S ′ , p : X → X ′ , g : T → T ′ such that the squares in this diagram are pullbacks: SS ′ T ′ X TX ′ ii ′ o ′ of gp p ∗ H = H ′ p ∗ .We need the squares to be pullbacks so that in Lemma 6.6.1 we can black-box morphisms ofopen Markov processes. In Lemma 6.4.2 we show that horizontally composing these morphismspreserves this pullback property. But to do this, we need the horizontal arrows in these squares tobe injections. This explains the conditions in Definitions 6.2.4 and 6.3.4.As an example, consider the following diagram: a b b c inputs outputs
668 47
This is a way of drawing an open Markov process S i ( X, H ) o T where X = { a, b , b , c } , S and T are one-element sets, i maps the one element of S to a , and o maps the one element of T to c .We can read off the infinitesimal stochastic operator H : R X → R X from this diagram and obtain H = −
15 0 0 08 −
10 0 07 4 − . The resulting open master equation is ddt v a ( t ) v b ( t ) v b ( t ) v c ( t ) = −
15 0 0 08 −
10 0 07 4 − v a ( t ) v b ( t ) v b ( t ) v c ( t ) + I ( t )000 − O ( t ) . Here I is an arbitrary smooth function of time describing the inflow at the one point of S , and O is a similar function describing the outflow at the one point of T .Suppose we want to simplify this open Markov process by identifying the states b and b . Todo this we take X ′ = { a, b, c } and define p : X → X ′ by p ( a ) = a, p ( b ) = p ( b ) = b, p ( c ) = c. To construct the infinitesimal stochastic operator H ′ : R X ′ → R X ′ for the simplified open Markovprocess we need to choose a stochastic section s : R X ′ → R X for p , for example s = / / . b , we assume the original Markovprocess has a 1 / b and a 2 / b . The operator H ′ = p ∗ Hs is then H ′ = −
15 0 015 − . It may be difficult to justify the assumptions behind our choice of stochastic section, but theexample at hand has a nice feature: H ′ is actually independent of this choice. In other words, H is lumpable with respect to p . The reason is explained in Theorem 6.3.10. Suppose we partition X into blocks, each the inverse image of some point of X ′ . Then H is lumpable with respect to p ifand only if when we sum the rows in each block of H , all the columns within any given block ofthe resulting matrix are identical. This matrix is p ∗ H : H = −
15 0 0 08 −
10 0 07 4 − = ⇒ p ∗ H = −
15 0 0 015 − − . While coarse-graining is of practical importance even in the absence of lumpability, the lumpablecase is better behaved, so we focus on this case.So far we have described a morphism of Markov processes p : ( X, H ) → ( X ′ , H ′ ), but togetherwith identity functions on the inputs S and outputs T this defines a morphism of open Markovprocesses, going from the above open Markov process to this one: a b c inputs outputs The open master equation for this new coarse-grained open Markov process is ddt v a ( t ) v b ( t ) v c ( t ) = −
15 0 015 − v a ( t ) v b ( t ) v c ( t ) + I ( t )00 − O ( t ) . In Section 6.4 we construct a double category M ark with open Markov processes as horizontal1-cells and morphisms between these as 2-morphisms. This double category is our main object ofstudy. First, however, we should prove the results mentioned above. For this it is helpful to recalla few standard concepts: Definition 6.3.5. A is a collection of linear operators U ( t ) : V → V on a vector space V , one for each t ∈ [0 , ∞ ), such that(1) U (0) = 1 and(2) U ( s + t ) = U ( s ) U ( t ) for all s, t ∈ [0 , ∞ ). If V is finite-dimensional we say the collection U ( t )is continuous if t U ( t ) v is continuous for each v ∈ V .116 efinition 6.3.6. Let X be a finite set. A Markov semigroup is a continuous 1-parametersemigroup U ( t ) : R X → R X such that U ( t ) is stochastic for each t ∈ [0 , ∞ ). Lemma 6.3.7.
Let X be a finite set and U ( t ) : R X → R X a Markov semigroup. Then U ( t ) =exp( tH ) for a unique infinitesimal stochastic operator H : R X → R X , which is given by Hv = ddt U ( t ) v (cid:12)(cid:12)(cid:12)(cid:12) t =0 for all v ∈ R X . Conversely, given an infinitesimal stochastic operator H , then exp( tH ) = U ( t ) isa Markov semigroup.Proof. This is well known. For a proof that every continuous one-parameter semigroup of operators U ( t ) on a finite-dimensional vector space V is in fact differentiable and of the form exp( tH ) where Hv = ddt U ( t ) v (cid:12)(cid:12) t =0 , see Engel and Nagel [22, Sec. I.2]. For a proof that U ( t ) is then a Markovsemigroup if and only if H is infinitesimal stochastic, see Norris [33, Theorem 2.1.2]. Lemma 6.3.8.
Let U ( t ) : R X → R X be a differentiable family of stochastic operators defined for t ∈ [0 , ∞ ) and having U (0) = 1 . Then ddt U ( t ) (cid:12)(cid:12) t =0 is infinitesimal stochastic.Proof. Let H = ddt U ( t ) (cid:12)(cid:12) t =0 = lim t → + ( U ( t ) − /t . As U ( t ) is stochastic, its entries are nonnegativeand the column sum of any particular column is 1. Then the column sum of any particular columnof U ( t ) − U ( t ) − t ≥
0, as is ( U ( t ) − /t , from which it follows that lim t → + ( U ( t ) − U (0)) /t = H is infinitesimal stochastic. Lemma 6.3.9.
Let p : X → X ′ be a function between finite sets with a stochastic section s : R X ′ → R X , and let H : R X → R X be an infinitesimal stochastic operator. Then H ′ = p ∗ Hs : R X ′ → R X ′ is also infinitesimal stochastic.Proof. Lemma 6.3.7 implies that exp( tH ) is stochastic for all t ≥
0. For any map p : X → X ′ theoperator p ∗ : R X → R X ′ is easily seen to be stochastic, and s is stochastic by assumption. Thus, U ( t ) = p ∗ exp( tH ) s is stochastic for all t ≥
0. Differentiating, we conclude that ddt U ( t ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = ddt p ∗ exp( tH ) s (cid:12)(cid:12)(cid:12)(cid:12) t =0 = p ∗ exp( tH ) Hs | t =0 = p ∗ Hs is infinitesimal stochastic by Lemma 6.3.8.We can now give some conditions equivalent to lumpability. The third is widely found in theliterature [14] and the easiest to check in examples. It makes use of the standard basis vectors e j ∈ R X associated to the elements j of any finite set X . The surjection p : X → X ′ defines apartition on X where two states j, j ′ ∈ X lie in the same block of the partition if and only if p ( j ) = p ( j ′ ). The elements of X ′ correspond to these blocks. The third condition for lumpabilitysays that p ∗ H has the same effect on two basis vectors e j and e j ′ when j and j ′ are in the sameblock. As mentioned in the example above, this condition says that if we sum the rows in eachblock of H , all the columns in any given block of the resulting matrix p ∗ H are identical. Theorem 6.3.10.
Let p : X → X ′ be a surjection of finite sets and let H be a Markov process on X . Then the following conditions are equivalent: H is lumpable with respect to p .(2) There exists a linear operator H ′ : R X ′ → R X ′ such that p ∗ H = H ′ p ∗ .(3) p ∗ He j = p ∗ He j ′ for all j, j ′ ∈ X such that p ( j ) = p ( j ′ ) .When these conditions hold there is a unique operator H ′ : R X ′ → R X ′ such that p ∗ H = H ′ p ∗ , it isgiven by H ′ = p ∗ Hs for any stochastic section s of p , and it is infinitesimal stochastic.Proof. ( i ) = ⇒ ( iii ). Suppose that H is lumpable with respect to p . Thus, p ∗ Hs : R X ′ → R X ′ isindependent of the choice of stochastic section s : R X ′ → R X . Such a stochastic section is simplyan arbitrary linear operator that maps each basis vector e i ∈ R X ′ to a probability distribution on X supported on the set { j ∈ X : p ( j ) = i } . Thus, for any j, j ′ ∈ X with p ( j ) = p ( j ′ ) = i , we canfind stochastic sections s, s ′ : R X ′ → R X such that s ( e i ) = e j and s ′ ( e i ) = e j ′ . Since p ∗ Hs = p ∗ Hs ′ ,we have p ∗ He j = p ∗ Hs ( e i ) = p ∗ Hs ′ ( e i ) = p ∗ He j ′ . ( iii ) = ⇒ ( ii ). Define H ′ : R X ′ → R X ′ on basis vectors e i ∈ R X ′ by setting H ′ e i = p ∗ He j for any j with p ( j ) = i . Note that H ′ is well-defined: since p is a surjection such j exists, and since H is lumpable, H ′ is independent of the choice of such j . Next, note that for any j ∈ X , if we let p ( j ) = i we have p ∗ He j = H ′ e i = H ′ p ∗ e j . Since the vectors e j form a basis for R X , it follows that p ∗ H = H ′ p ∗ .( ii ) = ⇒ ( i ). Suppose there exists an operator H ′ : R X ′ → R X ′ such that p ∗ H = H ′ p ∗ . Choosesuch an operator; then for any stochastic section s for p we have p ∗ Hs = H ′ p ∗ s = H ′ . It follows that p ∗ Hs is independent of the stochastic section s , so H is lumpable with respect to p .Suppose that any, hence all, of conditions ( i ) , ( ii ) , ( iii ) hold. Suppose that H ′ : R X ′ → R X ′ is anoperator with p ∗ H = H ′ p ∗ . Then the argument in the previous paragraph shows that H ′ = p ∗ Hs for any stochastic section s of p . Thus H ′ is unique, and by Lemma 6.3.9 it is infinitesimalstochastic. One of the main results of a joint work with Baez [2] is the construction of a double category M ark of open Markov processes, The pieces of the double category M ark work as follows:(1) An object is a finite set.(2) A vertical 1-morphism f : S → S ′ is a function.(3) A horizontal 1-cell is an open Markov process S ( X, H ) T . i o In other words, it is a pair of injections S i X o T together with a Markov process H on X . 1184) A 2-morphism is a morphism of open Markov processes SS ′ T ′ .( X, H ) T ( X ′ , H ′ ) i i ′ o ′ o f gp In other words, it is a triple of maps f, p, g such that these squares are pullbacks: SS ′ T ′ , X TX ′ i i ′ o ′ o f gp and H ′ p ∗ = p ∗ H .Composition of vertical 1-morphisms in M ark is straightforward. So is vertical composition of2-morphisms, since we can paste two pullback squares and get a new pullback square. Compositionof horizontal 1-cells is a bit more subtle. Given open Markov processes S ( X, H ) T, i o T ( Y, G ) U i o (6.1)we first compose their underlying cospans using a pushout: X + T YX j : : ✈✈✈✈✈✈✈✈✈ Y k d d ❍❍❍❍❍❍❍❍❍ S i ; ; ✇✇✇✇✇✇✇✇ T o d d ■■■■■■■■■■ i : : ✉✉✉✉✉✉✉✉✉✉ U o c c ●●●●●●●●● Since monomorphisms are stable under pushout in a topos, the legs of this new cospan are againinjections, as required. We then define the composite open Markov process to be S ( X + T Y, H ⊙ G ) U ji ko where H ⊙ G = j ∗ Hj ∗ + k ∗ Gk ∗ . (6.2)Here we use both pullbacks and pushforwards along the maps j and k , as defined in Definition6.2.5. To check that H ⊙ G is a Markov process on X + T Y we need to check that j ∗ Hj ∗ and k ∗ Gk ∗ , and thus their sum, are infinitesimal stochastic:119 emma 6.4.1. Suppose that f : X → Y is any map between finite sets. If H : R X → R X isinfinitesimal stochastic, then f ∗ Hf ∗ : R Y → R Y is infinitesimal stochastic.Proof. Using Definition 6.2.5, we see that the matrix elements of f ∗ and f ∗ are given by( f ∗ ) ji = ( f ∗ ) ij = ( f ( j ) = i i ∈ Y , j ∈ X . Thus, f ∗ Hf ∗ has matrix entries( f ∗ Hf ∗ ) ii ′ = X { j,j ′ : f ( j )= i,f ( j ′ )= i ′ } H jj ′ . To show that f ∗ Hf ∗ is infinitesimal stochastic we need to show that its off-diagonal entries arenonnegative and its columns sum to zero. By the above formula, these follow from the same factsfor H .Another formula for horizontal composition is also useful. Given the composable open Markovprocesses in Equation (6.1) we can take the copairing of the maps j : X → X + T Y and k : Y → X + T Y and get a map ℓ : X + Y → X + T Y . Then H ⊙ G = ℓ ∗ ( H ⊕ G ) ℓ ∗ (6.3)where H ⊕ G : R X + Y → R X + Y is the direct sum of the operators H and G . This is easy to checkfrom the definitions.Horizontal composition of 2-morphisms is even subtler: Lemma 6.4.2.
Suppose that we have horizontally composable 2-morphisms as follows: SS ′ T ′ ( X, H ) T TT ′ U ′ ( Y, G ) U ( X ′ , H ′ ) ( Y ′ , G ′ ) i i ′ o ′ o f gp i o g i ′ o ′ hq Then there is a 2-morphism SS ′ U ′ ( X + T Y, H ⊙ G ) U ( X ′ + T ′ Y ′ , H ′ ⊙ G ′ ) i o f hp + g qi ′ o ′ hose underlying diagram of finite sets is SS ′ X X + T Y Y UX ′ X ′ + T ′ Y ′ Y ′ U ′ , i j k o f p + g q hi ′ j ′ k ′ o ′ where j, k, j ′ , k ′ are the canonical maps from X, Y, X ′ , Y ′ , respectively, to the pushouts X + T Y and X ′ + T ′ Y ′ .Proof. To show that we have defined a 2-morphism, we first check that the squares in the abovediagram of finite sets are pullbacks. Then we show that ( p + g q ) ∗ ( H ⊙ G ) = ( H ′ ⊙ G ′ )( p + g q ) ∗ .For the first part, it suffices by the symmetry of the situation to consider the left square. Wecan write it as a pasting of two smaller squares: SS ′ X X + T YX ′ X ′ + T ′ Y ′ i jf p p + g qi ′ j ′ By assumption the left-hand smaller square is a pullback, so it suffices to prove this for the right-hand one. For this we use that fact that
FinSet is a topos and thus an adhesive category [30, 31],and consider this commutative cube: TT ′ X + T YXY X ′ + T ′ Y ′ X ′ Y ′ o i o ′ i ′ p jk p + g qj ′ k ′ g q By assumption the top and bottom faces are pushouts, the two left-hand vertical faces are pullbacks,and the arrows o ′ and i ′ are monic. In an adhesive category, this implies that the two right-handvertical faces are pullbacks as well. One of these is the square in question.To show that ( p + g q ) ∗ ( H ⊙ G ) = ( H ′ ⊙ G ′ )( p + g q ) ∗ , we again use the above cube. Because itstwo right-hand vertical faces commute, we have( p + g q ) ∗ j ∗ = j ′∗ p ∗ and ( p + g q ) ∗ k ∗ = k ′∗ q ∗ H ⊙ G we obtain( p + g q ) ∗ ( H ⊙ G ) = ( p + g q ) ∗ ( j ∗ Hj ∗ + k ∗ Gk ∗ )= ( p + g q ) ∗ j ∗ Hj ∗ + ( p + g q ) ∗ k ∗ Gk ∗ = j ′∗ p ∗ Hj ∗ + k ′∗ q ∗ Gk ∗ . By assumption we have p ∗ H = H ′ p ∗ and q ∗ G = G ′ q ∗ so we can go a step further, obtaining( p + g q ) ∗ ( H ⊙ G ) = j ′∗ H ′ p ∗ j ∗ + k ′∗ G ′ q ∗ k ∗ . Because the two right-hand vertical faces of the cube are pullbacks, Lemma 6.4.3 below impliesthat p ∗ j ∗ = j ′∗ ( p + g q ) ∗ and q ∗ k ∗ = k ′∗ ( p + g q ) ∗ . Using these, we obtain( p + g q ) ∗ ( H ⊙ G ) = j ′∗ H ′ j ′∗ ( p + g q ) ∗ + k ′∗ G ′ k ′∗ ( p + g q ) ∗ = ( j ′∗ H ′ j ′∗ + k ′∗ G ′ k ′∗ )( p + g q ) ∗ = ( H ′ ⊙ G ′ )( p + g q ) ∗ completing the proof.The following lemma is reminiscent of the Beck–Chevalley condition for adjoint functors: Lemma 6.4.3.
Given a pullback square in
FinSet : A BDC g fk h the following square of linear operators commutes: R A R B R D R C g ∗ f ∗ k ∗ h ∗ Proof.
Choose v ∈ R B and c ∈ C . Then( g ∗ f ∗ ( v ))( c ) = X { a : g ( a )= c } v ( f ( a )) , ( k ∗ h ∗ ( v ))( c ) = X { b : h ( b )= k ( c ) } v ( b ) , g ∗ f ∗ = k ∗ h ∗ it suffices to show that f restricts to a bijection f : { a ∈ A : g ( a ) = c } ∼ −→ { b ∈ B : h ( b ) = k ( c ) } . On the one hand, if a ∈ A has g ( a ) = c then b = f ( a ) has h ( b ) = h ( f ( a )) = k ( g ( a )) = k ( c ), sothe above map is well-defined. On the other hand, if b ∈ B has h ( b ) = k ( c ), then by the definitionof pullback there exists a unique a ∈ A such that f ( a ) = b and g ( a ) = c , so the above map is abijection. Theorem 6.4.4.
There exists a double category M ark as defined above.Proof. Let M ark , the category of objects, consist of finite sets and functions. Let M ark , thecategory of arrows, consist of open Markov processes and morphisms between these: SS ′ T ′ .( X, H ) T ( X ′ , H ′ ) i i ′ o ′ o f gp To make M ark into a double category we need to specify the identity-assigning functor u : M ark → M ark , the source and target functors s, t : M ark → M ark , and the composition functor ⊙ : M ark × M ark M ark → M ark . These are given as follows.For a finite set S , u ( S ) is given by S ( S, S ) S S S where 0 S is the zero operator from R S to R S . For a map f : S → S ′ between finite sets, u ( f ) isgiven by S ( S, S ) SS ′ S ′ ( S ′ , S ′ ) f ff s and t map a Markov process S i ( X, H ) o T to S and T ,respectively, and they map a morphism of open Markov processes SS ′ T ′ ( X, H ) T ( X ′ , H ′ ) i i ′ o ′ o f gp to f : S → S ′ and g : T → T ′ , respectively. The composition functor ⊙ maps the pair of openMarkov processes S ( X, H ) T T ( Y, G ) U i o i o to their composite S ( X + T Y, H ⊙ G ) U ji ko defined as in Equation (6.2), and it maps the pair of morphisms of open Markov processes SS ′ T ′ ( X, H ) T TT ′ U ′ ( Y, G ) U ( X ′ , H ′ ) ( Y ′ , G ′ ) i i ′ o ′ o f gp i o g i ′ o ′ hq to their horizontal composite as defined as in Lemma 6.4.2.It is easy to check that u, s and t are functors. To prove that ⊙ is a functor, the main thing weneed to check is the interchange law. Suppose we have four morphisms of open Markov processesas follows: SS ′ T ′ ( X, H ) T TT ′ U ′ ( Y, G ) U ( X ′ , H ′ ) ( Y ′ , G ′ ) S ′ S ′′ T ′′ ( X ′ , H ′ ) T ′ T ′ ( Y ′ , G ′ ) U ′ T ′′ U ′′ ( X ′′ , H ′′ ) ( Y ′′ , G ′′ ) f gp g hqf ′ g ′ p ′ g ′ h ′ q ′ SS ′ U ′ S ′′ U ′′ ,( X + T Y, H ⊙ G ) U ( X ′ + T ′ Y ′ , H ′ ⊙ G ′ ) S ′ ( X ′ + T ′ Y ′ , H ′ ⊙ G ′ ) U ′ ( X ′′ + T ′′ Y ′′ , H ′′ ⊙ G ′′ ) f hp + g qf ′ h ′ p ′ + g ′ q ′ and then composing vertically gives SS ′′ U ′′ .( X + T Y, H ⊙ G ) U ( X ′′ + T ′′ Y ′′ , H ′′ ⊙ G ′′ ) f ′ ◦ f h ′ ◦ h ( p ′ + g ′ q ′ ) ◦ ( p + g q ) Composing vertically gives S ( X, H ) T T ( Y, G ) US ′′ T ′′ T ′′ U ′′ ,( X ′′ , H ′′ ) ( Y ′′ , G ′′ ) f ′ ◦ f g ′ ◦ gp ′ ◦ p g ′ ◦ g h ′ ◦ hq ′ ◦ q and then composing horizontally gives SS ′′ U ′′ .( X + T Y, H ⊙ G ) U ( X ′′ + T ′′ Y ′′ , H ′′ ⊙ G ′′ ) f ′ ◦ f h ′ ◦ h ( p ′ ◦ p ) + ( g ′ ◦ g ) ( q ′ ◦ q ) The only apparent difference between the two results is the map in the middle: one has ( p ′ + g ′ q ′ ) ◦ ( p + g q ) while the other has ( p ′ ◦ p ) + ( g ′ ◦ g ) ( q ′ ◦ q ). But these are in fact the same map, so theinterchange law holds. 125he functors u, s, t and ◦ obey the necessary relations su = 1 = tu and the relations saying that the source and target of a composite behave as they should. Lastly,we have three natural isomorphisms: the associator, left unitor, and right unitor, which arise fromthe corresponding natural isomorphisms for the double category of finite sets, functions, cospans offinite sets, and maps of cospans. The triangle and pentagon equations hold in M ark because theydo in this simpler double category [18].Next we give M ark a symmetric monoidal structure. We call the tensor product ‘addition’.Given objects S, S ′ ∈ M ark we define their sum S + S ′ using a chosen coproduct in FinSet .The unit for this tensor product in M ark is the empty set. We can similarly define the sumof morphisms in M ark , since given maps f : S → T and f ′ : S ′ → T ′ there is a natural map f + f ′ : S + S ′ → T + T ′ . Given two objects in M ark : S ( X , H ) T S ( X , H ) T i o i o we define their sum to be S + S ( X + X , H ⊕ H ) T + T i + i o + o where H ⊕ H : R X + X → R X + X is the direct sum of the operators H and H . The unit forthis tensor product in M ark is ∅ ( ∅ , ∅ ) ∅ where 0 ∅ : R ∅ → R ∅ is the zero operator. Finally,given two morphisms in M ark : S S ′ T ′ S ′ T ′ ( X , H ) T S ( X , H ) T ( X ′ , H ′ ) ( X ′ , H ′ ) i o f g i ′ o ′ p i o f g i ′ o ′ p we define their sum to be S + S S ′ + S ′ T ′ + T ′ . ( X + X , H ⊕ H ) T + T ( X ′ + X ′ , H ′ ⊕ H ′ ) i + i o + o f + f g + g i ′ + i ′ o ′ + o ′ p + p We complete the description of M ark as a symmetric monoidal double category in the proof of thistheorem: Theorem 6.4.5.
The double category M ark can be given a symmetric monoidal structure with theabove properties. roof. First we complete the description of M ark and M ark as symmetric monoidal categories.The symmetric monoidal category M ark is just the category of finite sets with a chosen coproductof each pair of finite sets providing the symmetric monoidal structure. We have described the tensorproduct in M ark , which we call ‘addition’, so now we need to introduce the associator, unitors,and braiding, and check that they make M ark into a symmetric monoidal category.Given three objects in M ark S ( X , H ) T S ( X , H ) T S ( X , H ) T tensoring the first two and then the third results in ( S + S ) + S (( X + X ) + X , ( H ⊕ H ) ⊕ H ) ( T + T ) + T whereas tensoring the last two and then the first results in S + ( S + S ) ( X + ( X + X ) , H ⊕ ( H ⊕ H )) T + ( T + T ). The associator for M ark is then given as follows: ( S + S ) + S (( X + X ) + X , ( H ⊕ H ) ⊕ H ) ( T + T ) + T ( X + ( X + X ) , H ⊕ ( H ⊕ H )) S + ( S + S ) T + ( T + T ) a aa where a is the associator in ( FinSet , +). If we abbreviate an object S ( X, H ) T of M ark as( X, H ), and denote the associator for M ark as α , the pentagon identity says that this diagramcommutes: ((( X , H ) ⊕ ( X , H )) ⊕ ( X , H )) ⊕ ( X , H )(( X , H ) ⊕ ( X , H )) ⊕ (( X , H ) ⊕ ( X , H ))( X , H ) ⊕ (( X , H ) ⊕ (( X , H ) ⊕ ( X , H )))( X , H ) ⊕ ((( X , H ) ⊕ ( X , H )) ⊕ ( X , H ))(( X , H ) ⊕ (( X , H ) ⊕ ( X , H ))) ⊕ ( X , H ) α αα ⊕ ( X ,H ) α ( X ,H ) ⊕ α which is clearly true. Recall that the monoidal unit for M ark is given by ∅ ( ∅ , ∅ ) ∅ . The leftand right unitors for M ark , denoted λ and ρ , are given respectively by the following 2-morphisms: ∅ + SS T S T ( ∅ + X, ∅ ⊕ H ) ∅ + T S + ∅ ( X + ∅ , H ⊕ ∅ ) T + ∅ ( X, H ) (
X, H ) ℓ ℓℓ r rr ℓ and r are the left and right unitors in FinSet . The left and right unitors and associator for M ark satisfy the triangle identity: (( X, H ) ⊕ ( ∅ , ∅ )) ⊕ ( Y, G ) (
X, H ) ⊕ ( Y, G ) (
X, H ) ⊕ (( ∅ , ∅ ) ⊕ ( Y, G )). ρ ⊕ ⊕ λα The braiding in M ark is given as follows: S + S S + S T + T ( X , H ) ⊕ ( X , H ) T + T ( X , H ) ⊕ ( X , H ) b S ,S b T ,T b X ,X where b is the braiding in ( FinSet , +). It is easy to check that the braiding in M ark is its owninverse and obeys the hexagon identity, making M ark into a symmetric monoidal category.The source and target functors s, t : M ark → M ark are strict symmetric monoidal functors, asrequired. To make M ark into a symmetric monoidal double category we must also give it two otherpieces of structure. One, called χ , says how the composition of horizontal 1-cells interacts withthe tensor product in the category of arrows. The other, called µ , says how the identity-assigningfunctor u relates the tensor product in the category of objects to the tensor product in the categoryof arrows. We now define these two isomorphisms.Given horizontal 1-cells S ( X , H ) T T ( Y , G ) U S ( X , H ) T T ( Y , G ) U the horizontal composites of the top two and the bottom two are given, respectively, by S ( X + T Y , H ⊙ G ) U S ( X + T Y , H ⊙ G ) U . ‘Adding’ the left two and right two, respectively, we obtain S + S ( X + X , H ⊕ H ) T + T T + T ( Y + Y , G ⊕ G ) U + U . Thus the sum of the horizontal composites is S + S (( X + T Y ) + ( X + T Y ) , ( H ⊙ G ) ⊕ ( H ⊙ G )) U + U while the horizontal composite of the sums is S + S (( X + X ) + T + T ( Y + Y ) , ( H ⊕ H ) ⊙ ( G ⊕ G )) U + U . χ between these is S + S S + S U + U (( X , H ) ⊙ ( Y , G )) ⊕ (( X , H ) ⊙ ( Y , G )) U + U (( X , H ) ⊕ ( X , H )) ⊙ (( Y , G ) ⊕ ( Y , G )) S + S U + U ˆ χ where ˆ χ is the bijectionˆ χ : ( X + T Y ) + ( X + T Y ) → ( X + X ) + T + T ( Y + Y )obtained from taking the colimit of the diagram S X T Y U S X T Y U in two different ways. We call χ ‘globular’ because its source and target 1-morphisms are identities.We need to check that χ indeed defines a 2-isomorphism in M ark .To do this, we need to show that(( H ⊕ H ) ⊙ ( G ⊕ G )) ˆ χ ∗ = ˆ χ ∗ (( H ⊙ G ) ⊕ ( H ⊙ G )) . (6.4)To simplify notation, let K = ( X + T Y ) + ( X + T Y ) and K ′ = ( X + X ) + T + T ( Y + Y ) sothat ˆ χ : K ∼ → K ′ . Let q : X + X + Y + Y → K, q ′ : X + X + Y + Y → K ′ be the canonical maps coming from the definitions of K and K ′ as colimits, and note that q ′ = ˆ χq by the universal property of the colimit. A calculation using Equation (6.3) implies that( H ⊙ G ) ⊕ ( H ⊙ G ) = q ∗ (( H ⊕ H ) ⊕ ( G ⊕ G )) q ∗ and similarly ( H ⊕ H ) ⊙ ( G ⊕ G ) = q ′∗ (( H ⊕ H ) ⊕ ( G ⊕ G )) q ′∗ . Together these facts give( H ⊕ H ) ⊙ ( G ⊕ G ) = ˆ χ ∗ q ∗ (( H ⊕ H ) ⊕ ( G ⊕ G )) q ∗ ˆ χ ∗ = ˆ χ ∗ (( H ⊙ G ) ⊕ ( H ⊙ G ) ˆ χ ∗ . and since ˆ χ is a bijection, ˆ χ ∗ is the inverse of ˆ χ ∗ , so Equation (6.4) follows.129or the other globular 2-isomorphism, if S and T are finite sets, then u ( S + T ) is given by S + T ( S + T, S + T ) S + T S + T S + T while u ( S ) ⊕ u ( T ) is given by S + T ( S + T, S ⊕ T ) S + T S + 1 T S + 1 T so there is a globular 2-isomorphism µ between these, namely the identity 2-morphism. All thecommutative diagrams in the definition of symmetric monoidal double category [37] can be checkedin a straightforward way. If one prefers to work with bicategories as opposed to double categories, then one can lift theabove symmetric monoidal double category M ark to a symmetric monoidal bicategory Mark usinga result of Shulman. This bicategory
Mark will have:(1) finite sets as objects,(2) open Markov processes as morphisms,(3) morphisms of open Markov processes as 2-morphisms.To do this, we need to check that the symmetric monoidal double category M ark is isofibrant—meaning fibrant on vertical 1-morphisms which happen to be isomorphisms. See the Appendix fordetails. Definition 6.4.6.
Let D be a double category. Then the horizontal bicategory of D , which wedenote as H ( D ), is the bicategory with(1) objects of D as objects,(2) horizontal 1-cells of D as 1-morphisms,(3) globular 2-morphisms of D (i.e., 2-morphisms with identities as their source and target) as2-morphisms,and vertical and horizontal composition, identities, associators and unitors arising from those in D . Lemma 6.4.7.
The symmetric monoidal double category M ark is isofibrant.Proof. In what follows, all unlabeled arrows are identities. To show that M ark is isofibrant, weneed to show that every vertical 1-isomorphism has both a companion and a conjoint [37]. Givena vertical 1-isomorphism f : S → S ′ , meaning a bijection between finite sets, then a companion of f is given by the horizontal 1-cell: S ( S ′ , S ′ ) S ′ f S ( S ′ , S ′ ) S ′ S ′ S ′ ( S ′ , S ′ ) S ( S, S ) SS S ′ ( S ′ , S ′ ) ff ff f such that vertical composition gives S ( S, S ) SS S ′ ( S ′ , S ′ ) S ′ ( S ′ , S ′ ) S ′ = SS ′ ( S, S ) S ( S ′ , S ′ ) S ′ ff ff f ff and horizontal composition gives S ( S, S ) SS S ′ ( S ′ , S ′ ) ( S ′ , S ′ )( S ′ , S ′ ) S ′ S ′ = S ( S ′ , S ′ ) S ′ S ( S ′ , S ′ ) S ′ ff f f ff A conjoint of f : S → S ′ is given by the horizontal 1-cell S ′ ( S ′ , S ′ ) S f together with two 2-morphisms S ′ ( S ′ , S ′ ) SS ′ S ′ ( S ′ , S ′ ) S ( S, S ) SS ′ S ( S ′ , S ′ ) f f f ff that satisfy equations analogous to the two above.131 heorem 6.4.8. The bicategory
Mark is a symmetric monoidal bicategory.Proof.
This follows immediately from Theorem 5.0.1 of Shulman: M ark is an isofibrant symmetricmonoidal double category, so we obtain the symmetric monoidal bicategory Mark as the horizontalbicategory of M ark . The general idea of ‘black-boxing’, as mentioned in Chapter 2, is to take a system and forgeteverything except the relation between its inputs and outputs, as if we had placed it in a blackbox and were unable to see its inner workings. Previous work of Baez and Pollard [10] constructeda black-boxing functor (cid:4) : Dynam → SemiAlgRel where
Dynam is a category of finite sets and‘open dynamical systems’ and
SemiAlgRel is a category of finite-dimensional real vector spaces andrelations defined by polynomials and inequalities. When we black-box such an open dynamicalsystem, we obtain the relation between inputs and outputs that holds in steady state.A special case of an open dynamical system is an open Markov process as defined in this chapter.Thus, we could restrict the black-boxing functor (cid:4) : Dynam → SemiAlgRel to a category
Mark withfinite sets as objects and open Markov processes as morphisms. Since the steady state behavior of aMarkov process is linear , we would get a functor (cid:4) : Mark → LinRel where
LinRel is the category offinite-dimensional real vector spaces and linear relations [6]. However, we will go further and defineblack-boxing on the double category M ark . This will exhibit the relation between black-boxingand morphisms between open Markov processes.The symmetric monoidal double category L inRel of linear relations introduced in this sectionwill serve as the codomain of a symmetric monoidal black-box double functor in Section 6.6. Thisdouble category L inRel will have:(1) finite-dimensional real vector spaces U, V, W, . . . as objects,(2) linear maps f : V → W as vertical 1-morphisms from V to W ,(3) linear relations R ⊆ V ⊕ W as horizontal 1-cells from V to W ,(4) squares V V W W R ⊆ V ⊕ V gf S ⊆ W ⊕ W obeying ( f ⊕ g ) R ⊆ S as 2-morphisms.The last item deserves some explanation. A preorder is a category such that for any pair of objects x, y there exists at most one morphism α : x → y . When such a morphism exists we usually write132 ≤ y . Similarly there is a kind of double category for which given any ‘frame’ A BC D
M gf N there exists at most one 2-morphism
A BC D ⇓ α M gf N filling this frame. For lack of a better term let us call this a degenerate double category. Item (4)implies that L inRel will be degenerate in this sense.In L inRel , composition of vertical 1-morphisms is the usual composition of linear maps, whilecomposition of horizontal 1-cells is the usual composition of linear relations. Since compositionof linear relations obeys the associative and unit laws strictly, L inRel will be a strict doublecategory. Since L inRel is degenerate, there is at most one way to define the vertical composite of2-morphisms U U V V ⇓ αW W ⇓ β = U U W W ⇓ βα R ⊆ U ⊕ U gff ′ T ⊆ W ⊕ W g ′ S ⊆ V ⊕ V R ⊆ U ⊕ U g ′ gf ′ f T ⊆ W ⊕ W so we need merely check that a 2-morphism βα filling the frame at right exists. This amounts tonoting that ( f ⊕ g ) R ⊆ S, ( f ′ ⊕ g ′ ) S ⊆ T = ⇒ ( f ′ ⊕ g ′ )( f ⊕ g ) R ⊆ T. Similarly, there is at most one way to define the horizontal composite of 2-morphisms V V W W ⇓ α V W ⇓ α ′ = V V W W ⇓ α ′ ◦ α R ⊆ V ⊕ V gf S ⊆ W ⊕ W R ′ ⊆ V ⊕ V hS ′ ⊆ W ⊕ W R ′ R ⊆ V ⊕ V f S ′ S ⊆ W ⊕ W h α ′ ◦ α exists, which amounts to noting that( f ⊕ g ) R ⊆ S, ( g ⊕ h ) R ′ ⊆ S ′ = ⇒ ( f ⊕ h )( R ′ R ) ⊆ S ′ S. Theorem 6.5.1.
There exists a strict double category L inRel with the above properties.Proof. The category of objects L inRel has finite-dimensional real vector spaces as objects andlinear maps as morphisms. The category of arrows L inRel has linear relations as objects andsquares V V W W R ⊆ V ⊕ V gf S ⊆ W ⊕ W with ( f ⊕ g ) R ⊆ S as morphisms. The source and target functors s, t : L inRel → L inRel areclear. The identity-assigning functor u : L inRel → L inRel sends a finite-dimensional real vectorspace V to the identity map 1 V and a linear map f : V → W to the unique 2-morphism V VW W . V ff W The composition functor ⊙ : L inRel × L inRel L inRel → L inRel acts on objects by the usualcomposition of linear relations, and it acts on 2-morphisms by horizontal composition as describedabove. These functors can be shown to obey all the axioms of a double category. In particu-lar, because L inRel is degenerate, all the required equations between 2-morphisms, such as theinterchange law, hold automatically.Next we make L inRel into a symmetric monoidal double category. To do this, we first give L inRel the structure of a symmetric monoidal category. We do this using a specific choice of directsum for each pair of finite-dimensional real vector spaces as the tensor product, and a specific 0-dimensional vector space as the unit object. Then we give L inRel a symmetric monoidal structureas follows. Given linear relations R ⊆ V ⊕ W and R ⊆ V ⊕ W , we define their direct sum by R ⊕ R = { ( v , v , w , w ) : ( v , w ) ∈ R , ( v , w ) ∈ R } ⊆ V ⊕ V ⊕ W ⊕ W . Given two 2-morphisms in L inRel : V V W W V ′ V ′ W ′ W ′ ⇓ α ′ ⇓ α R ⊆ V ⊕ V gf S ⊆ W ⊕ W R ′ ⊆ V ′ ⊕ V ′ g ′ f ′ S ′ ⊆ W ′ ⊕ W ′ V ⊕ V ′ V ⊕ V ′ W ⊕ W ′ W ⊕ W ′ ⇓ α ⊕ α ′ R ⊕ R ′ ⊆ V ⊕ V ′ ⊕ V ⊕ V ′ g ⊕ g ′ f ⊕ f ′ S ⊕ S ′ ⊆ W ⊕ W ′ ⊕ W ⊕ W ′ because L inRel is degenerate. To show that α ⊕ α ′ exists, we need merely note that( f ⊕ g ) R ⊆ S, ( f ′ ⊕ g ′ ) R ′ ⊆ S ′ = ⇒ ( f ⊕ f ′ ⊕ g ⊕ g ′ )( R ⊕ R ′ ) ⊆ S ⊕ S ′ . Theorem 6.5.2.
The double category L inRel can be given the structure of a symmetric monoidaldouble category with the above properties.Proof. We have described L inRel and L inRel as symmetric monoidal categories. The sourceand target functors s, t : L inRel → L inRel are strict symmetric monoidal functors. The requiredglobular 2-isomorphisms χ and µ are defined as follows. Given four horizontal 1-cells R ⊆ U ⊕ V , R ⊆ V ⊕ W ,S ⊆ U ⊕ V , S ⊆ V ⊕ W , the globular 2-isomorphism χ : ( R ⊕ S )( R ⊕ S ) ⇒ ( R R ) ⊕ ( S S ) is the identity 2-morphism U ⊕ U W ⊕ W U ⊕ U W ⊕ W . ( R ⊕ S )( R ⊕ S ) 11 ( R R ) ⊕ ( S S ) The globular 2-isomorphism µ : u ( V ⊕ W ) ⇒ u ( V ) ⊕ u ( W ) is the identity 2-morphism V ⊕ W V ⊕ WV ⊕ W V ⊕ W . V ⊕ W
11 1 V ⊕ W All the commutative diagrams in the definition of symmetric monoidal double category [37] canbe checked straightforwardly. In particular, all diagrams of 2-morphisms commute automaticallybecause L inRel is degenerate. 135 .5.1 A bicategory of linear relations We can also promote the symmetric monoidal double category L inRel of linear relations fromthe previous section to a symmetric monoidal bicategory LinRel of linear relations due to Shulman’sTheorem 5.0.1 by showing L inRel is isofibrant. Lemma 6.5.3.
The symmetric monoidal double category L inRel is isofibrant.Proof. Let f : X → Y be a linear isomorphism between finite-dimensional real vector spaces. Defineˆ f to be the linear relation given by the linear isomorphism f and define 2-morphisms in L inRel X YY Y X XX Yα f ⇓ f α ⇓ ˆ ff
11 11 f ˆ f where α f and f α , the unique fillers of their frames, are identities. These two 2-morphisms and ˆ f satisfy the required equations, and the conjoint of f is given by reversing the direction of ˆ f , whichis just f − : Y → X . It follows that L inRel is isofibrant. Theorem 6.5.4.
There exists a symmetric monoidal bicategory
LinRel with(1) finite-dimensional real vector spaces as objects,(2) linear relations R ⊆ V ⊕ W as morphisms from V to W ,(3) inclusions R ⊆ S between linear relations R, S ⊆ V ⊕ W as 2-morphisms.Proof. Apply Shulman’s result, Theorem 5.0.1, to the isofibrant symmetric monoidal double cate-gory L inRel to obtain the symmetric monoidal bicategory LinRel as the horizontal edge bicategoryof L inRel . In this section we present the main result of the chapter which is a symmetric monoidal doublefunctor (cid:4) : M ark → L inRel . We proceed as follows:(1) On objects: for a finite set S , we define (cid:4) ( S ) to be the vector space R S ⊕ R S .(2) On horizontal 1-cells: for an open Markov process S i ( X, H ) o T , we define its black-boxing as in Definition 6.2.7: (cid:4) ( S i ( X, H ) o T ) = { ( i ∗ ( v ) , I, o ∗ ( v ) , O ) : v ∈ R X , I ∈ R S , O ∈ R T and H ( v ) + i ∗ ( I ) − o ∗ ( O ) = 0 } . (3) On vertical 1-morphisms: for a map f : S → S ′ , we define (cid:4) ( f ) : R S ⊕ R S → R S ′ ⊕ R S ′ to bethe linear map f ∗ ⊕ f ∗ . 136hat remains to be done is define how (cid:4) acts on 2-morphisms of M ark . This describes therelation between steady state input and output concentrations and flows of a coarse-grained openMarkov process in terms of the corresponding relation for the original process: Lemma 6.6.1.
Given a 2-morphism S ( X, H ) T ( X ′ , H ′ ) S ′ T ′ , f gi oi ′ o ′ p in M ark , there exists a (unique) 2-morphism (cid:4) ( S ) (cid:4) ( T ) (cid:4) ( S ′ ) (cid:4) ( T ′ ) (cid:4) ( S i ( X, H ) o T ) (cid:4) ( g ) (cid:4) ( f ) (cid:4) ( S ′ i ′ ( X ′ , H ′ ) o ′ T ′ ) in L inRel .Proof. Since L inRel is degenerate, if there exists a 2-morphism of the claimed kind it is automat-ically unique. To prove that such a 2-morphism exists, it suffices to prove( i ∗ ( v ) , I, o ∗ ( v ) , O ) ∈ V = ⇒ ( f ∗ i ∗ ( v ) , f ∗ ( I ) , g ∗ o ∗ ( v ) , g ∗ ( O )) ∈ W where V = (cid:4) ( S i ( X, H ) o T ) = { ( i ∗ ( v ) , I, o ∗ ( v ) , O ) : v ∈ R X , I ∈ R S , O ∈ R T and H ( v ) + i ∗ ( I ) − o ∗ ( O ) = 0 } and W = (cid:4) ( S ′ i ′ ( X ′ , H ′ ) o ′ T ′ ) = { ( i ′∗ ( v ′ ) , I ′ , o ′∗ ( v ′ ) , O ′ ) : v ′ ∈ R X ′ , I ′ ∈ R S ′ , O ′ ∈ R T ′ and H ′ ( v ′ ) + i ′∗ ( I ′ ) − o ′∗ ( O ′ ) = 0 } . To do this, assume ( i ∗ ( v ) , I, o ∗ ( v ) , O ) ∈ V , which implies that H ( v ) + i ∗ ( I ) − o ∗ ( O ) = 0 . (6.5)Since the commuting squares in α are pullbacks, Lemma 6.4.3 implies that f ∗ i ∗ = i ′∗ p ∗ , g ∗ o ∗ = o ′∗ p ∗ . Thus ( f ∗ i ∗ ( v ) , f ∗ ( I ) , g ∗ o ∗ ( v ) , g ∗ ( O )) = ( i ′∗ p ∗ ( v ) , f ∗ ( I ) , o ′∗ p ∗ ( v ) , g ∗ ( O ))137nd this is an element of W as desired if H ′ p ∗ ( v ) + i ′∗ f ∗ ( I ) − o ′∗ g ∗ ( O ) = 0 . (6.6)To prove Equation (6.6), note that H ′ p ∗ ( v ) + i ′∗ f ∗ ( I ) − o ′∗ g ∗ ( O ) = p ∗ H ( v ) + p ∗ i ∗ ( I ) − p ∗ o ∗ ( O )= p ∗ ( H ( v ) + i ∗ ( I ) − o ∗ ( O ))where in the first step we use the fact that the squares in α commute, together with the fact that H ′ p ∗ = p ∗ H . Thus, Equation (6.5) implies Equation (6.6).The following result is a special case of a result by Pollard and Baez on black-boxing opendynamical systems [10]. To make this chapter self-contained we adapt the proof to the case athand: Lemma 6.6.2.
The black-boxing of a composite of two open Markov processes equals the compositeof their black-boxings.Proof.
Consider composable open Markov processes S i −→ ( X, H ) o ←− T, T i ′ −→ ( Y, G ) o ′ ←− U. To compose these, we first form the pushout X + T YX j : : ✈✈✈✈✈✈✈✈✈ Y k d d ❍❍❍❍❍❍❍❍❍ S i ; ; ✇✇✇✇✇✇✇✇ T o d d ■■■■■■■■■■ i ′ : : ✉✉✉✉✉✉✉✉✉✉ U o ′ c c ●●●●●●●●● Then their composite is S ji −→ ( X + T Y, H ⊙ G ) ko ′ ←− U where H ⊙ G = j ∗ Hj ∗ + k ∗ Gk ∗ . To prove that (cid:4) preserves composition, we first show that (cid:4) ( Y, G ) (cid:4) ( X, H ) ⊆ (cid:4) ( X + T Y, H ⊙ G ) . Thus, given ( i ∗ ( v ) , I, o ∗ ( v ) , O ) ∈ (cid:4) ( X, H ) , ( i ′∗ ( v ′ ) , I ′ , o ′∗ ( v ′ ) , O ′ ) ∈ (cid:4) ( Y, G )with o ∗ ( v ) = i ′∗ ( v ′ ) , O = I ′ i ∗ ( v ) , I, o ′∗ ( v ′ ) , O ′ ) ∈ (cid:4) ( X + T Y, H ⊙ G ) . To do this, it suffices to find w ∈ R X + T Y such that( i ∗ ( v ) , I, o ′∗ ( v ′ ) , O ′ ) = (( ji ) ∗ ( w ) , I, ( ko ′ ) ∗ ( w ) , O ′ )and w is a steady state of ( X + T Y, H ⊙ G ) with inflows I and outflows O ′ .Since o ∗ ( v ) = i ′∗ ( v ′ ) , this diagram commutes: R X v > > ⑦⑦⑦⑦⑦⑦⑦ Y v ′ _ _ ❅❅❅❅❅❅❅ T o ` ` ❅❅❅❅❅❅❅❅ i ′ ? ? ⑦⑦⑦⑦⑦⑦⑦⑦ so by the universal property of the pushout there is a unique map w : X + T Y → R such that thiscommutes: X X + T Y YT R w i ′ o v ′ v j k (6.7)This simply says that because the functions v and v ′ agree on the ‘overlap’ of our two open Markovprocesses, we can find a function w that restricts to v on X and v ′ on Y .We now prove that w is a steady state of the composite open Markov process with inflows I andoutflows O ′ : ( H ⊙ G )( w ) + ( ji ) ∗ ( I ) − ( ko ′ ) ∗ ( O ′ ) = 0 . (6.8)To do this we use the fact that v is a steady state of S i ( X, H ) o T with inflows I and outflows O : H ( v ) + i ∗ ( I ) − o ∗ ( O ) = 0 (6.9)and v ′ is a steady state of T i ′ ( Y, G ) o ′ U with inflows I ′ and outflows O ′ : G ( v ′ ) + i ′∗ ( I ′ ) − o ′∗ ( O ′ ) = 0 . (6.10)We push Equation (6.9) forward along j , push Equation (6.10) forward along k , and sum them: j ∗ ( H ( v )) + ( ji ) ∗ ( I ) − ( jo ) ∗ ( O ) + k ∗ ( G ( v ′ )) + ( ki ′ ) ∗ ( I ′ ) − ( ko ′ ) ∗ ( O ′ ) = 0 . O = I ′ and jo = ki ′ , two terms cancel, leaving us with j ∗ ( H ( v )) + ( ji ) ∗ ( I ) + k ∗ ( G ( v ′ )) − ( ko ′ ) ∗ ( O ′ ) = 0 . Next we combine the terms involving the infinitesimal stochastic operators H and G , with the helpof Equation (6.7) and the definition of H ⊙ G : j ∗ ( H ( v )) + k ∗ ( G ( v ′ )) = ( j ∗ Hj ∗ + k ∗ Gk ∗ )( w )= ( H ⊙ G )( w ) . (6.11)This leaves us with ( H ⊙ G )( w ) + ( ji ) ∗ ( I ) − ( ko ′ ) ∗ ( O ′ ) = 0which is Equation (6.8), precisely what we needed to show.To finish showing that (cid:4) is a functor, we need to show that (cid:4) ( X + T Y, H ⊙ G ) ⊆ (cid:4) ( Y, G ) (cid:4) ( X, H ) . So, suppose we have (( ji ) ∗ ( w ) , I, ( ko ′ ) ∗ ( w ) , O ′ ) ∈ (cid:4) ( X + T Y, H ⊙ G ) . We need to show (( ji ) ∗ ( w ) , I, ( ko ′ ) ∗ ( w ) , O ′ ) = ( i ∗ ( v ) , I, o ′∗ ( v ′ ) , O ′ ) (6.12)where ( i ∗ ( v ) , I, o ∗ ( v ) , O ) ∈ (cid:4) ( X, H ) , ( i ′∗ ( v ′ ) , I ′ , o ′∗ ( v ′ ) , O ′ ) ∈ (cid:4) ( Y, G )and o ∗ ( v ) = i ′∗ ( v ′ ) , O = I ′ . To do this, we begin by choosing v = j ∗ ( w ) , v ′ = k ∗ ( w ) . This ensures that Equation (6.12) holds, and since jo = ki ′ , it also ensures that o ∗ ( v ) = ( jo ) ∗ ( w ) = ( ki ′ ) ∗ ( w ) = i ′∗ ( v ′ ) . To finish the job, we need to find an element O = I ′ ∈ R T such that v is a steady state of ( X, H )with inflows I and outflows O and v ′ is a steady state of ( Y, G ) with inflows I ′ and outflows O ′ .Of course, we are given the fact that w is a steady state of ( X + T Y, H ⊙ G ) with inflows I andoutflows O ′ .In short, we are given Equation (6.8), and we seek O = I ′ such that Equations (6.9) and (6.10)hold. Thanks to our choices of v and v ′ , we can use Equation (6.11) and rewrite Equation (6.8) as j ∗ ( H ( v ) + i ∗ ( I )) + k ∗ ( G ( v ′ ) − o ′∗ ( O ′ )) = 0 . (6.13)Equations (6.9) and (6.10) say that H ( v ) + i ∗ ( I ) − o ∗ ( O ) = 0 G ( v ′ ) + i ′∗ ( I ′ ) − o ′∗ ( O ′ ) = 0 . (6.14)140ow we use the fact that X + T YX j : : ✈✈✈✈✈✈✈✈✈ Y k d d ❍❍❍❍❍❍❍❍❍ T o d d ■■■■■■■■■■ i ′ : : ✉✉✉✉✉✉✉✉✉✉ is a pushout. Applying the ‘free vector space on a finite set’ functor, which preserves colimits, thisimplies that R X + T Y R X j ∗ : : ✈✈✈✈✈✈✈✈✈✈ R Yk ∗ d d ❍❍❍❍❍❍❍❍❍❍ R To ∗ d d ❍❍❍❍❍❍❍❍❍❍ i ′∗ : : ✈✈✈✈✈✈✈✈✈✈ is a pushout in the category of vector spaces. Since a pushout is formed by taking first a coproductand then a coequalizer, this implies that R T (0 ,i ′∗ ) / / ( o ∗ , / / R X ⊕ R Y [ j ∗ ,k ∗ ] / / R X + T Y is a coequalizer. Thus, the kernel of [ j ∗ , k ∗ ] is the image of ( o ∗ , − (0 , i ′∗ ). Equation (6.13) saysprecisely that ( H ( v ) + i ∗ ( I ) , G ( v ′ ) − o ′∗ ( O ′ )) ∈ ker([ j ∗ , k ∗ ]) . Thus, it is in the image of ( o ∗ , − (0 , i ′∗ ). In other words, there exists some element O = I ′ ∈ R T such that ( H ( v ) + i ∗ ( I ) , G ( v ′ ) − o ′∗ ( O ′ )) = ( o ∗ ( O ) , − i ′∗ ( I ′ )) . This says that Equations (6.9) and (6.10) hold, as desired.This is the main result of the paper on coarse-graining open Markov processes [2]:
Theorem 6.6.3.
There exists a symmetric monoidal double functor (cid:4) : M ark → L inRel with thefollowing behavior:(1) Objects: (cid:4) sends any finite set S to the vector space R S ⊕ R S .(2) Vertical 1-morphisms: (cid:4) sends any map f : S → S ′ to the linear map f ∗ ⊕ f ∗ : R S ⊕ R S → R S ′ ⊕ R S ′ .(3) Horizontal 1-cells: (cid:4) sends any open Markov process S i ( X, H ) o T to the linear relationgiven in Definition 6.2.7: (cid:4) ( S i ( X, H ) o T ) = { ( i ∗ ( v ) , I, o ∗ ( v ) , O ) : H ( v ) + i ∗ ( I ) − o ∗ ( O ) = 0 for some I ∈ R S , v ∈ R X , O ∈ R T } .
4) 2-Morphisms: (cid:4) sends any morphism of open Markov processes S ( X, H ) T ( X ′ , H ′ ) S ′ T ′ f gi oi ′ o ′ p to the 2-morphism in L inRel given in Lemma 6.6.1: (cid:4) ( S ) (cid:4) ( T ) (cid:4) ( S ′ ) (cid:4) ( T ′ ) . (cid:4) ( S i ( X, H ) o T ) (cid:4) ( g ) (cid:4) ( f ) (cid:4) ( S ′ i ′ ( X ′ , H ′ ) o ′ T ′ ) Proof.
First we must define functors (cid:4) : M ark → L inRel and (cid:4) : M ark → L inRel . Thefunctor (cid:4) is defined on finite sets and maps between these as described in (i) and (ii) of thetheorem statement, while (cid:4) is defined on open Markov processes and morphisms between theseas described in (iii) and (iv). Lemma 6.6.1 shows that (cid:4) is well-defined on morphisms betweenopen Markov processes; given this is it easy to check that (cid:4) is a functor. One can verify that (cid:4) and (cid:4) combine to define a double functor (cid:4) : M ark → L inRel : the hard part is checking thathorizontal composition of open Markov processes is preserved, but this was shown in Lemma 6.6.2.Horizontal composition of 2-morphisms is automatically preserved because L inRel is degenerate.To make (cid:4) into a symmetric monoidal double functor we need to make (cid:4) and (cid:4) into symmetricmonoidal functors, which we do using these extra structures: • an isomorphism in L inRel between { } and (cid:4) ( ∅ ), • a natural isomorphism between (cid:4) ( S ) ⊕ (cid:4) ( S ′ ) and (cid:4) ( S + S ′ ) for any two objects S, S ′ ∈ M ark , • an isomorphism in L inRel between the unique linear relation { } → { } and (cid:4) ( ∅ ( ∅ , ∅ ) ∅ ), and • a natural isomorphism between (cid:4) (( S ( X, H ) T ) ⊕ (cid:4) ( S ′ ( X ′ , H ′ ) T ′ )and (cid:4) ( S + S ′ ( X + X ′ , H ⊕ H ′ ) T + T ′ )for any two objects S ( X, H ) T , S ′ ( X ′ , H ′ ) T ′ of M ark .There is an evident choice for each of these extra structures, and it is straightforward to checkthat they not only make (cid:4) and (cid:4) into symmetric monoidal functors but also meet the extrarequirements for a symmetric monoidal double functor listed in Hansen and Shulman’s paper [28],which may also be found in Definition A.2.14. In particular, all diagrams of 2-morphisms commuteautomatically because L inRel is degenerate. 142 .6.1 A corresponding functor of bicategories We have symmetric monoidal bicategories
Mark and
LinRel , both of which come from dis-carding the vertical 1-morphisms of the symmetric monoidal double categories M ark and L inRel ,respectively. Morally, we should be able to do something similar to the symmetric monoidaldouble functor (cid:4) : M ark → L inRel to obtain a symmetric monoidal functor of bicategories (cid:4) : Mark → LinRel , and indeed we can by a result of Hansen and Shulman [28].
Theorem 6.6.4 ([28, Thm. 6.17]) . There exists a symmetric monoidal functor (cid:4) : Mark → LinRel that maps:(1) any finite set S to the finite-dimensional real vector space (cid:4) ( S ) = R S ⊕ R S ,(2) any open Markov process S i ( X, H ) o T to the linear relation from (cid:4) ( S ) to (cid:4) ( T ) givenby the linear subspace (cid:4) ( S i ( X, H ) o T ) = { ( i ∗ ( v ) , I, o ∗ ( v ) , O ) : H ( v ) + i ∗ ( I ) − o ∗ ( O ) = 0 } ⊆ R S ⊕ R S ⊕ R T ⊕ R T , (3) any morphism of open Markov processes SS T ( X, H ) T ( X ′ , H ′ ) i i ′ o ′ o S T p to the inclusion (cid:4) ( X, H ) ⊆ (cid:4) ( X ′ , H ′ ) . Proof.
This was proved by Hansen and Shulman [28, Theorem 6.17], by applying a more generalresult [28, Theorem 5.11] to the strong symmetric monoidal double functor (cid:4) : M ark → L inRel ofTheorem 6.6.3. 143 hapter 7 Possible future work
In this final chapter before the Appendix, I will touch on a few possible avenues in which thework in this thesis can be improved. The three main results are the contents of Chapter 3, Chapter4 and Chapter 6.Chapter 3 presents the results regarding the foot-replaced double categories formalism. Weshowed how to build a symmetric monoidal double category L C sp ( X ) from an adjoint functor L : A → X between categories with finite colimits. One possible generalization would be to let L be a ‘2-adjoint’ between two 2-categories A and X with finite ‘2-colimits’. In the conjecturedsymmetric monoidal double category L C sp ( X ) obtained from this 2-adjoint L , composing twohorizontal 1-cells—two cospans in X —would involve taking ‘2-pushouts’, which involve the typicalpushout square commuting not on the nose but only up to isomorphism.We can also generalize foot-replaced double categories. The idea of replacing the category ofobjects of a double category X with some other category A is easily transferable to even higherlevel categorifications. For example, if X is a ‘triple category’, which would involve a category X of objects, a category X of arrows and a category X of ‘faces’, we could replace the category ofobjects X with some other category A , or even both the category of objects X and category ofarrows X with some double category A in the event that the pair ( X , X ) form a double category.One version of a triple category due to Grandis and Par`e [26] is an ’intercategory’ which is, roughlyspeaking, a pair of double categories sharing a common ‘side category’.Chapter 4 explores improvements to Fong’s original conception of decorated cospans [23]. Here,the main insight was to not consider a set of decorations but a category of decorations. Evenfurther generalizations could be made here by replacing the finitely cocartesian category A with afinitely 2-cocartesian 2-category A and viewing Cat as a 3-category and defining an appropriatefunctor F : A → Cat . In this framework, we could then decorate objects with ‘higher level stuff ’[11], such as a decoration that makes a 2-category C into a monoidal 2-category ( C , ⊗ , frameworks themselves, but each framework issuitable to applications not mentioned in this thesis. Biological sciences, economics and even socialsciences are bound to have situations which can be modeled by either of the above frameworks.Anytime a concept or an idea can be thought of as a set equipped with some extra structure,decorated cospans are lurking in the background, and very often a trivial form of this structure iscaptured by a left adjoint.Chapter 6 applies double categories to coarse-graining open Markov processes. Here, the Markovprocesses we consider are really finite state Markov chains , but more general Markov processes can144e considered. Moreover, more general forms of coarse-graining outside of lumpability can also beconsidered, but would require a different definition of 2-morphism in the resulting double category.In a ‘triple category’ of coarse-grainings, 3-morphisms would then represent maps between twodifferent ways of applying a coarse-graining to a Markov process. This idea would not be wellsuited for the double category of coarse-grainings presented here, as the category of arrows M ark is locally posetal, meaning that there is at most one coarse-graining as we have defined it [2] betweentwo open Markov processes. 145 ppendix A Definitions
A.1 Everyday categories
This is a thesis largely about applications of double categories in network theory. The mostobvious place to start is with the following question: What is a category?
Definition A.1.1. A category C consists of a collection of objects denoted Ob( C ) and a collectionof morphisms denoted Mor( C ) such that:(1) every morphism f ∈ Mor( C ) has a source object s ( f ) ∈ Ob( C ) and a target object t ( f ) ∈ Ob( C ). A morphism f with source x and target y we denote as f : x → y , and we denote thecollection of all morphisms with source x and target y by hom( x, y ) or hom C ( x, y ).(2) Given a morphism f : x → y and a morphism g : y → z , there exists a composite morphism gf : x → z . In other words, for any triple of objects x, y, z ∈ Ob( C ), there is a well-definedmap ◦ : hom( x, y ) × hom( y, z ) → hom( x, z )called composition .(3) Composition of morphisms is associative, meaning that given three composable morphisms f, g, h ∈ Mor( C ) we have h ( gf ) = ( hg ) f .(4) Every object x ∈ Ob( C ) has an identity morphism 1 x : x → x such that for any morphism f : x → y , we have f x = f = 1 y f. If both Ob( C ) and Mor( C ) are sets, we say that C is a small category . If for every pair ofobjects x, y ∈ Ob( C ) we have that hom( x, y ) is a set, we say that C is a locally small category .Here are some examples:(1) The primordial example of a category is Set of sets and functions.(2) The category
Grp of groups and group homomorphisms.(3) The category
Top of topological spaces and continuous maps.1464) The category
Mat ( k ) of natural numbers and n × m matrices with entries in a field (or moregenerally, a ring or rig) k with composition given by matrix multiplication.(5) Every monoid is a locally small category with a single object whose morphisms are given bythe elements of the monoid.(6) The category Cat of categories and functors.(7) The category
Vect of vector spaces and linear maps.(8) The category
Diff of smooth manifolds and smooth maps.(9) The category
Rel of sets and relations.(10) The category
PreOrd of preordered sets and monotone functions.(11) The category
Graph of (directed) graphs and graph morphisms, which are pairs of functionspreserving the source and target of each edge.(12) Any set S gives rise to a category S whose objects are the elements of the set S containingonly identity morphisms.(13) There is a category 1 with only one object ⋆ and only an identity morphism 1 ⋆ .Even though a category is usually named after its objects, it is the morphisms of a categorythat are the real stars of the show. In fact, we can ‘do away’ with all the objects as the collectionof all identity morphisms tell us precisely what the objects of a category are.Any sort of mathematical gizmo is boring and pointless to study unless that mathematical gizmocan ‘talk’ to other similar mathematical gizmos via maps between the two. So, how do categoriestalk to each other? Definition A.1.2.
Given categories C and D , a functor F : C → D consists of a mapOb( F ) : Ob( C ) → Ob( D ) and a map Mor( F ) : Mor( C ) → Mor( D ) respecting source and target,meaning that s ( F ( f )) = F ( s ( f )) and t ( F ( f )) = F ( t ( f )), such that:(1) For any two composable morphisms f : x → y and g : y → z in C , we have F ( f ) F ( g ) = F ( f g ),and(2) For any object x ∈ C , we have F (1 x ) = 1 F ( x ) .We usually denote the maps Ob( F ) and Mor( F ) simply as F .Here are some examples:(1) For any category C , there is an identity functor 1 C : C → C that maps every object andmorphism of C to itself.(2) There is a forgetful functor R : Grp → Set , which we call R as it is a right adjoint, that mapsany group G to its underlying set U ( G ) and any group homomorphism f : G → G ′ to itsunderlying function U ( f ) : U ( G ) → U ( G ′ ).1473) For any category C , there is a functor ! : C → C to the oneobject ⋆ of 1 and any morphism in C to the only morphism 1 ⋆ of 1.(4) There is a functor F : Set → Cat which maps any set S to the discrete category on S whoseobjects are given by elements of S and whose only morphisms are identity morphisms.(5) Given categories C and D and an object d ∈ D , there is a functor F d : C → D called the constant functor at d which maps every object C to the object d in D and every morphism of C to the morphism 1 d .Functors may look a little similar to functions in that they are maps between objects that weare interested in. However, in the same way that the morphisms are the real stars of the showin a category, one could make the same argument that it is functors that are the real stars ofcategory theory: after all, a category C is ultimately determined by the identity functor 1 C on thatcategory. But we will not go down that road. The real fun of category theory starts when we startto consider maps between maps . Our first examples of a map between maps, which are also one ofthe main reasons that Eilenberg and Mac Lane invented category theory in the 1940’s, are naturaltransformations. Definition A.1.3.
Let F : C → D and G : C → D be functors. Then a natural transformation α : F ⇒ G consists of a family of morphisms α x : F ( x ) → G ( x ) indexed by the objects of C suchthat for any morphism f : x → y in C , the following naturality square commutes. F ( x ) F ( y ) G ( x ) G ( y ) F ( f ) G ( f ) α x α y We call α x the component of α at x . If each map α x is an isomorphism, then we say that α : F ⇒ G is a natural isomorphism .Here are some examples of natural transformations:(1) For any functor F : C → D , there is an identity natural transformation 1 : F ⇒ F in whichthe component at each object x is the identity 1 F ( x ) . This is a natural isomorphism.(2) Given a functor F d : C → D which is constant at some object d ∈ D and another functor G : C → D , a natural transformation α : F d ⇒ G is a cone over D , which consists of a familyof morphisms α x : d → G ( x ) which make a cone-like commutative diagram in which all thetop triangular faces commute. dG ( x ) G ( y ) G ( z ) α x α y α z Grp denote the category of groups and group homomorphisms,
AbGrp the category ofabelian groups and group homomorphisms and Ab : Grp → AbGrp the functor sending eachgroup to its abelianization, namely Ab ( G ) := G/ [ G, G ] where [
G, G ] is the commutator sub-group of G . Then there is a natural transformation π : 1 Grp ⇒ Ab where the component ateach group is given by π G : G → Ab ( G ). For any group homomorphism f : G → H , thefollowing square commutes. G H Ab ( G ) Ab ( H ) f Ab ( f ) π G π H This is not a natural isomorphism.(4) Given a field k and a finite dimensional vector space V over k , there is a canonical isomorphism α V : V → V ∗∗ from the vector space V to its double dual. This gives a natural transformation α : 1 FinVect k ⇒ ∗∗ where ∗∗ : FinVect k → FinVect k is the functor sending each finite dimensionalvector space V to its double dual V ∗∗ . The following square then commutes for every linearmap L : V → W of finite dimensional k -vector spaces. V WV ∗∗ W ∗∗ LL ∗∗ α V α W This is a natural isomorphism if all the vector spaces are finite dimensional. If we allowfor infinite dimensional vector spaces, we still have a natural transformation, but each map α V : V → V ∗∗ is no longer an isomorphism.(5) Given commutative rings R and S and a ring homomorphism f : R → S , the ring homomor-phism f : R → S restricts to a group homomorphism f × : R × → S × where R × denotes thegroup of units of the commutative ring R . This defines a functor × : CommRing → AbGrp .There are also well-known groups of linear transformations GL n ( R ) and GL n ( S ), and everyring homomorphism f : R → S induces a map GL n ( f ) : GL n ( R ) → GL n ( S ) given by appli-cation of f to every entry of H ∈ GL n ( R ). This defines another functor GL n : CommRing → AbGrp . There is then a natural transformation det : GL n ⇒ × where given H ∈ GL n ( R ),det R ( H ) is the determinant of H . The following square commutes for every ring homomor-149hism f : R → S . GL n ( R ) GL n ( S ) R × S × GL n ( f ) f × det R det S Definition A.1.4.
Given a two categories A and X and two functors going in opposite directionsbetween the two: A X LR we say that L and R are adjoint , with L the left adjoint and R the right adjoint , if for every a ∈ A and x ∈ X there is a natural isomorphismhom X ( L ( a ) , x ) ∼ = hom A ( a, R ( x )) . A.1.1 Monoidal categories and monoidal functors
Next we introduce ‘monoidal’ categories, which are largely the kinds of categories that this thesisis about. Roughly speaking, a monoidal category is a category with a binary operation in whichwe can multiply or ‘tensor’ two objects in the category much like we can multiply two objects in amonoid.
Definition A.1.5. A monoidal category is a category C equipped with the extra structure of:(1) a functor ⊗ : C × C → C called the tensor product of C ,(2) an object I ∈ C called the (monoidal) unit of C ,(3) for any three objects a, b, c ∈ C , a natural isomorphism called the associator α : (( − ) ⊗ ( − )) ⊗ ( − ) ∼ −→ ( − ) ⊗ (( − ) ⊗ ( − ))whose components are of the form α a,b,c : ( a ⊗ b ) ⊗ c ∼ −→ a ⊗ ( b ⊗ c )(4) for any object c , a natural isomorphism called the left unitor λ : ( I ⊗ ( − )) ∼ −→ ( − )whose components are of the form λ c : I ⊗ c ∼ −→ c c , a natural isomorphism called the right unitor ρ : (( − ) ⊗ I ) ∼ −→ ( − )whose components are of the form ρ c : c ⊗ I ∼ −→ c such that the following two diagrams commute, giving equations called the pentagon identity : (( a ⊗ b ) ⊗ c ) ⊗ d ( a ⊗ b ) ⊗ ( c ⊗ d ) a ⊗ ( b ⊗ ( c ⊗ d ))( a ⊗ ( b ⊗ c )) ⊗ d a ⊗ (( b ⊗ c ) ⊗ d ) α a ⊗ b,c,d α a,b,c ⊗ d α a,b,c ⊗ d α a,b ⊗ c,d a ⊗ α b,c,d and the triangle identity : ( a ⊗ I ) ⊗ b a ⊗ b a ⊗ ( I ⊗ b ) ρ a ⊗ b a ⊗ λ b α a, C ,b Sometimes we abbreviate a monoidal category C with tensor product ⊗ and monoidal unit 1 C as( C , ⊗ , C ). Some examples of monoidal categories which are relevant in this thesis are the following:(1) The category Set together with the tensor product given by cartesian product and monoidalunit given by a singleton { ⋆ } .(2) If C is a category with finite colimits, then C is monoidal with the tensor product given bybinary coproducts and monoidal unit given by an initial object 0.(3) The large category Cat together with the tensor product given by the product of two categoriesand monoidal unit given by a terminal category 1.Sometimes there is a relationship between the two tensor products a ⊗ b and b ⊗ a for two objects a and b in a monoidal category ( C , ⊗ , I ). Definition A.1.6. A braided monoidal category is a monoidal category ( C , ⊗ , I ) equippedwith a natural isomorphism β a,b : a ⊗ b ∼ −→ b ⊗ a braiding such that the following hexagons commute. ( a ⊗ b ) ⊗ c a ⊗ ( b ⊗ c )( b ⊗ a ) ⊗ c ( b ⊗ c ) ⊗ ab ⊗ ( a ⊗ c ) b ⊗ ( c ⊗ a ) α a,b,c β a,b ⊗ c β a,b ⊗ c α b,a,c α b,c,a b ⊗ β a,c a ⊗ ( b ⊗ c ) ( a ⊗ b ) ⊗ ca ⊗ ( c ⊗ b ) c ⊗ ( a ⊗ b )( a ⊗ c ) ⊗ b ( c ⊗ a ) ⊗ b α − a,b,c a ⊗ β b,c β a ⊗ b,c α − a,c,b α − c,a,b β a,c ⊗ b All of the above examples of monoidal categories are in fact braided monoidal categories. Some-times the braiding β is its own inverse, which finally brings us to: Definition A.1.7. A symmetric monoidal category is a braided monoidal category ( C , ⊗ , I )such that for any two objects a and b of C , the braiding β is its own inverse, meaning that β b,a β a,b = 1 a ⊗ b . All of the above examples are in fact symmetric monoidal categories. What about maps betweenvarious such categories?
Definition A.1.8.
Let ( C , ⊗ , I C ) and ( D , ⊗ , I D ) be monoidal categories. A (lax) monoidal func-tor is a functor F : C → D such that:(1) there exists an morphism µ : I D → F ( I C ) and(2) for every pair of objects a and b of C , there exists a natural transformation µ a,b : F ( a ) ⊗ F ( b ) → F ( a ⊗ b )which make the following diagrams commute: ( F ( a ) ⊗ F ( b )) ⊗ F ( c ) F ( a ) ⊗ ( F ( b ) ⊗ F ( c )) F ( a ⊗ b ) ⊗ F ( c ) F ( a ) ⊗ F ( b ⊗ c ) F (( a ⊗ b ) ⊗ c ) F ( a ⊗ ( b ⊗ c )) a ′ µ a,b ⊗ F ( c ) F ( a ) ⊗ µ b,c µ a ⊗ b,c µ a,b ⊗ c F ( a ) ( a ) ⊗ I D F ( a ) F ( a ) ⊗ F ( I C ) F ( a ⊗ I C ) I D ⊗ F ( a ) F ( I C ) ⊗ F ( a ) F ( a ) F ( I C ⊗ a ) F ( a ) ⊗ µ F ( ρ a ) ρ ′ F ( a ) µ a,I C µ ⊗ F ( a ) µ I C ,a λ ′ F ( a ) F ( λ a ) The monoidal functor F is called strong if the morphism µ and natural transformation µ − , − arean isomorphism and natural isomorphism, respectively, and the monoidal functor F is called oplax or colax if F : C op → D op is a lax monoidal functor. Definition A.1.9.
A (possibly lax or oplax) monoidal functor F : C → D is a braided monoidalfunctor if C and D are braided monoidal categories and the following diagram commutes. F ( a ) ⊗ F ( b ) F ( b ) ⊗ F ( a ) F ( a ⊗ b ) F ( b ⊗ a ) µ a,b µ b,a β ′ F ( β ) Definition A.1.10.
A (possibly lax or oplax) symmetric monoidal functor is a braidedmonoidal functor F : C → D between symmetric monoidal categories. Definition A.1.11.
Given monoidal functors F : ( C , ⊗ , C ) → ( D , ⊗ , D ) and G : ( C , ⊗ , C ) → ( D , ⊗ , D ), a monoidal natural transformation α : F ⇒ G is a transformation α : F ⇒ G suchthat the following diagrams commute. F ( x ) ⊗ F ( y ) G ( x ) ⊗ G ( y ) F ( x ⊗ y ) G ( x ⊗ y ) I D F ( I C ) G ( I C ) µ α I C µ ′ α x ⊗ α y α x ⊗ y µ x,y µ ′ x,y A monoidal transformation α is braided monoidal or symmetric monoidal if the functors F and G are braided monoidal or symmetric monoidal, respectively. A.1.2 Colimits
Definition A.1.12.
Given an arbitrary category C , a diagram in the category C is given by afunctor F : D → C . 153ere, the category D serves as the ‘shape’ of the diagram in the category C . Definition A.1.13.
Given a diagram F : D → C in C , a limit of the diagram F , denoted lim F ,is given by an object which we also denote by lim F , together with with a family of morphisms φ i : lim F → F ( d i ) for every i ∈ D such that for any morphism f : d i → d j in D , we have that F ( f ) φ i = φ j . Moreover, the object lim F together with the family of morphisms { φ i : i ∈ D } are universal among such, meaning that given another object c together with a family of morphisms ψ i : c → F ( d i ) such that F ( f ) ψ i = ψ j , there exists a unique morphism θ : c → lim F such that ψ i = φ i θ for every i ∈ D . A limit is finite if the category D is finite. Then, a colimit is just alimit in the opposite category, meaning that given a functor F : D → C , a colimit of F , denoted bycolim F , is given by a limit of F op : D op → C op .Limits and colimits are only unique up to a unique isomorphism, hence the usage of the indefinitearticles ‘a’ and ‘an’ rather than the definite article ‘the’.We largely work with finite colimits in this thesis, and so the examples presented next will beof such. The most famous examples of finite colimits are easily the following:(1) initial objects(2) binary coproducts(3) coequalizers(4) pushoutsIn fact, a category C has finite colimits iff C has an initial object and pushouts iff C has binarycoproducts and coequalizers. We discuss pushouts in the next section, but let us briefly introducethe other three famous finite colimits. Definition A.1.14. An initial object F : ∅ → C .Unraveling what this means, it means that an initial object is an object 0 in C together with anempty family of morphisms satisfying no properties such that for any other object c together withan empty family of morphisms satisfying no properties, there exists a unique morphism ! c : 0 → c which satisfies no properties. In other words, it is just an object 0 of C with a unique morphism toany other object of C . If C = Set , then 0 = ∅ , and surely there is a unique function ! S : ∅ → S forany set S . Definition A.1.15. A binary coproduct is a colimit of a functor F : { ⋆ ⋆ } → C where { ⋆ ⋆ } denotes the category with two objects and only identity morphisms.Unraveling what this means, given two objects c and c in C , a binary coproduct of c and c is an object which we denote as c + c together with two morphisms φ c : c → c + c and φ c : c → c + c such that for any other object c also with morphisms ψ : c → c and ψ : c → c ,154here exists a unique morphism θ : c + c → c such that ψ i = θφ i for i = 1 , c c c + c c φ φ ∃ ! θψ ψ In other words, such an object c + c and morphisms ( φ , φ ) are initial among such. A typicalexample of a binary coproduct is the disjoint union of two sets together with the natural injectionmaps of each set into the disjoint union, or the direct sum of two vector spaces V and V togetherwith the maps ((1 V , , (0 , V )) into the direct sum. Definition A.1.16. A coequalizer is a colimit of a functor F : { ⋆ ⇒ ⋆ } → C where { ⋆ ⇒ ⋆ } denotes the category with two objects, two morphisms from one object to the other, and twoidentity morphisms.Unraveling what this means, given two morphisms f, g : c → c ′ in C , a coequalizer of f and g is an object coeq( f, g ) together with a morphism φ : c ′ → coeq( f, g ) such that φf = φg. Such anobject and morphism are universal among such, meaning that given another object ˆ c and morphism ψ : c ′ → ˆ c such that ψf = ψg , there exists a unique morphism θ : coeq( f, g ) → ˆ c such that θφ = ψ . c c ′ coeq( f, g )ˆ c fg φψ ∃ ! θ In other words, such an object coeq( f, g ) and morphism θ are initial among such. An example ofa coequalizer is in the category Grp : given any group homomorphism f : G → H , there is always aunique group homomorphism 0 : G → H which sends every element of G to the identity element of H , in which case coeq( f,
0) = ker( f ). Definition A.1.17. A span in any category C is a diagram of the form: ba a i o A pushout is a colimit of a span, or equivalently, a colimit of a functor F : { ⋆ ← ⋆ → ⋆ } → C where { ⋆ ← ⋆ → ⋆ } denotes the category with three objects and two non-identity morphisms witha common source and distinct targets. 155nraveling what this means, a pushout of the above span is an object a + b a together with apair of maps j : a → a + b a and k : a → a + b a making the induced square commute, meaningthat ji = ko . Such an object and pair of maps are universal among such, meaning that givenanother object q and maps j ′ : a → q and k ′ : a → q such that j ′ i = k ′ o , there exists a unique ψ : a + b a → q such that j ′ = ψj and k ′ = ψk . ba a a + b a q i oj k ! ψj ′ k ′ In other words, a pushout is initial among such triples ( j ′ , k ′ , q ).We compose cospans by taking pushouts. In other words, given two composable cospans ba a a b ′ a i o i ′ o ′ we take the pushout of the span formed by the morphisms o and i ′ ba a b ′ a b + a b ′ i o i ′ o ′ j kji ko ′ and then the resulting cospan is given by taking the composite of the outer morphisms leading upto the apex. b + a b ′ a a ji ko ′ A.2 Double categories
Definition A.2.1.
Given a category A with finite limits, a category internal to A consists of:1561) an object of objects a ∈ A (2) an object of morphisms a ∈ A (3) source and target assigning morphisms s, t : a → a (4) an identity assigning morphism i : a → a (5) a composition assigning morphism c : a × a a → a such that the following square is a pullback a × a a a a a p p st and that the following diagrams commute: a a a a ii ts which specifies the source and target of an identity morphism, a × a a a × a a a a a a a a cp ss cp tt which say that the source and target of a composite of morphisms are the source and target of thefirst and second morphisms, respectively, a × a a × a a a × a a a × a a a × cc × cc which says that composition of morphisms is strictly associative, and a × a a a × a a a × a a a i × a × a icp p which says how the left and right unit laws are compatible with composition.157n the previous and following definitions, we do not really need all finite limits; it is enough forthe stated pullbacks to exist. Definition A.2.2.
Any 2-category (see Definition A.3.2) has an underlying category with the sameobjects and morphisms, and we say that a 2-category has finite limits if its underlying categorydoes. Given a 2-category A with finite limits, a pseudocategory object in A consists of thesame data as a category object internal to the underlying category of A , except that the followingdiagrams commute up to isomorphism. a × a a × a a a × a a a × a a a α ⇒ × cc × cc a × a a a × a a a × a a a λ ⇒ ρ ⇒ i × a × a icp p The isomorphisms α, λ and ρ satisfy the pentagon and triangle identities of a monoidal category. Definition A.2.3. A strict double category is a category object internal to Cat (which is acategory with finite limits).
Definition A.2.4. A (pseudo) double category is a pseudocategory object internal to Cat (which is a 2-category with finite limits).In a nutshell, a strict double category is a category internal to the category
Cat of categories andfunctors, similar to how an ordinary small category is a category internal to the category
Set of setsand functions. What this means is that instead of having a set of objects and a set of morphisms,we will instead have a category of objects and a category or morphisms. There are various kindsof double categories one can consider depending on how strict we are with the internalizations;whereas
Set is a mere category,
Cat is a 2-category which allows us to consider a triple composite ofmorphisms up to a 2-morphism. Internalizing a category object in the ordinary category
Cat leadsto what are typically known as strict double categories, whereas internalizing a category objectin
Cat viewed as a 2-category, also known as a pseudocategory object, leads to pseudo doublecategories, where the left and right unitors and associators no longer hold on-the-nose but only upto isomorphism. These latter pseudo double categories are the ones that we are primarily interestedin. It is helpful to have the following picture in mind. A double category has 2-morphisms shapedlike this:
A BC D ⇓ a M gf N
A, B, C and D objects or , f and g vertical 1-morphisms , M and N horizontal1-cells and a a . Note that a vertical 1-morphism is a morphism between 0-cells anda 2-morphism is a morphism between horizontal 1-cells. We denote both vertical 1-morphisms andhorizontal 1-cells using single arrows, namely ‘ → ’. We follow the notation of Shulman [37] with thefollowing definitions. Definition A.2.5. A pseudo double category D , or double category for short, consists of acategory of objects D and a category of arrows D with the following functors U : D → D S, T : D ⇒ D ⊙ : D × D D → D (where the pullback is taken over D T −→ D S ←− D )such that S ( U A ) = A = T ( U A ) S ( M ⊙ N ) = SNT ( M ⊙ N ) = T M equipped with natural isomorphisms α : ( M ⊙ N ) ⊙ P ∼ −→ M ⊙ ( N ⊙ P ) λ : U B ⊙ M ∼ −→ Mρ : M ⊙ U A ∼ −→ M such that S ( α ) , S ( λ ) , S ( ρ ) , T ( α ) , T ( λ ) and T ( ρ ) are all identities and that the coherence axiomsof a monoidal category are satisfied. Following the notation of Shulman, objects of D are called or objects and morphisms of D are called vertical 1-morphisms . Objects of D arecalled horizontal 1-cells and morphisms of D are called . The morphisms of D ,which are vertical 1-morphisms, will be denoted f : A → C and we denote a horizontal 1-cell M with S ( M ) = A, T ( M ) = B by M : A → B . Then a 2-morphism a : M → N of D with S ( a ) = f, T ( a ) = g would look like: A BC D ⇓ a M gf N
The horizontal and vertical composition of 2-morphisms together obey a ‘middle-four’ interchangelaw, or simply, interchange law, expressing the compatibility of horizontal and vertical composition159ith each other. Specifically, given four 2-morphisms as such:
A BC D B ED F ⇓ a ⇓ bC DG H D FH I ⇓ a ′ ⇓ b ′ M gf N O hg PN g ′ f ′ Q P h ′ g ′ R the following equality holds, where ⊙ denotes horizontal composition and juxtaposition denotesvertical composition. ( a ′ ⊙ b ′ )( a ⊙ b ) = ( a ′ a ) ⊙ ( b ′ b )The key difference between a strict double category and a pseudo double category is that ina pseudo double category, horizontal composition is associative and unital only up to naturalisomorphism. The natural isomorphisms α, λ and ρ are identities in a strict double category. Letus look at a few examples.If C is any category, there exists a strict double category Sq( C ), where ‘Sq’ denotes ‘square’,which has:(1) objects given by those of C ,(2) vertical 1-morphisms given by morphisms of C ,(3) horizontal 1-cells also given by morphisms of C , and(4) 2-morphisms as commutative squares in C .Composition of horizontal 1-cells coincides with composition of morphisms in C and both the hori-zontal and vertical composite of 2-morphisms is given by composing the edges of the commutativesquares.If C is a category with pushouts, then an example of a pseudo double category, and probablythe most important example of a double category in this thesis, is given by C sp ( C ), where “ C sp ”denotes “cospan”, which has:(1) objects as those of C ,(2) vertical 1-morphisms as morphisms of C ,(3) horizontal 1-cells as cospans in C , and 1604) 2-morphisms as maps of cospans in C which are given by commutative diagrams of the form: a ba ′ b ′ a a ′ i o gf i o h Composition of vertical 1-morphisms and the vertical composite of 2-morphisms is given by com-position of morphisms in C , and composition of horizontal 1-cells and the horizontal composite of2-morphisms is given by pushouts in C a ba ′ b ′ a a ′ ⊙ a ca ′ c ′ a a ′ = a b + a ca ′ b ′ + a ′ c ′ a a ′ i o gf i o h i o kh i o l Jψi Jψo g + h kf Jψi Jψo l where ψ is the natural map into a coproduct and J is the natural map from a coproduct to apushout, for example, ψ : b → b + c and J : b + c → b + a c . More about this double category andothers similar to it may be found in the work of Niefield [32].The pseudo double categories that we are interested in all share a certain ‘lifting’ propertybetween the vertical 1-morphisms and horizontal 1-cells. Definition A.2.6.
Let D be a double category and f : A → B a vertical 1-morphism. A compan-ion of f is a horizontal 1-cell ˆ f : A → B together with 2-morphisms A BB B b ff U B ⇓ and A AA B U A f b f ⇓ such that the following equations hold. A AA BB B f f U A U B ⇓⇓ b f = A AB B f fU A U B ⇓ U f and AA AB BBA B f U A ˆ f ˆ f ⇓ ⇓⇓ λ ˆ f ˆ f U B = A A BA B U A ˆ f f ⇓ ρ f (A.1)A conjoint of f , denoted ˇ f : B → A , is a companion of f in the double category D h · op obtainedby reversing the horizontal 1-cells, but not the vertical 1-morphisms, of D . Definition A.2.7.
We say that a double category is fibrant if every vertical 1-morphism has botha companion and a conjoint and isofibrant if every vertical 1-isomorphism has both a companionand a conjoint. 161he property of isofibrancy in a double category is key as we are primarily interested in symmet-ric monoidal double categories and bicategories, and it is precisely the property of isofibrancy thatallows us to lift the portion of the monoidal structure of a symmetric monoidal double categorythat resides in the category of objects, such as the unitors, associators and braidings, to obtain asymmetric monoidal bicategory using a result of Shulman [37].Next, we define the kinds of maps between double categories.
Definition A.2.8.
Let A and B be pseudo double categories. A lax double functor is a functor F : A → B that takes items of A to items of B of the corresponding type, respecting verticalcomposition in the strict sense and the horizontal composition up to an assigned comparison φ .This means that we have functors F : A → B and F : A → B such that the following equationsare satisfied: S ◦ F = F ◦ ST ◦ F = F ◦ T Sometimes for brevity, we will omit the subscripts and simply say F ; as to whether we mean F or F will be clear from context. Furthermore, every object a is equipped with a special globular 2-morphism φ a : 1 F ( a ) → F (1 a ) (the identity comparison ), and every composable pair of horizontal1-cells N ⊙ N is equipped with a special globular 2-morphism φ ( N , N ) : F ( N ) ⊙ F ( N ) → F ( N ⊙ N ) (the composition comparison ), in a coherent way. This means that the followingdiagrams commute.(1) For a horizontal composite, β ⊙ α , F ( A ) | F ( N ) / / (cid:15) (cid:15) F ( α ) F ( B ) (cid:15) (cid:15) | F ( N ) / / F ( β ) F ( C ) (cid:15) (cid:15) F ( A ) | F ( N ) / / φ ( N ,N )1 (cid:15) (cid:15) F ( B ) | F ( N ) / / F ( C ) (cid:15) (cid:15) F ( A ′ ) | F ( N ) / / φ ( N ,N )1 (cid:15) (cid:15) F ( B ′ ) | F ( N ) / / F ( C ′ ) (cid:15) (cid:15) = F ( A ) | F ( N ⊙ N ) / / (cid:15) (cid:15) F ( β ⊙ α ) F ( C ) (cid:15) (cid:15) F ( A ′ ) | F ( N ⊙ N ) / / F ( C ′ ) F ( A ′ ) | F ( N ⊙ N ) / / F ( C ′ ) . (A.2)(2) For a horizontal 1-cell N : A → B , the following diagrams are commutative (under horizontalcomposition). F ( N ) ⊙ F ( A ) F ( N ) F ( N ) ⊙ F (1 A ) F ( N ⊙ A ) 1 F ( B ) ⊙ F ( N ) F (1 B ) ⊙ F ( N ) F ( N ) F (1 B ⊙ N ) ⊙ φ A F ρρ F ( N ) φ ( N, A ) φ B ⊙ φ (1 B , N ) λ F ( N ) F λ (3) For consecutive horizontal 1-cells N , N and N , the following diagram is commutative.162 F ( N ) ⊙ F ( N )) ⊙ F ( N ) a ′ / / φ ( N ,N ) ⊙ (cid:15) (cid:15) F ( N ) ⊙ ( F ( N ) ⊙ F ( N )) ⊙ φ ( N ,N ) (cid:15) (cid:15) F ( N ⊙ N ) ⊙ F ( N ) φ ( N ⊙ N ,N ) (cid:15) (cid:15) F ( N ) ⊙ F ( N ⊙ N ) φ ( N ,N ⊙ N ) (cid:15) (cid:15) F (( N ⊙ N ) ⊙ N ) F a / / F ( N ⊙ ( N ⊙ N ))We say the double functor F is strict if the comparison constraints φ a and φ N ,N are identi-ties, strong if the comparison constrains are globular isomorphisms, pseudo if the comparisonconstraints are isomorphisms, and oplax if the comparison constraints go in the opposite direction.We can also consider maps between maps of double functors, also known as double transforma-tions. These are only used in Section 3.4 of this thesis. Definition A.2.9. A double transformation α : F ⇒ G between two double functors F : A → B and G : A → B consists of two natural transformations α : F ⇒ G and α : F ⇒ G such that forall horizontal 1-cells M we have that S ( α M ) = α S ( M ) and T ( α M ) = α T ( M ) and for composablehorizontal 1-cells M and N , we have that F ( a ) F ( b ) F ( c ) G ( a ) G ( c ) F ( a ) F ( c ) = F ( b ) G ( a ) G ( b ) G ( c ) ⇓ F M,N ⇓ α M ⊙ N F ( a ) G ( a ) F ( c ) G ( c ) ⇓ α M ⇓ α N ⇓ G M,N α a α c F ( M ) F ( M ⊙ N ) F ( N ) G ( M ⊙ N ) α a α b G ( N ) G ( M ) G ( M ⊙ N ) F ( M ) F ( N ) α c F ( a ) F ( a ) G ( a ) G ( a ) F ( a ) F ( a ) = G ( a ) G ( a ) ⇓ F U ⇓ α U a F ( a ) G ( a ) F ( a ) G ( a ) ⇓ U α a ⇓ G U α a α a U F ( a ) F ( U a ) G ( U a ) α a U G ( a ) G ( U a ) U F ( a ) α a We call α the object component and α the arrow component of the double transformation α . A.2.1 Monoidal double categories
Let
Dbl denote the 2-category of double categories, double functors and double transformations.One can check that
Dbl has finite products, and in any 2-category with finite products we candefine a ‘pseudomonoid’ or a ‘weak’ monoid, which is a categorified analogue of a monoid in which163he left and right unitors and associators are not identities but natural isomorphisms. It is the2-categorical structure of
Dbl , or more generally, any 2-category with finite limits, that enablesus to do this. For example, a pseudomonoid in
Cat is a monoidal category. We are primarilyconcerned with the (weak) monoidal double categories in which the associators and left and rightunitors are natural isomorphisms.
Definition A.2.10.
Let ( C , ⊗ , I ) be a monoidal category. A monoid internal to C consists ofan object M ∈ C together with a morphism m : M ⊗ M → M for multiplication and a morphism i : I → M for the multiplicative identity satisfying the associative law: ( M ⊗ M ) ⊗ M M ⊗ ( M ⊗ M ) M ⊗ MM ⊗ M M α ⊗ mm ⊗ m m and left and right unit laws: I ⊗ M M M ⊗ IM ⊗ M λ ρi ⊗ m ⊗ i A pseudomonoid internal to a monoidal 2-category ( C , ⊗ , I ) consists of an object M togetherwith a morphism m : M ⊗ M → M and a morphism i : I → M such that the above diagramscommute up to specified 2-isomorphisms: ( M ⊗ M ) ⊗ M M ⊗ ( M ⊗ M ) M ⊗ MM ⊗ M M ⇒ A α ⊗ mm ⊗ m m I ⊗ M M M ⊗ IM ⊗ M ⇒ L ⇒ R λ ρi ⊗ m ⊗ i Furthermore, the 2-isomorphisms
A, L and R are required to satisfy two equations which can befound in the work of Day and Street [19]. 164 efinition A.2.11. A braided pseudomonoid is a pseudomonoid M equipped with the extrastructure of a braiding isomorphism β : ⊗ ∼ = ⊗ ◦ t where t is the ‘twist’ isomorphism t : M ⊗ M → M ⊗ M that together with the associators make the usual hexagons of a braided monoidal category com-mute. A symmetric pseudomonoid is a braided pseudomonoid such that the braiding isomor-phism β : ⊗ ∼ = ⊗ ◦ t is self-inverse. Definition A.2.12. A monoidal double category is a pseudomonoid in the monoidal 2-category Dbl .Explicitly, a monoidal double category is a double category equipped with double functors ⊗ : D × D → D and I : ∗ → D where ∗ is the terminal double category, along with invertible doubletransformations called the associator : A : ⊗ ◦ (1 D × ⊗ ) ⇒ ⊗ ◦ ( ⊗ × D ) , left unitor : L : ⊗ ◦ (1 D × I ) ⇒ D , and right unitor : R : ⊗ ◦ ( I × D ) ⇒ D satisfying the pentagon axiom and triangle axioms of a monoidal category.This is a very nice and compact definition which encapsulates the structure of a monoidal doublecategory. Unraveling this a bit, this means that:(1) D and D are both monoidal categories.(2) If I is the monoidal unit of D , then U I is (coherently isomorphic to) the monoidal unit of D .(3) The functors S and T are strict monoidal, meaning that S ( M ⊗ N ) = SM ⊗ SN and T ( M ⊗ N ) = T M ⊗ T N and S and T also preserve the associativity and unit constraints.(4) We have globular isomorphisms χ : ( M ⊗ N ) ⊙ ( M ⊗ N ) ∼ −→ ( M ⊙ M ) ⊗ ( N ⊙ N )and µ : U A ⊗ B ∼ −→ ( U A ⊗ U B )which arise from weakly-commuting squares:165 D × D D ) × ( D × D D ) D × D D D × D D D × D D × D D D ⇒ µ ⇒ χ ⊙ × ⊙ ⊙⊗ × D ⊗⊗ U × U ⊗⊗ U expressing the weak commutativity of ⊗ with the functors U and ⊙ .These globular isomorphisms χ and µ make the following diagrams commute:(5) The following diagrams commute expressing that ⊗ : D × D → D is a pseudo double functor. (( M ⊗ N ) ⊙ ( M ⊗ N )) ⊙ ( M ⊗ N ) (( M ⊙ M ) ⊗ ( N ⊙ N )) ⊙ ( M ⊗ N )( M ⊗ N ) ⊙ (( M ⊗ N ) ⊙ ( M ⊗ N )) (( M ⊙ M ) ⊙ M ) ⊗ (( N ⊙ N ) ⊙ N )( M ⊗ N ) ⊙ (( M ⊙ M ) ⊗ ( N ⊙ N )) ( M ⊙ ( M ⊙ M )) ⊗ ( N ⊙ ( N ⊙ N )) α ⊙ χ χα ⊗ αχ ⊙ χ ( M ⊗ N ) ⊙ U C ⊗ D M ⊗ N ( M ⊗ N ) ⊙ ( U C ⊗ U D )( M ⊙ U C ) ⊗ ( N ⊙ U D ) ⊙ µρ ρ ⊗ ρ χ U A ⊗ B ⊙ ( M ⊗ N ) M ⊗ N ( U A ⊗ U B ) ⊙ ( M ⊗ N )( U A ⊙ M ) ⊗ ( U B ⊙ N ) µ ⊙ λ λ ⊗ λ χ (6) The following diagrams commute expressing the associativity isomorphism for ⊗ is a trans-formation of double categories. (( M ⊗ N ) ⊗ P ) ⊙ (( M ⊗ N ) ⊗ P ) ( M ⊗ ( N ⊗ P )) ⊙ ( M ⊗ ( N ⊗ P ))(( M ⊗ N ) ⊙ ( M ⊗ N )) ⊗ ( P ⊙ P ) ( M ⊙ M ) ⊗ (( N ⊗ P ) ⊙ ( N ⊗ P ))(( M ⊙ M ) ⊗ ( N ⊙ N )) ⊗ ( P ⊙ P ) ( M ⊙ M ) ⊗ (( N ⊙ N ) ⊗ ( P ⊙ P )) χχ ⊗ χ ⊗ χα ⊙ αα ( A ⊗ B ) ⊗ C U A ⊗ ( B ⊗ C ) U A ⊗ B ⊗ U C U A ⊗ U B ⊗ C ( U A ⊗ U B ) ⊗ U C U A ⊗ ( U B ⊗ U C ) µµ ⊗ µ ⊗ µU α α (7) The following diagrams commute expressing that the unit isomorphisms for ⊗ are transfor-mations of double categories. ( M ⊗ U I ) ⊙ ( N ⊗ U I ) M ⊙ N ( M ⊙ N ) ⊗ ( U I ⊙ U I )( M ⊙ N ) ⊗ U I r ⊙ r χ ⊗ ρr U A ⊗ I U A ⊗ U I U A µU r r ( U I ⊗ M ) ⊙ ( U I ⊗ N ) M ⊙ N ( U I ⊙ U I ) ⊗ ( M ⊙ N ) U I ⊗ ( M ⊙ N ) ℓ ⊙ ℓ χ λ ⊗ ℓ U I ⊗ A U I ⊗ U A U A µU ℓ ℓ Thus we define a monoidal double category to be a pseudomonoid object weakly internal to the2-category
Dbl of double categories, double functors and double transformations. In other words,a monoidal double category is a pseudomonoid internal to categories weakly internal to
Cat . Butbeware: this is not the same as a category weakly internal to the 2-category
MonCat of monoidalcategories, strong monoidal functors and monoidal natural transformations. In a monoidal doublecategory, the functors S and T are strict monoidal. In a category weakly internal to MonCat ,they would only need to be strong monoidal.
Definition A.2.13. A braided monoidal double category is a braided pseudomonoid internalto Dbl .This means that a braided monoidal double category is a monoidal double category categoryequipped with an invertible double transformation β : ⊗ ⇒ ⊗ ◦ τ called the braiding , where τ : D × D → D × D is the twist double functor sending pairs in theobject and arrow categories to the same pairs in the opposite order. The braiding is required tosatisfy the usual two hexagon identities [34, Sec. XI.1]. If the braiding is self-inverse we say that D is a symmetric pseudomonoid internal to Dbl and that D is a symmetric monoidal doublecategory .Unraveling this a bit, we get that a braided monoidal double category is a monoidal doublecategory such that: 1678) D and D are braided monoidal categories.(9) The functors S and T are strict braided monoidal functors.(10) The following diagrams commute expressing that the braiding is a transformation of doublecategories. ( M ⊙ M ) ⊗ ( N ⊙ N )( M ⊗ N ) ⊙ ( M ⊗ N ) ( N ⊙ N ) ⊗ ( M ⊙ M )( N ⊗ M ) ⊙ ( N ⊗ M ) χ β χβ ⊙ β U A ⊗ U B U B ⊗ U A U A ⊗ B U B ⊗ A β µ U β µ Finally, a symmetric monoidal double category is a braided monoidal double category D such that:(11) D and D are symmetric monoidal categories. A.2.2 Monoidal double functors and transformations
We also have maps between symmetric monoidal double categories, which, just as maps betweenordinary symmetric monoidal categories, can come in three flavors according to direction of thecomparison maps φ ( − , − ) . Definition A.2.14. A (strong) monoidal lax double functor F : C → D between monoidaldouble categories C and D is a lax double functor F : C → D such that • F and F are (strong) monoidal functors, meaning that there exists(1) an isomorphism ǫ : 1 D → F (1 C )(2) a natural isomorphism θ A,B : F ( A ) ⊗ F ( B ) → F ( A ⊗ B ) for all objects A and B of C (3) an isomorphism δ : U D → F ( U C )(4) a natural isomorphism ν M,N : F ( M ) ⊗ F ( N ) → F ( M ⊗ N ) for all horizontal 1-cells N and M of C such that the following diagrams commute: for objects A, B and C of C ,( F ( A ) ⊗ F ( B )) ⊗ F ( C ) α ′ / / θ A,B ⊗ (cid:15) (cid:15) F ( A ) ⊗ ( F ( B ) ⊗ F ( C )) ⊗ θ B,C (cid:15) (cid:15) F ( A ⊗ B ) ⊗ F ( C ) θ A ⊗ B,C (cid:15) (cid:15) F ( A ) ⊗ F ( B ⊗ C ) θ A,B ⊗ C (cid:15) (cid:15) F (( A ⊗ B ) ⊗ C ) F α / / F ( A ⊗ ( B ⊗ C ))168 ( A ) ⊗ D F ( A ) F ( A ) ⊗ F (1 C ) F ( A ⊗ C ) 1 D ⊗ F ( A ) F (1 C ) ⊗ F ( A ) F ( A ) F (1 C ⊗ A ) ⊗ ǫ F ( r A ) r F ( A ) θ A, C ǫ ⊗ θ C ,A ℓ F ( A ) F ( ℓ A ) and for horizontal 1-cells N , N and N of C ,( F ( N ) ⊗ F ( N )) ⊗ F ( N ) α ′ / / ν N ,N ⊗ (cid:15) (cid:15) F ( N ) ⊗ ( F ( N ) ⊗ F ( N )) ⊗ ν N ,N (cid:15) (cid:15) F ( N ⊗ N ) ⊗ F ( N ) ν N ⊗ N ,N (cid:15) (cid:15) F ( N ) ⊗ F ( N ⊗ N ) ν N ,N ⊗ N (cid:15) (cid:15) F (( N ⊗ N ) ⊗ N ) F α / / F ( N ⊗ ( N ⊗ N )) F ( N ) ⊗ U D F ( N ) F ( N ) ⊗ F ( U C ) F ( N ⊗ U C ) U D ⊗ F ( N ) F ( U C ) ⊗ F ( N ) F ( N ) F ( U C ⊗ N ) ⊗ δ F ( r N ) r F ( N ) ν N ,U C δ ⊗ ν U C ,N ℓ F ( N ) F ( ℓ N ) • S F = F S and T F = F T are equations between monoidal functors, and • the composition and unit comparisons φ ( N , N ) : F ( N ) ⊙ F ( N ) → F ( N ⊙ N ) and φ A : U F ( A ) → F ( U A ) are monoidal natural transformations. • The following diagrams commute expressing that θ and ν together constitute a transformationof double categories: ( F ( M ) ⊗ F ( N )) ⊙ ( F ( M ) ⊗ F ( N )) F ( M ⊗ N ) ⊙ F ( M ⊗ N )( F ( M ) ⊙ F ( M )) ⊗ ( F ( N ) ⊙ F ( N )) F (( M ⊗ N ) ⊙ ( M ⊗ N )) F ( M ⊙ M ) ⊗ F ( N ⊙ N ) F (( M ⊙ M ) ⊗ ( N ⊙ N )) χ ′ φ M ,M ⊗ φ N ,N φ M ⊗ N ,M ⊗ N F ( χ ) ν M ,M ⊙ ν N ,N ν M ⊙ M ,N ⊙ N F ( A ) ⊗ F ( B ) U F ( A ⊗ B ) U F ( A ) ⊗ U F ( B ) F ( U A ⊗ B ) F ( U A ) ⊗ F ( U B ) F ( U A ⊗ U B ) µφ A ⊗ φ B φ A ⊗ B µU θ A,B ν F ( A ) , F ( B ) The monoidal lax double functor is braided if F and F are braided monoidal functors and symmetric if they are symmetric monoidal functors, and lax monoidal or oplax monoidal ifinstead of the isomorphisms and families of natural isomorphisms in items (1)-(4), we merely havemorphisms and natural transformations going in the appropriate directions. If the double functor F : C → D is a double functor between isofibrant symmetric monoidal double categories, alsoknown as ‘symmetric monoidal framed bicategories’ [38], instead of θ and ν together constituting atransformation of double categories, it suffices that the comparison and unit constraints F M,N and F c be monoidal natural transformations. Definition A.2.15.
Given monoidal double functors ( F , φ ) , ( G , ψ ) : C → D , a monoidal doubletransformation α : F ⇒ G is a double transformation α such that both the object component α : F ⇒ G and arrow component α : F ⇒ G are monoidal natural transformations. Thismeans that the following equations hold: F ( a ) ⊗ F ( c ) F ( b ) ⊗ F ( d ) G ( a ⊗ c ) G ( b ⊗ d ) F ( a ⊗ c ) F ( b ⊗ d ) = G ( a ) ⊗ G ( c ) G ( b ) ⊗ G ( d ) ⇓ φ M,N ⇓ α M ⊗ N F ( a ) ⊗ F ( c ) G ( a ⊗ c ) F ( b ) ⊗ F ( d ) G ( b ⊗ d ) ⇓ α M ⊗ α N ⇓ ψ M,N φ a,c α a ⊗ c φ b,d α b ⊗ d F ( M ) ⊗ F ( N ) F ( M ⊗ N ) G ( M ⊗ N ) α a ⊗ α c ψ a,c G ( M ) ⊗ G ( N ) G ( M ⊗ N ) F ( M ) ⊗ F ( N ) α b ⊗ α d ψ b,d D D G (1 C ) G (1 C ) F (1 C ) F (1 C ) = G (1 C ) G (1 C ) ⇓ φ ⇓ α U C D D ⇓ ψ φ α C φ α C U D F ( U C ) G ( U C ) ψ G ( U C ) U D ψ A.3 Bicategories and 2-categories
Definition A.3.1. A bicategory C is a double category (see Definition A.2.5) C = ( C , C ) suchthat the category of objects C is discrete, meaning that C contains only identity morphisms. Ina bicategory, we refer to the objects of C , which are horizontal 1-cells, as morphisms .170nraveling this a bit, a bicategory C consists of:(1) a collection of objects a, b, c, d, . . . ,(2) for every pair of objects a and b , a category hom C ( a, b ), called the hom category of a and b , where objects are called morphisms from a to b and whose morphisms are called - morphisms ,(3) for every object a , a functor 1 a : 1 → hom C ( a, a ) which picks out the identity morphism forthe object a and for every triple of objects a, b and c , a functor ◦ : hom C ( a, b ) × hom C ( b, c ) → hom C ( a, c ) for composition,(4) for every pair of objects a and b and morphism f ∈ hom C ( a, b ), a natural isomorphism λ : 1 b f ⇒ f called the left unitor and a natural isomorphism ρ : f a ⇒ f called the right unitor ,(5) for every quadruple of objects a, b, c and d , a natural isomorphism α : ◦ (1 × ◦ ) ⇒ ◦ ( ◦ × ◦ (1 × ◦ ) , ◦ ( ◦ ×
1) : hom C ( a, b ) × hom C ( b, c ) × hom C ( c, d ) → hom C ( a, d )such that the left and right unitors satisfy the triangle identity and the associator satisfies thepentagon identity. Definition A.3.2. A is a bicategory C in which the left and right unitors andassociators are identity 2-morphisms.Equivalently, a 2-category is a strict double category in which the category of objects is discrete.The primordial example of a 2-category is Cat , the 2-category of categories, functors and naturaltransformations: natural transformations make up the morphisms in each hom category hom C ( a, b ).A 2-category is sometimes called a ‘strict’ 2-category and a bicategory a ‘weak’ 2-category. Strict 2-categories along with double categories were first discovered by Ehresmann [20, 21], and bicategoriesare due to B´enabou [13]. Definition A.3.3.
Given a 2-morphism α : f ⇒ g : c → d and a morphism h : b → c in a 2-category C : b c d ⇓ α h fg left whiskering of α by h , denoted by 1 h ⊙ α , is given by the horizontal composite of the2-morphism α with the identity 2-morphism of h : b c d ⇓ h ⇓ α hh fg Right whiskering is defined analogously.
A.3.1 Pseudofunctors and pseudonatural transformations
Definition A.3.4.
Given bicategories C and D , a pseudofunctor F : C → D consists of:(1) for each object c ∈ C , an object F ( c ) ∈ D ,(2) for each category C ( c, c ′ ), a functor F : C ( c, c ′ ) → D ( F ( c ) , F ( c ′ )),(3) for each object c ∈ C , a 2-isomorphism F c : 1 F ( c ) ⇒ F (1 c )(4) for every triple of objects a, b, c ∈ C and pair of composable morphisms f : a → b and g : b → c in C , a 2-isomorphism F f,g : F ( f ) F ( g ) ⇒ F ( f g ) natural in f and g such that the following diagrams commute: ( F ( f ) F ( g )) F ( h ) F ( f )( F ( g ) F ( h )) F ( fg ) F ( h ) F ( f ) F ( gh ) F (( fg ) h ) F ( f ( gh )) a ′ F f,g ⊙ F ( h ) F ( f ) ⊙ F g,h F fg,h F f,gh F ( a ) F ( f )1 F ( a ) F ( f ) F ( f ) F (1 a ) F ( f a ) 1 F ( b ) F ( f ) F (1 b ) F ( f ) F ( f ) F (1 b f ) F ( f ) ⊙ F a F ( r f ) r ′ F ( f ) F f, a F b ⊙ F ( f ) F b ,f ℓ ′ F ( f ) F ( ℓ f ) Here, all of the arrows in the diagrams are given by 2-morphisms in D , a, ℓ, r denote the associator,left and right unitors for morphism composition in C , similarly a ′ , ℓ ′ , r ′ denote the associator,left and right unitors for morphism composition in D , juxtaposition is used to denote morphismcomposition in both C and D and ⊙ denotes whiskering in D (see Definition A.3.3). Definition A.3.5.
Given two pseudofunctors
F, G : A → B , a pseudonatural transformation σ consists of: 1721) for each object a ∈ A , a morphism σ a : F ( a ) → G ( a ) in B and(2) for each morphism f : a → b in A , an invertible natural 2-morphism σ f : G ( f ) σ a ∼ −→ σ b F ( f )in B which is compatible with composition and identities.Let [ A , Cat ] ps denote the 2-category of pseudofunctors, pseudonatural transformations and‘modifications’ from an ordinary category A viewed as a 2-category with trivial 2-morphisms. Wecall [ A , Cat ] ps the , as an indexed category is a contravari-ant pseudofunctor into Cat . A lax monoidal pseudofunctor F : A → B between monoidalbicategories [39] is then a pseudofunctor equipped with pseudonatural transformations with com-ponents µ a,b : F ( a ) ⊗ F ( b ) ∼ −→ F ( a ⊗ b )and µ : 1 B → F (1 A )together with coherent invertible modifications for associativity and unitality. This is also knownas a weak monoidal pseudofunctor. A symmetric lax monoidal pseudofunctor is then a laxmonoidal pseudofunctor between symmetric monoidal bicategories together with invertible modifi-cations F ( β ) µ a,b ∼ −→ µ b,a β ′ . 173 ibliography [1] D. Andrieux, Bounding the coarse graining error in hidden Markov dynamics, Appl. Math. Lett. (2012), 1734–1739. Available at arXiv:1104.1025. (Referred to on page 107.)[2] J. C. Baez and K. Courser, Coarse-graining open Markov processes, Theor. Appl. Categ. Theor. Appl. Categ. Theor. Appl. Categ. Theor. Appl.Categ. (2018), 1158–1222. Available at arXiv:1504.05625. (Referred to on page 2, 4, 6, 10,27, 87, 99, 100, 107.)[8] J. C. Baez, B. Fong and B. S. Pollard. A compositional framework for Markov processes. J.Math. Phys. (2016). Available at arXiv:1508.06448. (Referred to on page 4, 6, 87, 107, 109,110, 113.)[9] J. C. Baez and J. Master, Open Petri nets. Available at arXiv:1808.05415. (Referred to on page4, 10, 11, 30, 87, 91, 93.)[10] J. C. Baez and B. S. Pollard, A compositional framework for reaction networks, Rev. Math.Phys. (2017), 1750028. Available at arXiv:1704.02051. (Referred to on page 4, 6, 30, 87, 89,101, 109, 110, 113, 132, 138.)[11] J. C. Baez and M. Shulman, Lectures on n -categories and cohomology, in Towards HigherCategories , J. Baez and P. May (eds.), Springer, Berlin, 2010, 1–68. Available at arXiv:0608420.(Referred to on page 144.) 17412] J. C. Baez, D. Weisbart and A. Yassine, Open systems in classical mechanics. In preparation.(Referred to on page 6.)[13] J. B´enabou, Introduction to bicategories,
Reports of the Midwest Category Seminar , J.B´enabou et al (eds.), Springer Lecture Notes in Mathematics , New York, 1967, pp. 1–77.(Referred to on page 171.)[14] P. Buchholz, Exact and ordinary lumpability in finite Markovchains, Journal of Applied Probability (1994), 59–75. Available athttp://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.1632. (Referred to on page107, 113, 114, 117.)[15] D. Cicala, Rewriting structured cospans. Available at arXiv:2001.09029. (Referred to on page26.)[16] D. Cicala and C. Vasilakopoulou, On adjoints and fibrations. In preparation. (Referred to onpage 73.)[17] F. Clerc, H. Humphrey and P. Panangaden, Bicategories of Markov processes, in Models,Algorithms, Logics and Tools , Lecture Notes in Computer Science , Springer, Berlin,2017, pp. 112–124. (Referred to on page 110.)[18] K. Courser, A bicategory of decorated cospans,
Theor. Appl. Categ. (2017), 995–1027. Alsoavailable at arXiv:1605.08100. (Referred to on page 3, 13, 22, 103, 126.)[19] B. Day and R. Street, Monoidal bicategories and Hopf algebroids, Adv. Math. (1997),99–157. (Referred to on page 164.)[20] C. Ehresmann, Cat´egories structur´ees III: Quintettes et applications covariantes,
Cah. Top.G´eom. Diff. (1963), 1–22. (Referred to on page 171.)[21] C. Ehresmann, Cat´egories et Structures,
Dunod, Paris, 1965. (Referred to on page 171.)[22] K.-J. Engel and R. Nagel,
One-Parameter Semigroups for Linear Evolution Equations ,Springer, Berlin, 1999. (Referred to on page 117.)[23] B. Fong, Decorated cospans,
Theor. Appl. Categ. (2015), 1096–1120. Available atarXiv:1502.00872. (Referred to on page 4, 6, 43, 144.)[24] B. Fong, The Algebra of Open and Interconnected Systems , Ph.D. thesis, University of Oxford,2016. Available at arXiv:1609.05382. (Referred to on page 107.)[25] L. de Francesco Albasini, N. Sabadini and R. F. C. Walters, The compositional construction ofMarkov processes,
Appl. Cat. Str. (2011), 425–437. Available at arXiv:0901.2434. (Referredto on page 111.)[26] M. Grandis and R. Par`e, Intercategories, Theor. Appl. Categ. (2015), 1215–1255. Availableat arXiv:1412.0144. (Referred to on page 144.)17527] J. Gray, Fibred and cofibred categories, in Proceedings of the Conference on CategoricalAlgebra: La Jolla 1965 , S. Eilenberg et al (eds.), New York, 1966, pp. 21–83. (Referred to onpage 73.)[28] L. W. Hansen and M. Shulman, Constructing symmetric monoidal bicategories functorially.Available at arXiv:1901.09240. (Referred to on page 102, 104, 110, 142, 143.)[29] C. Hermida,
Fibrations, logical predicates and indeterminates , Ph.D. Thesis, University ofEdinburgh, 1993. (Referred to on page 72.)[30] S. Lack and P. Soboci´nski, Adhesive categories, in
International Conference on Foundations ofSoftware Science and Computation Structures: FOSSACS 2004 , I. Walukiewicz (ed.), Springer,Berlin, 2004, pp. 273–288. Available at http://users.ecs.soton.ac.uk/ps/papers/adhesive.pdf.(Referred to on page 121.)[31] S. Lack and P. Soboci´nski, Toposes are adhesive, in
Graph Transformations: ICGT 2006 , A.Corradi et al (eds.), Lecture Notes in Computer Science , Springer, Berlin, pp. 184–198.Available at http://users.ecs.soton.ac.uk/ps/papers/toposesAdhesive.pdf. (Referred to on page121.)[32] S. Niefield, Span, cospan, and other double categories,
Theor. Appl. Categ. (2012), 729–742.Available at arXiv:1201.3789. (Referred to on page 22, 161.)[33] J. R. Norris, Markov Chains , Cambridge U. Press, Cambridge, 1998. (Referred to on page112, 117.)[34] S. Mac Lane,
Categories for the Working Mathematician , Springer, Berlin, 2013. (Referred toon page 14, 167.)[35] J. Moeller and C. Vasilakopoulou, Monoidal Grothendieck construction,
Theor. Appl. Categ. (2020), 1159–1207. Available at arXiv:1809.00727. (Referred to on page 6, 7, 14, 72.)[36] B. S. Pollard, Open Markov Processes and Reaction Networks , Ph.D. thesis, University ofCalifornia at Riverside, 2017. Available at arXiv:1709.09743. (Referred to on page 107.)[37] M. Shulman, Constructing symmetric monoidal bicategories. Available at arXiv:1004.0993.(Referred to on page 3, 94, 95, 110, 130, 135, 159, 162.)[38] M. Shulman, Framed bicategories and monoidal fibrations,
Theor. Appl. Categ. (2008),650–738. Available at arXiv:0706.1286. (Referred to on page 69, 70, 72, 82, 105, 170.)[39] M. Stay, Compact closed bicategories, Theor. Appl. Categ.31