aa r X i v : . [ m a t h . C T ] J a n UNIVERSITY OF CALIFORNIARIVERSIDEThe Grothendieck Construction in Categorical Network TheoryA Dissertation submitted in partial satisfactionof the requirements for the degree ofDoctor of PhilosophyinMathematicsbyJoseph Patrick MoellerDecember 2020Dissertation Committee:
Dr. John C. Baez, ChairpersonDr. Wee Liang GanDr. Carl Mautneropyright byJoseph Patrick Moeller2020 he Dissertation of Joseph Patrick Moeller is approved:
Committee Chairperson
University of California, Riverside cknowledgments
First of all, I owe all of my achievements to my wife, Paola. I couldn’t have gotten herewithout my parents: Daniel, Andrea, Tonie, Maria, and Luis, or my siblings: Danielle,Anthony, Samantha, David, and Luis.I would like to thank my advisor, John Baez, for his support, dedication, and his uniqueand brilliant style of advising. I could not have become the researcher I am under another’sinstruction. I would also like to thank Christina Vasilakopoulou, whose kindness, energy,and expertise cultivated a deeper appreciation of category theory in me. My experiencewas also greatly enriched by my academic siblings: Daniel Cicala, Kenny Courser, BrandonCoya, Jason Erbele, Jade Master, Franciscus Rebro, and Christian Williams, and by mycohort: Justin Davis, Ethan Kowalenko, Derek Lowenberg, Michel Manrique, and MichaelPierce.I would like to thank the UCR math department. Professors from whom I learned a tonof algebra, topology, and category theory include Julie Bergner, Vyjayanthi Chari, Wee-Liang Gan, Jos´e Gonzalez, Jacob Greenstein, Carl Mautner, Reinhard Schultz, and SteffanoVidussi. Special thanks goes to the department chair Yat-Sun Poon, as well as MargaritaRoman, Randy Morgan, and James Marberry, and many others who keep the whole thingtogether.The material in Chapter 2 consists of work from both
Network models joint with JohnBaez, John Foley, and Blake Pollard [BFMP20]. Chapter 3 consists of work done in mypaper
Noncommutative network models [Moe20]. Chapter 4 arose from
Network modelsfrom Petri nets with catalysts joint with Baez and Foley [BFM19]. Chapter 5 consists ofjoint work with Christina Vasilakopoulou appearing in our paper
Monoidal Grothendieckconstruction [MV20]. Part of this work was performed with funding from a subcontract withMetron Scientific Solutions working on DARPA’s Complex Adaptive System Compositionand Design Environment (CASCADE) project.iv o Teresa Danielle Moeller. v BSTRACT OF THE DISSERTATION
The Grothendieck Construction in Categorical Network TheorybyJoseph Patrick MoellerDoctor of Philosophy, Graduate Program in MathematicsUniversity of California, Riverside, December 2020Dr. John C. Baez, ChairpersonIn this thesis, we present a flexible framework for specifying and constructing operadswhich are suited to reasoning about network construction. The data used to present theseoperads is called a network model , a monoidal variant of Joyal’s combinatorial species. Theconstruction of the operad required that we develop a monoidal lift of the Grothendieckconstruction. We then demonstrate how concepts like priority and dependency can berepresented in this framework. For the former, we generalize Green’s graph products ofgroups to the context of universal algebra. For the latter, we examine the emergence ofmonoidal fibrations from the presence of catalysts in Petri nets.vi ontents
A Monoidal Categories 100
A.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100A.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103vii.3 Monoid Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105A.4 The Eckmann–Hilton Argument . . . . . . . . . . . . . . . . . . . . . . . . 107A.5 Characterizing (co)cartesian monoidal categories . . . . . . . . . . . . . . . 108
B Monoidal 2-Categories and Pseudomonoids 114
B.1 Monoidal 2-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114B.2 Pseudomonoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
C Fibrations and Indexed Categories 120
C.1 Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120C.2 Indexed Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122C.3 The Grothendieck Construction . . . . . . . . . . . . . . . . . . . . . . . . . 124C.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
D Species and Operads 130
D.1 Combinatorial Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130D.2 Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Bibliography 136 viii hapter 1
Introduction
Search and Rescue
Imagine that you have a network of boats, planes, and drones tasked with rescuingsailors who have fallen overboard in a hurricane. You want to be able to task these agentsto search certain areas for survivors in an intelligent way. You do not want to waste timeand resources by double searching some areas while other areas get neglected. Also, if oneof the searchers gets taken out by the storm, you must update the tasking so that otheragents can cover the areas which the downed agent has yet to search, as well as recordingthat there is a new known person in need of rescue.In 2015, DARPA launched a program called Complex Adaptive System Compositionand Design Environment, or CASCADE. The goal of this program was to write softwarethat would be able to handle this sort of tasking of agents in a network in a flexible andresponsive way. The bulk of this thesis was developed while I was working on this projectwith Metron Scientific Solutions Inc., developing a mathematically principled foundationaround which this software could be designed. John Baez, John Foley, Blake Pollard, andI developed the theory of network models to address this challenge [BFMP20].
Network Operads
Large complex networks can be viewed as being built up from small simple pieces. Thissort of many-to-one composition is perfectly suited to being modeled using operads . Whilea category can be described as a system of composition for a collection of arrows which havea specified input type and a specified output type, an operad is a system of compositionfor a collection of trees which have a specified family of input types and a single specifiedoutput type.
7→ 7→
1e use the word “operations” instead of “trees”, hence the name operad . Like categories,operads were originally developed in algebraic topology [May72, BV73]. Also like categories,operads have since found applications elsewhere, including physics and computer science[MSS02, M´en15]. We include a review of the basics of operads needed for this thesis inAppendix D.2.In a network operad, the operations describe ways of sticking together a collection ofnetworks to form a new larger network. To get a network operad, we treat a network as oneof these operations and define the composition as overlaying a bunch of small networks ontop of a large base network. For example, in the following picture, we are considering simplegraphs as a sort of network. On the left, we are starting with a base network consisting ofnine nodes and four edges, and we are attempting to attach more edges by overlaying threesmaller graphs. The result of the operadic composition is on the right. = This example is fairly elementary, and it is probably not too difficult for someone com-fortable with the notions to define this operad. However, it is not just simple graphs thatone needs when talking about managing and tasking complex networks of various sorts ofagents with various forms of communication and capabilities. One could continue replicatingthe procedure for constructing network operads for each type of network whenever needed,but this is an inefficient strategy. Instead, we devised a general recipe for constructing suchan operad for a given network type, and a general method for specifying a network type inan efficient way, using what we call a network model . All of this is done in the language ofcategory theory, so we also have a theory of how morphisms between network models givemorphisms between their operads.
Constructing Network Operads
There is a well-known trick for extracting an operad from any symmetric monoidal cat-egory. An operation in the operad is defined to be a morphism from a tensor product of afinite family of objects to a single object. This is called the underlying operad of the symmet-ric monoidal category. So now we have shifted the problem of defining an operad where the2perations are networks to defining a symmetric monoidal category where the morphismsare networks. To achieve this, we can use the famous
Grothendieck construction —thoughwe need to enhance it to suit our purposes.
Monoidal Grothendieck Construction
The Grothendieck construction is a well-known trick for turning a family of categoriesindexed by the objects of some other category into a single category in an intelligent way[Gro71]. What we really would like is that morphisms in the indexing category translateinto morphisms in our total category between objects from the corresponding indices. Theclassic example is the family of categories
Mod R of R -modules, indexed by the objects of Ring , the category of rings. Sometimes, one would like to talk about a single category ofmodules over all possible rings to study the interactions between such modules. The naivething to do would be to just take the coproduct, defining
Mod = a R ∈ Ring
Mod R . However, in this category an R -module and an S -module would have no morphisms betweenthem. This runs counter to the goal of having a single category for reasoning about theinteractions of modules over potentially different rings. If f : R → S is a ring homomor-phism, there is a way of turning S -modules into R -modules using f , called pullback . If M is an S -module, m ∈ M , and r ∈ R , pulling back M along f defines an R -module structureon the underlying abelian group of M . We define the action of r ∈ R on m ∈ M by thefollowing formula. r · m = f ( r ) · m This construction turns out to give a functor f ∗ : Mod S → Mod R . We should hope also that the data of these functors is included in the total category weconstruct. Indeed, the Grothendieck construction accomplishes precisely this.However, it is not simply a category that we need, but a symmetric monoidal category.So we built an enhanced version of the Grothendieck construction, which takes family ofcategories indexed by a symmetric monoidal category and constructs a symmetric monoidal category [BFMP20]. Christina Vasilakopoulou and I extended this modification to solve themonoidality problem in the Grothendieck correspondence [MV20].These two steps constitute the construction of the desired network operad: we startwith a monoidal indexed category, use the monoidal Grothendieck construction to producea symmetric monoidal category, and then take its underlying operad. This leads to anotherquestion though: what monoidal indexed categories should we feed into this constructionin order to produce network operads? 3 etwork Models
The answer is that we should take a monoidal version of Joyal’s combinatorial species[Joy81]. A combinatorial species is a functor F : FinBij → Set . One way of looking at thisis as a family of symmetric group actions, one for each natural number. Another way oflooking at it is as a particular type of indexed category, where there is a family of discretecategories (sets) indexed by the natural numbers, and functors (functions) between themcorresponding to the morphisms in
FinBij . So this is something to which we can apply theGrothendieck construction. The resulting total category is a groupoid which has all theelements in all the sets as the objects, and an isomorphism between these elements if theyare in the same orbit under the symmetric group action.Recall our example of a network operad where an operation is a simple graph. To buildthis, we can start with the species of simple graphs SG :
FinBij → Set . We give this thestructure of a lax monoidal functor (
FinBij , +) → ( Set , × ) by equipping it with a naturalmap SG( m ) × SG( n ) → SG( m + n ) given by disjoint union. We include the data of theoverlaying of graphs as a monoid structure on the set SG( n ) of simple graphs on n nodes.The product of two graphs on n nodes is another graph on n nodes given by identifyingcorresponding nodes, and including an edge wherever either of the original graphs had one.So now we have a lax symmetric monoidal functor (SG , ⊔ ) : ( FinBij , +) → ( Mon , × ). Wecall such a map a network model . When we take the Grothendieck construction of this, wetreat the monoids as one-object categories. By doing this, the resulting category has objectsgiven by finite sets, a morphism n → n is given by a simple graph on n nodes, compositionoverlays the graphs, and tensor sets them side by side. Constructing Network Models
Network operads are constructed from network models. How do we get our hands onsome network models? We know about a few examples of network models: simple graphs,directed graphs, multigraphs, colored vertices, etc. Ideally, we would have a (functorial)way to generate network models from some simple description of what we want a networkto look like.We can begin by examining the basic example: simple graphs. It consists of a familyof monoids SG( n ) where the elements are simple graphs on n nodes, with symmetric groupactions which permute the nodes, and a “disjoint union” operation ⊔ : SG( m ) × SG( n ) → SG( m + n ). The level-0 and level-1 monoids are both trivial. The first interesting one islevel-2, where the monoid is isomorphic to the Boolean truth values with the “or” operation.The rest of the monoids in this network model can be seen as built from SG(2). A simplegraph with n nodes has (cid:0) n (cid:1) places where it can either have or not have an edge. We candefine the monoid SG( n ) to be the product of (cid:0) n (cid:1) copies of SG(1), indexed by distinctpairs of nodes. This leads to the general construction: given a monoid M , let M ( n ) be themonoid given by the product (cid:0) n (cid:1) copies of M . Then the collection of these monoids M isa network model, where a network has an element of M between every pair of nodes, andoverlaying two networks simply requires performing the monoid operation at every pair ofnodes. This construction covers the example of simple graphs by design, but also includes4ultigraphs, directed graphs, graphs with colored edges, and many other examples. Noncommutative Network Models
Another property we wanted to be able to represent within the network operads frame-work was forms of communication which had a built-in limitation on the number of con-nections. This is a natural issue in the search and rescue domain problem [Moe20].There is no natural way to decide which edges not to include when the limit of con-nections is reached. This means that the network must have some extra data built intoit. In particular, it must remember the order in which the connections were added to eachnode. For this, we need the edge components of the constituent monoids to not commutewith each other. Due to a variant of the Eckmann-Hilton argument, edge components of anetwork model’s constituent monoid actually must commute with each other if they do notshare any of their nodes. This means the most we can ask for is that edge components ofthe network model do not commute with each other when edges have a node in common.We cannot simply take iterated products of the monoid as we did before because theedge components of the resulting monoids always commute with each other. We also cannotsimply take coproducts because the edge components do not commute with each other inway that are necessary for a network model. Therefore, we must have a mix of products andcoproducts depending on which edges share a node and which do not. Specifically, if twoedges share a node, then elements of the corresponding edge components of the monoid mustnot commute with each other, and if they do not share a node, they must commute witheach other. Such a monoid can be constructed using graph products of monoids , introducedfor groups in Elisabeth Green’s thesis [Gre90]. The idea is to produce a new monoid froma finite set of monoids by assigning them to the nodes in a graph, taking the coproduct ofthem all, then imposing commutativity relations between elements coming from monoidswhich had an edge between them in the graph.What indexing graph should we use though? We want a copy of the monoid for everypossible edge. So our indexing graph should have (cid:0) n (cid:1) nodes, one for every subset of cardi-nality 2. We want to impose commutativity between two edge components whenever thecorresponding edges do not share a node, so we add an edge for each pair of cardinality 2subsets which have empty intersection. This is precisely the definition of what are calledthe Kneser graphs ! The first few non-empty ones are depicted below.For a given monoid M , we thus define the corresponding network model to be thegraph product of M with itself indexed by the corresponding Kneser graph. In fact, this5onstruction gives the free network model on M , forming a left adjoint to the functor whichevaluates a network model at 2. This provides a solution to the problem of representingdegree limited networks in the language of network operads. This construction gives anetwork operad where the networks are graphs such that every vertex has degree ≤ N , andthe network does not take an edge if this limit would be exceeded. Petri Nets with Catalysts
Network models are also able to describe scenarios where there is an agent or agentsthat can manipulate and transport resources within the network [BFM19]. Baez, Foley,and I use a simple structure called a
Petri net to represent resources and processes thattransform them [BB18]. A Petri net can be drawn as a directed graph with vertices of twokinds: places or species , which we draw as yellow circles below, and transitions , which wedraw as blue squares:Petri nets are intended to model resources in a network of processes. Sometimes, we repre-sent the resources by a finite number of tokens in each place: ••• This is called a marking . We can then “run” the Petri net by repeatedly changing themarking using the transitions. For example, the above marking can change to this: •• and then this: ••• Thus, the places represent different types of resource, and the transitions describe ways thatone collection of resources of specified types can turn into another such collection.An agent might pick up a box and carry it over to a truck, and then drive the truckover to a new warehouse, and then unload the box. In this scenario, the gasoline in thetruck might be a resource that considered to be consumed by this process, but the agentis not. This qualitative difference between the agent as a resource and the gasoline as aresource leads to a quantitative difference. Specifically, the number of agents in this networkis never changing, but the number of gallons of gasoline is. What this means for the Petri6et model of this network is that there is no combination of transition firing that changethe number of agents. This gives us a fibration of the commutative monoidal category ofexecutions for the Petri net. However, unlike the monoidal fibrations described earlier, thefibres here are only pre monoidal in general, not quite monoidal. This gives an example ofa generalized network model, one where the monoids in the original definition are replacedwith categories.
Outline of the Thesis
I begin by laying out the theory of network models and network operads in Chapter 2.Section 2.2 and Section 2.3 contain basic definitions and examples. The construction of anetwork operad from a network model and several examples of algebras of network operadsare given in Section 2.4.In Chapter 3, more constructions of network models are given. The construction offree network models from a given monoid is detailed in Section 3.3. This depends on ageneralization of Green’s graph products given in Section 3.2. In section 3.4, an example ofan algebra for a noncommutative network model arising from limitations on communicationnetworks is given.Chapter 4 discusses the construction of network models from Petri nets with catalysts.In Section 4.2, the basic notions for the categorical treatment of Petri nets are recalled.Section 4.3 explains what it means for a Petri net to have catalysts. Section 4.4 describeshow catalysts induce a premonoidal fibration on the category of executions, and explainhow this gives an example of a generalized network model.I finish with a self-contained treatment of the monoidal Grothendieck construction inChapter 5. As the theoretical underpinning of the theory of network models, it is themost technically dense, and thus saved for the most enduring of readers. Section 5.2 andSection 5.3 describe monoidal fibrations and indexed categories. Section 5.4 details thecorresponding Grothendieck constructions for each monoidal variant. Section 5.6 discussesthe special case of when the base category is co/cartesian. In Section 5.5, we give a nuts-and-bolts description of the monoidal structures constructed in various scenarios. Section 5.7demonstrates the potential usefulness of the construction with examples from categoricalalgebra and dynamical systems.I wanted to include my own explanations and several references for preliminary materials,but did not want this to clutter the primary narrative of the thesis. I have included muchof this in several appendices. I discuss some of the basic theory of monoidal categories inAppendix A; monoidal 2-categories and Gray monoids in Appendix B; fibrations, indexedcategories, and the Grothendieck construction in Appendix C; and combinatorial speciesand operads in Appendix D. 7 hapter 2
Network Models
In this chapter, we study operads suited for designing networks. These could be networkswhere the vertices represent fixed or moving agents and the edges represent communicationchannels. More generally, they could be networks where the vertices represent entities ofvarious types, and the edges represent relationships between these entities, e.g. that oneagent is committed to take some action involving the other. The work done is this chapterarose from an example where the vertices represent planes, boats and drones involved ina search and rescue mission in the Caribbean [BFMP16, BFMP17]. However, even forthis one example, we want a flexible formalism that can handle networks of many kinds,described at a level of detail that the user is free to adjust.To achieve this flexibility, we introduce a general concept of network model . Simply put,a network model is a kind of network. Any network model gives an operad whose operationsare ways to build larger networks of this kind by gluing smaller ones. This operad has acanonical algebra where the operations act to assemble networks of the given kind. But italso has other algebras, where it acts to assemble networks of this kind equipped with extrastructure and properties . This flexibility is important in applications.What exactly is a kind of network? At the crudest level, we can model networks as sim-ple graphs. If the vertices are agents of some sort and the edges represent communicationchannels, this means we allow at most one channel between any pair of agents. However,simple graphs are too restrictive for many applications. If we allow multiple communica-tion channels between a pair of agents, we should replace simple graphs with multigraphs.Alternatively, we may wish to allow directed channels, where the sender and receiver havedifferent capabilities: for example, signals may only be able to flow in one direction. Thisrequires replacing simple graphs with directed graphs. To combine these features we coulduse directed multigraphs. It is also important to consider graphs with colored vertices, tospecify different types of agents, and colored edges, to specify different types of channels.This leads us to colored directed multigraphs. All these are examples of what we mean by akind of network. Even more complicated kinds, such as hypergraphs or Petri nets, are likelyto become important as we proceed. Thus, instead of separately studying all these kinds ofnetworks, we introduce a unified notion that subsumes all these variants: a network model .8amely, given a set C of vertex colors, a network model F is a lax symmetric monoidalfunctor F : S ( C ) → Cat , where S ( C ) is the free strict symmetric monoidal category on C and Cat is the category of small categories, considered with its cartesian monoidal structure.Unpacking this definition takes a little work. It simplifies in the special case where F takesvalues in Mon , the category of monoids. It simplifies further when C is a singleton, sincethen S ( C ) is the groupoid S , where objects are natural numbers and morphisms from m to n are bijections σ : { , . . . , m } → { , . . . , n } . If we impose both these simplifying assumptions,we have what we call a one-colored network model : a lax symmetric monoidal functor F : S → Mon . As we shall see, the network model of simple graphs is a one-colored networkmodel, and so are many other motivating examples.Joyal began an extensive study of functors F : S → Set , which are now commonly called species [Joy81, Joy86, BLL98]. Any type of extra structure that can be placed on finite setsand transported along bijections defines a species if we take F ( n ) to be the set of structuresthat can be placed on the set { , . . . , n } . From this perspective, a one-colored networkmodel is a species with some extra operations.This perspective is helpful for understanding what a one-colored network model F : S → Mon is actually like. If we call elements of F ( n ) networks with n vertices , then:1. Since F ( n ) is a monoid, we can overlay two networks with the same number ofvertices and get a new one. We denote this operation by ∪ : F ( n ) × F ( n ) → F ( n ) . For example: ∪ =
214 3 214 3 214 3
2. Since F is a functor, the group S n acts on the monoid F ( n ). Thus, for each σ ∈ S n ,we have a monoid automorphism that we call σ : F ( n ) → F ( n ) . For example, if σ = (2 3) ∈ S , then σ :
21 3 21 3
3. Since F is lax monoidal, we have an operation ⊔ : F ( m ) × F ( n ) → F ( m + n )We call this operation the disjoint union of networks. For example: ⊔ =
21 3 214 3 547 621 3 F : S → Mon . The last twoare present whenever we have a lax symmetric monoidal functor F : S → Set . When F is aone-colored network model we have all three—and unpacking the definition further, we seethat they obey some equations, which we list in Theorem 2.3. For example, the interchangelaw ( g ∪ g ′ ) ⊔ ( h ∪ h ′ ) = ( g ⊔ h ) ∪ ( g ′ ⊔ h ′ )holds whenever g, g ′ ∈ F ( m ) and h, h ′ ∈ F ( n ).In Section 2.2 we study one-colored network models more formally, and give manyexamples. In Section 2.2.1 we describe a systematic procedure for getting one-colorednetwork models from monoids. In Section 2.3 we study general network models and giveexamples of these. In Section 2.3.1 we describe a category NetMod of network models, andshow that the procedure for getting network models from monoids is functorial. We alsomake
NetMod into a symmetric monoidal category, and give examples of how to build newnetworks models by tensoring old ones.Our main result is that any network model gives a typed operad, also known as a colored operad or symmetric multicategory [Yau16]. A typed operad describes ways ofsticking together things of various types to get new things of various types. An algebra ofthe operad gives a particular specification of these things and the results of sticking themtogether. We review the definitions of operads and their algebras in Appendix D.2. A bitmore precisely, a typed operad O has: • a set T of types , • sets of operations O ( t , ..., t n ; t ) where t i , t ∈ T , • ways to compose any operation f ∈ O ( t , . . . , t n ; t )with operations g i ∈ O ( t i , . . . , t ik i ; t i ) (1 ≤ i ≤ n )to obtain an operation f ◦ ( g , . . . , g n ) ∈ O ( t i , . . . , t k , . . . , t n , . . . t nk n ; t ) , • and ways to permute the arguments of operations,which obey some rules [Yau16]. An algebra A of O specifies a set A ( t ) for each type t ∈ T such that the operations of O act on these sets. Thus, it has: • for each type t ∈ T , a set A ( t ) of things of type t , • ways to apply any operation f ∈ O ( t , . . . , t n ; t )to things a i ∈ A ( t i ) (1 ≤ i ≤ n )to obtain a thing α ( f )( a , . . . , a n ) ∈ A ( t ) . O F arising from a one-colored networkmodel F . The set of types is N , since we can think of ‘network with n vertices’ as a type.The sets of operations are given as follows: O F ( n , . . . , n k ; n ) = (cid:26) S n × F ( n ) if n + · · · + n k = n ∅ otherwise.The key idea here is that we can overlay a network in F ( n ) on the disjoint union of networkswith n , . . . , n k vertices and get a new network with n vertices as long as n + · · · n k = n .We can also permute the vertices; this accounts for the group S n . But the most importantfact is that networks serve as operations to assemble networks , thanks to our ability tooverlay them.Using this fact, we show in Example 2.32 that the operad O F has a canonical algebra A F whose elements are simply networks of the kind described by F : A F ( n ) = F ( n ) . In this algebra any operation( σ, g ) ∈ O F ( n , . . . , n k ; n ) = S n × F ( n )acts on a k -tuple of networks h i ∈ A F ( n i ) = F ( n i ) (1 ≤ i ≤ k )to give the network α ( σ, g )( h , . . . , h k ) = g ∪ σ ( h ⊔ · · · ⊔ h k ) ∈ A F ( n ) . In other words, we first take the disjoint union of the networks h i , then permute theirvertices with σ , and then overlay the network g .An example is in order, since the generality of the formalism may hide the simplicityof the idea. The easiest example of our theory is the network model for simple graphs. InExample 2.4 we describe a one-colored network model SG : S → Mon such that SG( n ) isthe collection of simple graphs with vertex set n = { , . . . , n } . Such a simple graph is reallya collection of 2-element subsets of n , called edges . Thus, we may overlay simple graphs g, g ′ ∈ SG( n ) by taking their union g ∪ g ′ . This operation makes SG( n ) into a monoid.Now consider an operation f ∈ O SG (3 , ,
2; 9). This is an element of S × SG(9): apermutation of the set { , . . . , } together with a simple graph having this set of vertices.If we take the permutation to be the identity for simplicity, this operation is just a simple11raph g ∈ SG(9). We can draw an example as follows:
The dashed circles indicate that we are thinking of this simple graph as an element of O (3 , ,
2; 9): an operation that can be used to assemble simple graphs with 3, 4, and 2vertices, respectively, to produce one with 9 vertices.Next let us see how this operation acts on the canonical algebra A SG , whose elementsare simple graphs. Suppose we have elements a ∈ A SG (3), a ∈ A SG (4) and a ∈ A SG (2): We can act on these by the operation f to obtain α ( f )( a , a , a ) ∈ A SG (9). It looks like12his: We have simply taken the disjoint union of a , a , and a and then overlaid g , obtaining asimple graph with 9 vertices.The canonical algebra is one of the simplest algebras of the operad O SG . We can definemany more interesting algebras for this operad. For example, we might wish to use this op-erad to describe communication networks where the communicating entities have locationsand the communication channels have limits on their range. To include location data, wecan choose A ( n ) for n ∈ N to be the set of all graphs with n vertices where each vertex isan actual point in the plane R . To handle range-limited communications, we could insteadchoose A ( n ) to be the set of all graphs with n vertices in R where an edge is permittedbetween two vertices only if their Euclidean distance is less than some specified value. Thisstill gives a well-defined algebra: when we apply an operation, we simply omit those edgesfrom the resulting graph that would violate this restriction.Besides the plethora of interesting algebras for the operad O SG , and useful homomor-phisms between these, one can also modify the operad by choosing another network model.This provides additional flexibility in the formalism. Different network models give differentoperads, and the construction of operads from network models is functorial, so morphismsof network models give morphisms of operads.In Section 2.4 we apply the machinery provided by Chapter 5 to build operads fromnetwork models. We also describe some algebras of these operads, and in Example 2.35 wediscuss an algebra whose elements are networks of range-limited communication channels. We begin with a special class of network models: those where the vertices of the networkhave just one color. To define these, we use S to stand for a skeleton of the groupoid offinite sets and bijections: Definition 2.1.
Let S , the symmetric groupoid , be the groupoid for which: • objects are natural numbers n ∈ N , 13 a morphism from m to n is a bijection σ : { , . . . , m } → { , . . . , n } and bijections are composed in the usual way.There are no morphisms in S from m to n unless m = n . For each n ∈ N , the endomor-phisms of n form the symmetric group S n . It is convenient to write n for the set { , . . . , n } ,so that a morphism σ : n → n in S is the same as a bijection σ : n → n .There is a functor + : S × S → S defined as follows. Given m, n ∈ N we let m + n be theusual sum, and given σ ∈ S m and τ ∈ S n , let σ + τ ∈ S m + n be as follows:( σ + τ )( j ) = ( σ ( j ) if j ≤ mτ ( j − m ) + m otherwise. (2.1)For objects m, n ∈ S , let B m,n be the block permutation of m + n which swaps the first m with the last n . For example B , : 7 → B give S the structure of a strict symmetric monoidalcategory. This follows as a special case of Proposition 2.16. Definition 2.2. A one-colored network model is a lax symmetric monoidal functor F : S → Mon . Here
Mon is the category with monoids as objects and monoid homomorphisms as mor-phisms, considered with its cartesian monoidal structure.Algebraically, a network model is a family of monoids { M n } n ∈ N each with a group actionof the corresponding symmetric group S n , such that the product of any two embed into theone indexed by the sum of their indices equivariantly, i.e. in a way which respects the groupaction: M m × M n ֒ → M m + n .Many examples of network models are given below. A pedestrian way to verify thatthese examples really are network models is to use the following result: Theorem 2.3.
A one-colored network model F : S → Mon is the same as: • a family of sets { F ( n ) } n ∈ N • distinguished identity elements e n ∈ F ( n ) • a family of overlay functions ∪ : F ( n ) × F ( n ) → F ( n ) • a bijection σ : F ( n ) → F ( n ) for each σ ∈ S n • a family of disjoint union functions ⊔ : F ( m ) × F ( n ) → F ( m + n ) satisfying the following equations: . e n ∪ g = g ∪ e n = g g ∪ ( g ∪ g ) = ( g ∪ g ) ∪ g σ ( g ∪ g ) = σg ∪ σg σe n = e n ( σ σ ) g = σ ( σ g ) ( g ∪ g ) ⊔ ( h ∪ h ) = ( g ⊔ h ) ∪ ( g ⊔ h ) g ) = g e m ⊔ e n = e m + n σg ⊔ τ h = ( σ + τ )( g ⊔ h ) g ⊔ ( g ⊔ g ) = ( g ⊔ g ) ⊔ g e ⊔ g = g ⊔ e = g B m,n ( h ⊔ g ) = g ⊔ h for g, g i ∈ F ( n ) , h, h i ∈ F ( m ) , σ, σ i ∈ S n , τ ∈ S m , and the identity of S n .Proof. Having a functor F : S → Mon is equivalent to having the first four items satisfyingEquations 1–6. The binary operation ∪ gives the set F ( n ) the structure of a monoid, with e n acting as the identity. Equation 1 tells us e n acts as an identity, and Equation 2 gives theassociativity of ∪ . Equations 3 and 4 tell us that σ is a monoid homomorphism. Equations5 and 6 say that the map ( σ, g ) σg defines an action of S n on F ( n ) for each n . All ofthese actions together give us the functor F : S → Mon .That the functor is lax monoidal is equivalent to having item 5 satisfying Equations 7–11. Equations 7 and 8 tell us that ⊔ is a family of monoid homomorphisms. Equation 9 tellsus that it is a natural transformation. Equation 10 tells us that the associativity hexagondiagram for lax monoidal functors commutes for F . Equation 11 implies the commutativityof the left and right unitor square diagrams. That the lax monoidal functor is symmetricis equivalent to Equation 12. It tells us that the square diagram for symmetric monoidalfunctors commutes for F .This is one of the simplest examples of a network model: Example 2.4 ( Simple graphs ) . Let a simple graph on a set V be a set of 2-elementsubsets of V , called edges . There is a one-colored network model SG : S → Mon such thatSG( n ) is the set of simple graphs on n .To construct this network model, we make SG( n ) into a monoid where the product ofsimple graphs g , g ∈ SG( n ) is their union g ∪ g . Intuitively speaking, to form their union,we ‘overlay’ these graphs by taking the union of their sets of edges. The simple graph on n with no edges acts as the unit for this operation. The groups S n acts on the monoidsSG( n ) by permuting vertices, and these actions define a functor SG : S → Mon .Given simple graphs g ∈ SG( m ) and h ∈ SG( n ) we define g ⊔ h ∈ SG( m + n ) to betheir disjoint union. This gives a monoid homomorphism ⊔ : SG( m ) × SG( n ) → SG( m + n )because ( g ∪ g ) ⊔ ( h ∪ h ) = ( g ⊔ h ) ∪ ( g ⊔ h ) . This in turn gives a natural transformation with components ⊔ m,n : SG( m ) × SG( n ) → SG( m + n ) , M by letting elements of Γ M ( n ) be maps from thecomplete graph on n to M . If we take M = B = { F, T } with ‘or’ as the monoid operation,this procedure gives the network model SG = Γ B . We explain this in Example 2.12.There are many other kinds of graph, and many of them give network models: Example 2.5 ( Directed graphs ) . Let a directed graph on a set V be a collection ofordered pairs ( i, j ) ∈ V such that i = j . These pairs are called directed edges . There isa network model DG : S → Mon such that DG( n ) is the set of directed graphs on n . As inExample 2.4, the monoid operation on DG( n ) is union. Example 2.6 ( Multigraphs ) . Let a multigraph on a set V be a multiset of 2-elementsubsets of V . If we define MG( n ) to be the set of multigraphs on n , then there are at leasttwo natural choices for the monoid operation on MG( n ). The most direct generalization ofSG of Example 2.4 is the network model MG : S → Mon with values (MG( n ) , ∪ ) where ∪ isnow union of edge multisets. That is, the multiplicity of { i, j } in g ∪ h is maximum of themultiplicity of { i, j } in g and the multiplicity of { i, j } in h . Alternatively, there is anothernetwork model MG + : S → Mon with values (MG( n ) , +) where + is multiset sum. That is, g + h obtained by adding multiplicities of corresponding edges. Example 2.7 ( Directed multigraphs ) . Let a directed multigraph on a set V be amultiset of ordered pairs ( i, j ) ∈ V such that i = j . There is a network model DMG : S → Mon such that DMG( n ) is the set of directed multigraphs on n with monoid operation theunion of multisets. Alternatively, there is a network model with values (DMG( n ) , +) where+ is multiset sum. Example 2.8 ( Hypergraphs ) . Let a hypergraph on a set V be a set of nonempty subsetsof V , called hyperedges . There is a network model HG : S → Mon such that HG( n ) is theset of hypergraphs on n . The monoid operation HG( n ) is union. Example 2.9 ( Graphs with colored edges ) . Fix a set B of edge colors and let SG : S → Mon be the network model of simple graphs as in Example 2.4. Then there is a networkmodel H : S → Mon with H ( n ) = SG( n ) B making the product of B copies of the monoid SG( n ) into a monoid in the usual way. In thismodel, a network is a B -tuple of simple graphs, which we may view as a graph with at mostone edge of each color between any pair of distinct vertices. We describe this constructionin more detail in Example 2.24.There are also examples of network models not involving graphs: Example 2.10 ( Partitions ) . A poset is a lattice if every finite subset has both an infimumand a supremum. If L is a lattice, then ( L, ∨ ) and ( L, ∧ ) are both monoids, where x ∨ y isthe supremum of { x, y } ⊆ L and x ∧ y is the infimum.16et P ( n ) be the set of partitions of the set n . This is a lattice where π ≤ π ′ if thepartition π is finer than π ′ . Thus, P ( n ) can be made a monoid in either of the two waysmentioned above. Denote these monoids as P ∨ ( n ) and P ∧ ( n ). These monoids extend togive two network models P ∨ , P ∧ : S → Mon . There is a systematic procedure that gives many of the network models we have seenso far. To do this, we take networks to be ways of labelling the edges of a complete graphby elements of some monoid M . The operation of overlaying two of these networks is thendescribed using the monoid operation.For example, consider the Boolean monoid B : that is, the set { F, T } with ‘inclusive or’as its monoid operation. A complete graph with edges labelled by elements of B can be seenas a simple graph if we let T indicate the presence of an edge between two vertices and F the absence of an edge. To overlay two simple graphs g , g with the same set of vertices wesimply take the ‘or’ of their edge labels. This gives our first example of a network model,Example 2.4.To formalize this we need some definitions. Given n ∈ N , let E ( n ) be the set of 2-elementsubsets of n = { , . . . , n } . We call the members of E ( n ) edges , since they correspond toedges of the complete graph on the set n . We call the elements of an edge e ∈ E ( n ) its vertices .Let M be a monoid. For n ∈ N , let Γ M ( n ) be the set of functions g : E ( n ) → M . Definethe operation ∪ : Γ M ( n ) × Γ M ( n ) → Γ M ( n ) by ( g ∪ g )( e ) = g ( e ) g ( e ) for e ∈ E ( n ). Definethe map ⊔ : Γ M ( m ) × Γ M ( n ) → Γ M ( m + n ) by( g ⊔ g )( e ) = g ( e ) if both vertices of e are ≤ mg ( e ) if both vertices of e are > m the identity of M otherwiseThe symmetric group S n acts on Γ M ( n ) by σ ( g )( e ) = g ( σ − ( e )). Theorem 2.11.
For each monoid M the data above gives a one-colored network model Γ M : S → Mon .Proof.
We can define Γ M as the composite of two functors, E : S → Inj and M − : Inj → Mon ,where
Inj is the category of sets and injections.The functor E : S → Inj sends each object n ∈ S to E ( n ), and it sends each morphism σ : n → n to the permutation of E ( n ) that maps any edge e = { x, y } ∈ E ( n ) to σ ( e ) = { σ ( x ) , σ ( y ) } . The category Inj does not have coproducts, but it is closed under coproductsin
Set . It thus becomes symmetric monoidal with + as its tensor product and the emptyset as the unit object. For any m, n ∈ S there is an injection µ m,n : E ( m ) + E ( n ) → E ( m + n )expressing the fact that a 2-element subset of either m or n gives a 2-element subset of m + n . The functor E : S → Inj becomes lax symmetric monoidal with these maps µ m,n giving the lax preservation of the tensor product.17he functor M − : Inj → Mon sends each set X to the set M X made into a monoid withpointwise operations, and it sends each function f : X → Y to the monoid homomorphism M f : M X → M Y given by( M f g )( y ) = (cid:26) g ( f − ( y )) if y ∈ im( f )1 otherwisefor any g ∈ M X . Using the natural isomorphisms M X + Y ∼ = M X × M Y and M ∅ ∼ = 1 thisfunctor can be made symmetric monoidal.As the composite of the lax symmetric monoidal functor E : S → Inj and the symmetricmonoidal functor M − : Inj → Mon , the functor Γ M : S → Mon is lax symmetric monoidal,and thus a network model. With the help of Theorem 2.3, it is easy to check that thisdescription of Γ M is equivalent to that in the theorem statement. Example 2.12 ( Simple graphs, revisited ) . Let B = { F, T } be the Boolean monoid. Ifwe interpret T and F as ‘edge’ and ‘no edge’ respectively, then Γ B is just SG, the networkmodel of simple graphs discussed in Example 2.4.Recall from Example 2.6 that a multigraph on the set n is a multisubset of E ( n ), orin other words, a function g : E ( n ) → N . There are many ways to create a network model F : S → Mon for which F ( n ) is the set of multigraphs on the set n , since N has many monoidstructures. Two of the most important are these: Example 2.13 ( Multigraphs with addition for overlaying ) . Let ( N , +) be N madeinto a monoid with the usual notion of addition as +. In this network model, overlaying twomultigraphs g , g : E ( n ) → N gives a multigraph g : E ( n ) → N with g ( e ) = g ( e ) + g ( e ). Infact, this notion of overlay corresponds to forming the multiset sum of edge multisets andΓ ( N , +) is the network model of multigraphs called MG + in Example 2.6. Example 2.14 ( Multigraphs with maximum for overlaying ) . Let ( N , max) be N madeinto a monoid with max as the monoid operation. Then Γ ( N , max) is a network model whereoverlaying two multigraphs g , g : E ( n ) → N gives a multigraph g : E ( n ) → N with g ( e ) = g ( e ) max g ( e ). For this monoid structure overlaying two copies of the same multigraphgives the same multigraph. In other words, every element in each monoid Γ ( N , max) ( n ) isidempotent and Γ ( N , max) is the network model of multigraphs called MG in Example 2.6. Example 2.15 ( Multigraphs with at most k edges between vertices ) . For any k ∈ N ,let B k be the set { , . . . , k } made into a monoid with the monoid operation ⊕ given by x ⊕ y = ( x + y ) min k and 0 as its unit element. For example, B is the trivial monoid and B is isomorphic to theBoolean monoid. There is a network model Γ B k such that Γ B k ( n ) is the set of multigraphson n with at most k edges between any two distinct vertices.18 .3 General Network Models The network models described so far allow us to handle graphs with colored edges, butnot with colored vertices. Colored vertices are extremely important for applications in whichwe have a network of agents of different types. Thus, network models will involve a set C of vertex colors in general. This requires that we replace S by the free strict symmetricmonoidal category generated by the color set C . Thus, we begin by recalling this category.For any set C , there is a category S ( C ) for which: • Objects are formal expressions of the form c ⊗ · · · ⊗ c n for n ∈ N and c , . . . , c n ∈ C . We denote the unique object with n = 0 as I . • There exist morphisms from c ⊗ · · · ⊗ c m to c ′ ⊗ · · · ⊗ c ′ n only if m = n , and in thatcase a morphism is a permutation σ ∈ S n such that c ′ σ ( i ) = c i for all i . • Composition is the usual composition of permutations.Note that elements of C can be identified with certain objects of S ( C ), namely theone-fold tensor products. We do this in what follows. Proposition 2.16. S ( C ) can be given the structure of a strict symmetric monoidal categorymaking it into the free strict symmetric monoidal category on the set C . Thus, if A isany strict symmetric monoidal category and f : C → Ob( A ) is any function from C toobjects of the A , there exists a unique strict symmetric monoidal functor F : S ( C ) → A with F ( c ) = f ( c ) for all c ∈ C .Proof. This is well-known; see for example Sassone [Sas94, Sec. 3] or Gambino and Joyal[GJ17, Sec. 3.1]. The tensor product of objects is ⊗ , the unit for the tensor product is I ,and the braiding( c ⊗ · · · ⊗ c m ) ⊗ ( c ′ ⊗ · · · ⊗ c ′ n ) → ( c ′ ⊗ · · · ⊗ c ′ n ) ⊗ ( c ⊗ · · · ⊗ c m )is the block permutation B m,n . Given f : C → Ob( A ), we define F : S ( C ) → A on objectsby F ( c ⊗ · · · ⊗ c n ) = f ( c ) ⊗ · · · ⊗ f ( c n ) , and it is easy to check that F is strict symmmetric monoidal, and the unique functor withthe required properties. Definition 2.17.
Let C be a set, called the set of vertex colors . A C -colored networkmodel is a lax symmetric monoidal functor F : S ( C ) → Cat . A network model is a C -colored network model for some set C .19f C has just one element, S ( C ) ∼ = S and a C -colored network model is a one-colorednetwork model in the sense of Definition 2.2. Here are some more interesting examples: Example 2.18 ( Simple graphs with colored vertices ) . There is a network model ofsimple graphs with C -colored vertices. To construct this, we start with the network modelof simple graphs SG : S → Mon given in Example 2.4. There is a unique function from C to the one-element set. By Proposition 2.16, this function extends uniquely to a strictsymmetric monoidal functor F : S ( C ) → S . An object in S ( C ) is formal tensor product of n colors in C ; applying F to this objectwe forget the colors and obtain the object n ∈ S . Composing F and SG, we obtain a laxsymmetric monoidal functor S ( C ) F −→ S SG −→ Mon which is the desired network model. We can use the same idea to ‘color’ any of the networkmodels in Section 2.2.Alternatively, suppose we want a network model of simple graphs with C -colored verticeswhere an edge can only connect two vertices of the same color. For this we take a cartesianproduct of C copies of the functor SG, obtaining a lax symmetric monoidal functorSG C : S C → Mon C . There is a function h : C → Ob( S C ) sending each c ∈ C to the object of S C that equals1 ∈ S in the c th place and 0 ∈ S elsewhere. Thus, by Proposition 2.16, h extends uniquelyto a strict symmetric monoidal functor H C : S ( C ) → S C . Furthermore, the product in
Mon gives a symmetric monoidal functorΠ :
Mon C → Mon . Composing all these, we obtain a lax symmetric monoidal functor S ( C ) H C −→ S C SG C −→ Mon C Π −→ Mon which is the desired network model.More generally, if we have a network model F c : S → Mon for each color c ∈ C , we canuse the same idea to create a network model: S ( C ) S C Mon C Mon H C Q c ∈ C F c Q in which the vertices of color c ∈ C partake in a network of type F c . Example 2.19 ( Petri nets ) . Petri nets are a kind of network widely used in computerscience, chemistry and other disciplines [BP17]. A
Petri net ( S, T, i, o ) is a pair of finite20ets and a pair of functions i, o : S × T → N . Let P ( m, n ) be the set of Petri nets ( m , n , i, o ).This becomes a monoid with product( m , n , i, o ) ∪ ( m , n , i ′ , o ′ ) = ( m , n , i + i ′ , o + o ′ )The groups S m × S n naturally act on these monoids, so we have a functor P : S → Mon . There are also ‘disjoint union’ operations ⊔ : P ( m, n ) × P ( m ′ , n ′ ) → P ( m + m ′ , n + n ′ )making P into a lax symmetric monoidal functor. In Example 2.18 we described a strictsymmetric monoidal functor H C : S ( C ) → S C for any set C . In the case of the 2-elementset this gives H : S (2) → S . We define the network model of Petri nets to be the composite S (2) H −→ S P −→ Mon . For each choice of the set C of vertex colors, we can define a category NetMod C of C -colored network models. However, it is useful to create a larger category NetMod containingall these as subcategories, since there are important maps between network models thatinvolve changing the vertex colors.
Definition 2.20.
For any set C , let NetMod C be the category for which: • an object is a C -colored network model, that is, a lax symmetric monoidal functor F : S ( C ) → Cat , • a morphism is a monoidal natural transformation between such functors: S ( C ) Cat FF ′ g and composition is the usual composition of monoidal natural transformations.In particular, NetMod is the category of one-colored network models. For an exampleinvolving this category, consider the network models built from monoids in Section 2.2.1.Any monoid M gives a one-colored network model Γ M for which an element of Γ M ( n ) is away of labelling the edges of the complete graph on n by elements of M . Thus, we shouldexpect any homomorphism of monoids f : M → M ′ to give a morphism of network modelsΓ f : Γ M → Γ M ′ for which Γ f ( n ) : Γ M ( n ) → Γ M ′ ( n )21pplies f to each edge label.Indeed, this is the case. As explained in the proof of Theorem 2.11, the network modelΓ M is the composite S E −→ Inj M − −→ Mon . The homomorphism f gives a natural transformation f − : M − ⇒ M ′− that assigns to any finite set X the monoid homomorphism f X : M X → M ′ X g f ◦ g. It is easy to check that this natural transformation is monoidal. Thus, we can whisker itwith the lax symmetric monoidal functor E to get a morphism of network models: S Inj Mon E M − M ′− f − and we call this Γ f : Γ M → Γ M ′ . Theorem 2.21.
There is a functor
Γ :
Mon → NetMod sending any monoid M to the network model Γ M and any homomorphism of monoids f : M → M ′ to the morphism of network models Γ f : Γ M → Γ M ′ .Proof. To check that Γ preserves composition, note that
S Inj Mon E M − M ′− f − M ′′− f ′− equals S Inj Mon E M − M ′′− ( f ′ f ) − since f ′− f − = ( f ′ f ) − . Similarly Γ preserves identities.22t has been said that category theory is the subject in which even the examples needexamples. So, we give an example of the above result: Example 2.22 ( Imposing a cutoff on the number of edges ) . In Example 2.13 wedescribed the network model of multigraphs MG + as Γ ( N , +) . In Example 2.15 we describeda network model Γ B k of multigraphs with at most k edges between any two distinct vertices.There is a homomorphism of monoids f : ( N , +) → B k n n min k and this induces a morphism of network modelsΓ f : Γ ( N , +) → Γ B k . This morphism imposes a cutoff on the number of edges between any two distinct vertices:if there are more than k , this morphism keeps only k of them. In particular, if k = 1, B k isthe Boolean monoid, and Γ f : MG + → SGsends any multigraph to the corresponding simple graph.One useful way to combine C -colored networks is by ‘tensoring’ them. This makes NetMod C into a symmetric monoidal category: Theorem 2.23.
For any set C , the category NetMod C can be made into a symmetricmonoidal category with the tensor product defined pointwise, so that for objects F, F ′ ∈ NetMod C we have ( F ⊗ F ′ )( x ) = F ( x ) × F ′ ( x ) for any object or morphism x in S ( C ) , and for morphisms φ, φ ′ in NetMod C we have ( φ ⊗ φ ′ ) x = φ x × φ ′ x for any object x ∈ S ( C ) .Proof. More generally, for any symmetric monoidal categories A and B , there is a symmetricmonoidal category SymMonCat ( A , B ) whose objects are lax symmetric monoidal functorsfrom A to B and whose morphisms are monoidal natural transformations, with the tensorproduct defined pointwise. The proof in the ‘weak’ case was given by Hyland and Power[HP02], and the lax case works the same way.If F, F ′ : S ( C ) → Mon then their tensor product again takes values in
Mon . There aremany interesting examples of this kind:
Example 2.24 ( Graphs with colored edges, revisited ) . In Example 2.9 we describednetwork models of simple graphs with colored edges. The above result lets us build thesenetwork models starting from more basic data. To do this we start with the network modelfor simple graphs, SG : S → Mon , discussed in Example 2.4. Fixing a set B of ‘edge colors’,23e then take a tensor product of copies of SG, one for each b ∈ B . The result is a networkmodel SG ⊗ B : S → Mon with SG ⊗ B ( n ) = SG( n ) B for each n ∈ N . Example 2.25 ( Combined networks ) . We can also combine networks of different kinds.For example, if DG : S → Mon is the network model of directed graphs given in Example 2.5and MG : S → Mon is the network model of multigraphs given in Example 2.6, thenDG ⊗ MG : S → Mon is another network model, and we can think of an element of (DG ⊗ MG)( n ) as a directedgraph with red edges together with a multigraph with blue edges on the set n .Next we describe a category NetMod of network models with arbitrary color sets, whichincludes all the categories
NetMod C as subcategories. To do this, first we introduce ‘color-changing’ functors. Recall that elements of C can be seen as certain objects of S ( C ), namelythe 1-fold tensor products. If f : C → C ′ is a function, there exists a unique strict symmetricmonoidal functor f ∗ : S ( C ) → S ( C ′ ) that equals f on objects of the form c ∈ C . This followsfrom Proposition 2.16.Next, we define an indexed category NetMod − : Set op → CAT that sends any set C to NetMod C and any function f : C → D to the functor that sends any D -colored networkmodel F : S ( D ) → Cat to the C -colored network model given by the composite S ( C ) f ∗ −→ S ( D ) F −→ Cat . Applying the Grothendieck construction (see Appendix C) to this indexed category, wedefine the category of network models to be
NetMod = Z NetMod − . In elementary terms,
NetMod has: • pairs ( C, F ) for objects, where C is a set and F : S ( C ) → Cat is a C -colored networkmodel. • pairs ( f, g ) : ( C, F ) → ( D, G ) for morphisms, where f : C → D is a function and g : F ⇒ G ◦ f ∗ is a morphism of network models. Example 2.26 ( Simple graphs with colored vertices, revisited ) . In Example 2.18 weconstructed the network model of simple graphs with colored vertices. We started with thenetwork model for simple graphs, which is a one-colored network model SG : S → Mon . Theunique function ! : C → ∗ : S ( C ) → S (1) ∼ = S .The network model of simple graphs with C -colored vertices is the composite S ( C ) ! ∗ −→ S SG −−→ Mon and there is a morphism from this to the network model of simple graphs, which has theeffect of forgetting the vertex colors. 24n fact,
NetMod can be understood as a subcategory of the following category:
Definition 2.27.
Let
SymMonICat be the category where: • objects are pairs ( C , F ) where C is a small symmetric monoidal category and F : C →
Cat is a lax symmetric monoidal functor, where
Cat is considered with its cartesianmonoidal structure. • morphisms from ( C , F ) to ( C ′ , F ′ ) are pairs ( G, g ) where G : C → C ′ is a lax symmetricmonoidal functor and g : F ⇒ F ′ ◦ G is a symmetric monoidal natural transformation: C Cat C ′ FG F ′ g We shall use this way of thinking in the next two sections to build operads from networkmodels. It must be said that
SymMonICat is naturally a 2-category where a 2-morphism ξ : ( G, g ) ⇒ ( G ′ , g ′ ) is a natural transformation ξ : G → G ′ such that C C ′ Cat = Cat . C C ′ G G ′ F G FF ′ g ′ ξ F ′ g and here we are considering its 1-dimensional truncation. The 2-dimensional structureis detailed in Appendix C, and utilized in Chapter 5. This lets us define 2-morphismsbetween network models, extending NetMod to a 2-category. We do not seem to need these2-morphisms in our applications, so we suppress 2-categorical considerations in most ofwhat follows.
Next we describe the operad associated to a network model. This construction is given intwo steps. For the first step, we can use the strict symmetric Grothendieck construction ofSection 5.4 to define a strict symmetric monoidal category R F from a given network model F : S ( C ) → Cat . For the second step, we then use the underlying operad construction(recalled in Proposition D.11) to build an operad O F . Definition 2.28.
Given a network model F : S ( C ) → Cat , define the network operad O F to be Op ( R F ). 25or the sake of the unfamiliar reader, we give a brief description of these constructionsin the specific context of network models, which does not assume prior knowledge. Werecall the ordinary Grothendieck construction in Appendix C, and Chapter 5 is entirelydedicated to studying the (braided/symmetric) monoidal variants of it. We give a nuts-and-bolts description of the symmetric monoidal category ( R F, ⊗ φ ) built from a networkmodel ( F, φ ) : ( S ( C ) , ⊗ ) → ( Mon , × ).The objects of R F correspond to objects of S ( C ), which are formal expressions of theform c ⊗ · · · ⊗ c n with n ∈ N and c i ∈ C . The morphisms of R F are pairs ( σ, g ) where σ : c ⊗ · · · ⊗ c n → c σ ⊗ · · · ⊗ c σn is a morphism in S ( C ), and g is an element of the monoid F ( c σ ⊗ · · · ⊗ c σn ). Composition is given by ( σ, g ) ◦ ( τ, h ) = ( στ, g · F σ ( h )). The tensorproduct of two objects is given by concatenation. The tensor product of two morphisms isgiven by ( σ, g ) ⊗ ( τ, h ) = ( σ ⊗ τ, φ ( g, h )). The unit object is ( I, φ ), where I is the monoidalunit for S ( C ) and φ is the unit laxator for F .For a one-object network model F , a more compact description of the category R F can be given by the following formula, where monoids and groups are being considered asone-object categories by default. Z F ∼ = a n ∈ N F ( n ) ⋊ S n The network operad O F is a typed operad where the types are ordered k -tuples of ele-ments of C . For objects x i , x of R F , the operations in O F are given by O F ( x , . . . , x n ; x ) = R F ( x ⊗ · · · ⊗ x n , x ).Now suppose that F is a one-colored network model, so that F : S → Mon . Then theobjects of S are simply natural numbers, so O F is an N -typed operad. Given n , . . . , n k , n ∈ N , we have O F ( n , . . . , n k ; n ) = hom R F ( n + · · · + n k , n ) . By definition, a morphism in this homset is a pair consisting of a bijection σ : n + · · · + n k → n and an element of the monoid F ( n ). So, we have O F ( n , . . . , n k ; n ) = (cid:26) S n × F ( n ) if n + · · · n k = n ∅ otherwise. (2.2)Here is the basic example: Example 2.29 ( Simple network operad ) . If SG : S → Mon is the network model of sim-ple graphs in Example 2.4, we call O SG the simple network operad . By Equation (2.2),an operation in O SG ( n , . . . , n k ; k ) is an element of S n together with a simple graph having n = { , . . . , n } as its set of vertices.The operads coming from other one-colored network models work similarly. For example,if DG : S → Mon is the network model of directed graphs from Example 2.5, then anoperation in O SG ( n , . . . , n k ; n ) is an element of S n together with a directed graph having n as its set of vertices.In Theorem 2.3 we gave a pedestrian description of one-colored network models. Wecan describe the corresponding network operads in the same style:26 heorem 2.30. Suppose F is a one-colored network model. Then the network operad O F is the N -typed operad for which the following hold:1. The sets of operations are given by O F ( n , . . . , n k ; n ) = (cid:26) S n × F ( n ) if n + · · · n k = n ∅ otherwise.2. Composition of operations is given as follows. Suppose that ( σ, g ) ∈ S n × F ( n ) = O F ( n , . . . , n k ; n ) and for ≤ i ≤ k we have ( τ i , h i ) ∈ S n i × F ( n i ) = O F ( n i , . . . , n ij i ; n i ) . Then ( σ, g ) ◦ (( τ , h ) , . . . , ( τ k , h k )) = ( σ ( τ + · · · + τ k ) , g ∪ σ ( h ⊔ · · · ⊔ h k )) where + is defined in Equation (2.1) , while ∪ and ⊔ are defined in Theorem 2.3.3. The identity operation in O F ( n ; n ) is (1 , e n ) , where is the identity in S n and e n isthe identity in the monoid F ( n ) .4. The right action of the symmetric group S k on O F ( n , . . . , n k ; n ) is given as follows.Given ( σ, g ) ∈ O F ( n , . . . , n k ; n ) and τ ∈ S k , we have ( σ, g ) τ = ( στ, g ) . Proof.
This is a straightforward combination of the underlying operad of a symmetricmonoidal category and the symmetric monoidal structure on R F .The construction of operads from symmetric monoidal categories described in Proposi-tion D.11 is functorial, so the construction of operads from network models is as well. Theorem 2.31.
The assignment of a network model F : S ( C ) → Cat to the operad O F = Op ( R G ) and a morphism of network models ( G, g ) : ( C, F ) → ( C ′ , F ′ G ′ ) to the operad mor-phism O G = Op ( b Γ) is a functor O : NetMod → Opd . Proof.
There is a functor R : NetMod → SymMonCat given by restricting the strict symmetric monoidal Grothendieck construction of Theo-rem 5.8 to
NetMod . Composing this with the functor Op : SymMonCat → Opd constructed in Proposition D.12 we obtain a functor O : NetMod → Opd with the propertiesstated in the theorem. Since these properties specify how O acts on objects and morphisms,it is unique. 27 .4.1 Algebras of network operads Our interest in network operads comes from their use in designing and tasking networksof mobile agents. The operations in a network operad are ways of assembling larger networksof a given kind from smaller ones. To describe how these operations act in a concretesituation we need to specify an algebra of the operad. The flexibility of this approach tosystem design takes advantage of the fact that a single operad can have many differentalgebras, related by homomorphisms.An algebra A of a typed operad O specifies a set A ( t ) for each type t ∈ T such that theoperations of O can be applied to act on these sets. That is, each algebra A specifies: • for each type t ∈ T , a set A ( t ), and • for any types t , . . . , t n , t ∈ T , a function α : O ( t , . . . , t n ; t ) → hom( A ( t ) × · · · × A ( t n ) , A ( t ))obeying some rules that generalize those for the action of a monoid on a set [Yau16]. Allthe examples in this section are algebras of network operads constructed from one-colorednetwork models F : S → Mon . This allows us to use Theorem 2.30, which describes O F explicitly.The most basic algebra of such a network operad O F is its ‘canonical algebra’, where itacts on the kind of network described by the network model F : Example 2.32 ( The canonical algebra ) . Let F : S → Mon be a one-colored networkmodel. Then the operad O F has a canonical algebra A F with A F ( n ) = F ( n )for each n ∈ N , the type set of O F . In this algebra any operation( σ, g ) ∈ O F ( n , . . . , n k ; n ) = S n × F ( n )acts on a k -tuple of elements h i ∈ A F ( n i ) = F ( n i ) (1 ≤ i ≤ k )to give α ( σ, g )( h , . . . , h k ) = g ∪ σ ( h ⊔ · · · ⊔ h k ) ∈ A ( n ) . Here we use Theorem 2.3, which gives us the ability to overlay networks using the monoidstructure ∪ : F ( n ) × F ( n ) → F ( n ), take their ‘disjoint union’ using maps ⊔ : F ( m ) × F ( m ′ ) → F ( m + m ′ ), and act on F ( n ) by elements of S n . Using the equations listed in this theoremone can check that α obeys the axioms of an operad algebra.When we want to work with networks that have more properties than those capturedby a given network model, we can equip elements of the canonical algebra with extraattributes. Three typical kinds of network attributes are vertex attributes, edge attributes,and ‘global network’ attributes. For our present purposes, we focus on vertex attributes.Vertex attributes can capture internal properties (or states) of agents in a network such astheir locations, capabilities, performance characteristics, etc.28 xample 2.33 ( Independent vertex attributes ) . For any one-colored network model F : S → Mon and any set X , we can form an algebra A X of the operad O F that consists ofnetworks whose vertices have attributes taking values in X . To do this, we define A X ( n ) = F ( n ) × X n . In this algebra, any operation( σ, g ) ∈ O F ( n , . . . , n k ; n ) = S n × F ( n )acts on a k -tuple of elements( h i , x i ) ∈ F ( n i ) × X n i (1 ≤ i ≤ k )to give α X ( σ, g ) = ( g ∪ σ ( h ⊔ · · · ⊔ h k ) , σ ( x , . . . , x k )) . Here ( x , . . . , x k ) ∈ X n is defined using the canonical bijection X n ∼ = k Y i =1 X n i when n + · · · + n k = n , and σ ∈ S n acts on X n by permutation of coordinates. In otherwords, α X acts via α on the F ( n i ) factors while permuting the vertex attributes X n in thesame way that the vertices of the network h ⊔ · · · ⊔ h k are permuted.One can easily check that the projections F ( n ) × X n → F ( n ) define a homomorphismof O F -algebras, which we call π X : A X → A. This homomorphism ‘forgets the vertex attributes’ taking values in the set X . Example 2.34 ( Simple networks with a rule obeyed by edges ) . Let O SG be thesimple network operad as explained in Example 2.29. We can form an algebra of theoperad O SG that consists of simple graphs whose vertices have attributes taking values insome set X , but where an edge is permitted between two vertices only if their attributesobey some condition. We specify this condition using a symmetric function p : X × X → B where B = { F, T } . An edge is not permitted between vertices with attributes ( x , x ) ∈ X × X if this function evaluates to F .To define this algebra, which we call A p , we let A p ( n ) ⊆ SG( n ) × X n be the set of pairs( g, x ) such that for all edges { i, j } ∈ g the attributes of the vertices i and j make p true: p ( x ( i ) , x ( j )) = T. There is a function τ p : A X ( n ) → A p ( n )29hat discards edges { i, j } for which p ( x ( i ) , x ( j )) = F . Recall that A X ( n ) = SG( n ) × X n ,and recall from Example 2.12 that we can regard SG( n ) as the set of functions g : E ( n ) → B .Then we define τ p by τ p ( g, x ) = ( g, x )where g { i, j } = (cid:26) g { i, j } if p ( x ( i ) , x ( j )) = TF if p ( x ( i ) , x ( j )) = F. We can define an action α p of O SG on the sets A p ( n ) with the help of this function. Namely,we take α p to be the composite O SG ( n , . . . , n k ; n ) × A p ( n ) × · · · × A p ( n k ) O SG ( n , . . . , n k ; n ) × A X ( n ) × · · · × A X ( n k ) A X ( n ) A p ( n ) α X τ p where the action α X was defined in Example 2.33. One can check that α p makes the sets A p ( n ) into an algebra of O SG , which we call A p . One can further check that the maps τ define a homomorphism of O SG -algebras, which we call τ p : A X → A p . Example 2.35 ( Range-limited networks ) . We can use the previous examples to modelrange-limited communications between entities in a plane. First, let X = R and form thealgebra A X of the simple network operad O SG . Elements of A X ( n ) are simple graphs withvertices in the plane.Then, choose a real number L ≥ d be the usual Euclidean distance functionon the plane. Define p : X × X → B by setting p ( x, y ) = T if d ( x, y ) ≤ L and p ( x, y ) = F otherwise. Elements of A p ( n ) are simple graphs with vertices in the plane such that no edgehas length greater than L . Example 2.36 ( Networks with edge count limits ) . Recall the network model formultigraphs MG + , defined in Example 2.6 and clarified in Example 2.13. An element ofMG + ( n ) is a multigraph on the set n , namely a function g : E ( n ) → N where E ( n ) is theset of 2-element subsets of n . If we fix a set X we obtain an algebra A X of O MG + asin Example 2.33. The set A X ( n ) consists of multigraphs on n where the vertices haveattributes taking values in X .Starting from A X we can form another algebra where there is an upper bound on howmany edges are allowed between two vertices, depending on their attributes. We specifythis upper bound using a symmetric function b : X × X → N .
30o define this algebra, which we call A b , let A b ( n ) ⊆ MG + ( n ) × X n be the set of pairs( g, x ) such that for all { i, j } ∈ E ( n ) we have g ( i, j ) ≤ b ( x ( i ) , x ( j )) . Much as in Example 2.34 there is function π : A X ( n ) → A b ( n )that enforces this upper bound: for each g ∈ A X ( n ) its image π ( g ) is obtained by reducingthe number of edges between vertices i and j to the minimum of g ( i, j ) and β ( i, j ): π ( g )( i, j ) = g ( i, j ) min β ( i, j ) . We can define an action α b of O MG on the sets A b ( n ) as follows: O MG ( n , . . . , n k ; n ) × A p ( n ) × · · · × A p ( n k ) O MG ( n , . . . , n k ; n ) × A X ( n ) × · · · × A X ( n k ) A X ( n ) A p ( n ) α X π One can check that α b indeed makes the sets A b ( n ) into an algebra of O MG + , which we call A b , and that the maps π p define a homomorphism of O MG + -algebras, which we call π p : A X → A b . Example 2.37 ( Range-limited networks, revisited ) . We can use Example 2.36 tomodel entities in the plane that have two types of communication channel, one of which hasrange L and another of which has a lesser range L < L . To do this, take X = R anddefine b : X × X → N by b ( x, y ) = d ( x, y ) > L L < d ( x, y ) ≤ L d ( x, y ) ≤ L Elements of A b ( n ) are multigraphs with vertices in the plane having no edges betweenvertices whose distance is > L , at most one edge between vertices whose distance is ≤ L but > L , and at most two edges between vertices whose distance is ≤ L .Moreover, the attentive reader may notice that the action α b of O MG + for this specificchoice of b factors through an action of O Γ B , where Γ B is the network model defined inExample 2.15. That is, operations O Γ B ( n , . . . , n k ; n ) = S n × Γ B ( n ) where Γ B ( n ) is theset of multigraphs on n with at most edges between vertices are sufficient to compose theserange-limited networks. This is due to the fact that the values of this b : X × X → N areat most 2. 31hese examples indicate that vertex attributes and constraints can be systematicallyadded to the canonical algebra to build more interesting algebras, which are related byhomomorphisms. Example 2.33 illustrates how adding extra attributes to the networksin some algebra A can give networks that are elements of an algebra A ′ equipped with ahomomorphism π : A ′ → A that forgets these extra attributes. Example 2.36 illustrateshow imposing extra constraints on the networks in some algebra A can give an algebra A ′ equipped with a homomorphism τ : A → A ′ that imposes these constraints: this works onlyif there is a well-behaved systematic procedure, defined by τ , for imposing the constraintson any element of A to get an element of A ′ .The examples given so far scarcely begin to illustrate the rich possibilities of networkoperads and their algebras. In particular, it is worth noting that all the specific examples ofnetwork models described here involve commutative monoids. However, noncommutativemonoids are also important. Suppose, for example, that we wish to model entities witha limited number of point-to-point communication interfaces—e.g. devices with a finitenumber p of USB ports. More formally, we wish to act on sets of degree-limited networks A deg ( n ) ⊂ SG( n ) × N n made up of pairs ( g, p ) such that the degree of each vertex i , deg( i ) , is at most the degree-limiting attribute of i : deg( i ) ≤ p ( i ). Na¨ıvely, we might attempt toconstruct a map τ deg : A N → A deg as in Example 2.36 to obtain an action of the simplenetwork operad O SG . However, this is turns out to be impossible. For example, if attemptto build a network from devices with a single USB port, and we attempt to connect multipleUSB cables to one of these devices, the relevant network operad must include a rule sayingwhich attempts, if any, are successful. Since we cannot prioritize links from some verticesover others—which would break the symmetry built into any network model—the order inwhich these attempts are made must be relevant. Since the monoids SG( n ) are commutative,they cannot capture this feature of the situation.The solution is to use a class of noncommutative monoids dubbed ‘graphic monoids’by Lawvere [Law89b]: namely, those that obey the identity aba = ab . These allow us toconstruct a one-colored network model Γ : S → Mon whose network operad O Γ acts on A deg .For our USB device example, the relation aba = ab means that first attempting to connectsome USB cables between some devices ( a ), second attempting to connect some furtherUSB cables ( b ), and third attempting to connect some USB cables precisely as attemptedin the first step ( a , again) has the same result as only performing the first two steps ( ab ).We explore more applications of noncommutativity in network models in Chapter 3.32 hapter 3 Noncommutative Network Models
In Theorem 2.11, we gave a functorial construction of a network model from a monoid,which we call the ordinary network model for weighted graphs . In this chapter, we providea different construction in order to realize a larger class of networks as algebras of networkoperads, which we call the free varietal network model for weighted graphs . In Section 3.4,we give an example of a family of networks which cannot form an algebra for any ordinarynetwork model for weighted graphs, but does for a varietal one. In this chapter, we give aconstruction for the free network model on a given monoid. This describes networks whichlook like the given monoid when you restrict to looking at the combinatorial behavior ata single pair of nodes. In Section 3.3, we give a concrete construction of a left adjoint tothe functor which evaluates a network model at its second level. This requires a categoricaltreatment and generalization of Green’s theory of products of groups indexed by a graph,(i.e. graph products of groups ) [Gre90], which we give in Section 3.2.This construction is designed to model networks which carry information on the edges.For example, with N a monoid under addition, Γ N is a network model for loopless undirectedmultigraphs where overlaying is given by adding the number of edges. A similar exampleis Γ B = SG. There is a monoid homomorphism N → B which sends all but 0 to T . Thisinduces a map of network models Γ N → Γ B . Essentially this map reduces the information ofa graph from the number of connections between each pair of vertices to just the existenceof any connection. Example 3.1 ( Algebra for range-limited communication).
Consider a communica-tion network where each node represents a boat and an edge between two nodes representsa working communication channel between the corresponding boats. Some forms of com-munication are restricted by the distance between those communicating. Assume that thereis a known maximal distance over which our boats can communicate. Networks of this sortform an algebra of the simple graphs operad in the following way.Let (
X, d ) be a metric space, and 0 ≤ L ∈ R . Our boats will be located at points inthis space. The operad O SG has an algebra ( A d,L , α ) defined as follows. The set A d,L ( n )is the set of pairs ( h, f ) where h ∈ SG( n ) is a simple graph and f : n → X is a functionsuch that if { v , v } is an edge in g then d ( f ( v ) , f ( v )) ≤ L . The number L represents the33aximal distance over which the boat’s communication channels operate. Notice that thiscondition does not demand that all connections within range must be made. An operation( σ, g ) ∈ O SG ( n , . . . , n k ; n ) acts on a k -tuple ( h i , f i ) ∈ A d,L ( n i ) by α ( σ, g )(( h , f ) , . . . , ( h k , f k )) = ( g ∪ σ ( h ⊔ · · · ⊔ h k ) , f ⊔ · · · ⊔ f k ) . Elements of this algebra are simple graphs in the space X with an upper limit on edgelengths. When an operation acts on one of these, it tries to put new edges into the graph,but fails to when the range limit is exceeded [BFMP20].A characteristic of the construction given in Theorem 2.11 is that elements of the re-sulting monoids that correspond to different edges automatically commute with each other.For example, for a monoid M , the fourth constituent monoid of the ordinary M networkmodel is Γ M (4) = M . Then the element ( m , , , , ,
0) represents a graph with one edgewith weight m ∈ M , the element (0 , m , , , ,
0) represents a graph with a different edgewith weight m ∈ M , and( m , , , , , ∪ (0 , m , , , ,
0) = ( m , m , , , , , m , , , , ∪ ( m , , , , , . This commutativity between edges means that networks given by ordinary networkmodels cannot record information about the order in which edges were added to it. Theability to record such information about a network is desirable, for example, if one wishesto model networks which have a limit on the number of connections each agent can maketo other agents.The degree of a vertex in a simple graph is the number of edges which include thatvertex. The degree of a graph is the maximum degree of its vertices. A graph is said tohave degree bounded by k , or simply bounded degree , if the degree of each vertex isless than or equal to k . Let B k ( n ) denote the set of networks with n vertices and degreebound k . One might guess that the family of such networks could form an algebra for thesimple graphs operad. Question.
Does the collection of networks of bounded degree form an algebra of a networkoperad? If so, is there such an algebra which is useful in applications?Specifically, can networks of bounded degree form an algebra of O SG , the simple graphoperad? Setting two graphs next to each other will not change the degree of any of thevertices. Overlaying them almost definitely will, which makes defining an action of SG( n )on B k ( n ) less obvious.Ordinary network models are not sufficient to model this type of network because thegraph monoids it produced could not remember the order that edges were added into anetwork. Even if M is a noncommutative monoid, since Γ M is a product of several copiesof M , one for each pair of vertices, it cannot distinguish the order that two different edgestouching v were added to a network if their other endpoints are different.Instead of taking the product of (cid:0) n (cid:1) copies of M , we consider taking the coproduct, soas not to impose any commutativity relations between the edges. Since the lax structure34ap ⊔ : F ( m ) × F ( n ) → F ( m + n ) associated to a network model F : S → Mon must be amonoid homomorphism, then( a ⊔ b ) ∪ ( c ⊔ d ) = ( a ∪ c ) ⊔ ( b ∪ d ) . In particular, if we let ∅ denote the the identity of F ( n ) for any n , then( a ⊔ ∅ ) ∪ ( ∅ ⊔ b ) = ( a ∪ ∅ ) ⊔ ( ∅ ∪ b )= ( ∅ ∪ a ) ⊔ ( b ∪ ∅ )= ( ∅ ⊔ b ) ∪ ( a ⊔ ∅ ) . This is reminiscent of the Eckmann–Hilton argument (see Appendix A), but notice that thedomains of the operations ∪ and ⊔ are not the same. This equation says that elements whichcorrespond to disjoint edges must commute with each other. Simply taking the coproductof (cid:0) n (cid:1) copies of M cannot give the constituent monoids of a network model.For a collection of monoids { M i } i ∈ I , elements of the product monoid which come fromdifferent components always commute with each other. In the coproduct, they never do.A graph product (in the sense of Green [Gre90]) of such a collection allows one to imposecommutativity between certain components and not others by indicating such relations via asimple graph. The calculation above shows that the constituent monoids of a network modelmust satisfy certain partial commutativity relations. We use graph products to constructa family of monoids with the right amount of commutativity to both answer the questionabove and satisfy the conditions of being a network model. The following theorems areproven in Section 3.3. Theorem 3.2.
The functor
NetMod → Mon defined by F F ( ) has a left adjoint Γ − , Mon : Mon → NetMod . The fact that this construction is a left adjoint tells us that the network models con-structed are ones in which the only relations that hold are those that follow from the definingaxioms of network models.A variety of monoids is the class of all monoids satisfying a given set of identities.For example,
Mon has subcategories
CMon of commutative monoids and
GMon of graphicmonoids which are varieties of monoids satisfying the equations ab = ba and aba = ab respectively. Given a variety of monoids V , let NetMod V be the subcategory of NetMod consisting of V -valued network models. We recreate graph products in varieties of monoidsto obtain a more general result. Theorem 3.3.
The functor
NetMod V → V defined by F F ( ) has a left adjoint Γ − , V : V →
NetMod V . In particular, if V = CMon , since products and coproducts are the same in
CMon , theordinary M network model and the CMon varietal M network model are also the same.35ote that this does not indicate that Γ − , V is a complete generalization of Γ − from Theorem2.11, since Γ M is not an example of Γ − , V when M is not commutative.The ordinary construction for a network model given a monoid M has constituentmonoids given by finite cartesian powers of M . To include the networks described in thequestion above into the theory of network models, we must construct a network model froma given monoid which does not impose as much commutativity as the ordinary constructiondoes, specifically among elements corresponding to different edges. The first attempt at asolution is to use coproducts instead of products. However, in this section we saw that wecannot create the constituent monoids of a network model simply by taking them to becoproducts of M instead of products. There must be some commutativity between differentedges, specifically between edges which do not share a vertex.Given a monoid M , we want to create a family of monoids indexed by N , the n th ofwhich looks like a copy of M for each edge in the complete graph on n , has minimal commu-tativity relations between these edge components, but does have commutativity relationsbetween disjoint edges. Partial commutativity like this can be described with Green’s graphproducts, which we describe in Section 3.2.1. The type of graph which describes disjoint-ness of edges in a graph as we need is called a Kneser graph , which we describe in Section3.2.2. Besides concerning ourselves with relations between edge components, sometimes wealso want the constituent monoids in a network model to obey certain relations which M obeys. In Section 3.2.4 we describe varieties of monoids and a construction which producesmonoids in a chosen variety. In Section 3.3 we prove this construction is functorial, and inSection 3.4 we use this construction to give a positive answer to the question. This section is dedicated to constructing the constituent monoids for the network modelswe want. In this section there are two different ways that graphs are being used. It isimportant that the reader does not get these confused. One way is the graphs which areelements of the constituent monoids of the network models we are constructing. The otherway we use graphs is to index the
Green product (which we define in Section 3.2.1) todescribe commutativity relations in the constituent monoids of the network models we areconstructing.A network model is essentially a family of monoids with properties similar to the simplegraphs example, so we think of the elements of these monoids as graphs, and we think of theoperation as overlaying the graphs. These monoids have partial commutativity relationsthey must satisfy, as we see in Section 2.2. The graphs we use in the Green product, theKneser graphs, are there to describe the partial commutativity in the constituent monoids.
Given a family of monoids { M v } v ∈ V indexed by a set V , there are two obvious waysto combine them to get a new monoid, the product and the coproduct. From an algebraicperspective, a significant difference between these two is whether or not elements thatcame from different components commute with each other. In the product they do. In36he coproduct they do not. Green products , or commonly graph products , of groups wereintroduced in 1990 by Green [Gre90], and later generalized to monoids by Veloso da Costa[Vel01]. The idea provides something of a sliding scale of relative commutativity betweencomponents. We follow [FK09] in the following definitions.By a simple graph G = ( V, E ), we mean a set V which we call the set of vertices, and aset E ⊆ (cid:0) V (cid:1) , which we call the set of edges. A map of simple graphs f : ( V, E ) → ( V ′ , E ′ )is a function f : V → V ′ such that if { u, v } ∈ E then { f ( u ) , f ( v ) } ∈ E ′ . Let SimpleGph denote the category of simple graphs and maps of simple graphs.For a set V , a family of monoids { M v } v ∈ V , and a simple graph G = ( V, E ), the G Green product (or simply
Green product when unambiguous) of { M v } v ∈ V , denoted G ( M v ), is G ( M v ) = a v ∈ V M v ! /R G where R G is the congruence generated by the relation { ( mn, nm ) | m ∈ M v , n ∈ M u , u, v are adjacent in G } where the operation in the free product is denoted by concatenation. If G is the completegraph on n vertices, then G ( M v ) ∼ = Q M v . If G is the n -vertex graph with no edges, then G ( M v ) ∼ = ` M v .We call each M v a component of the Green product. Elements of G ( M v ) are written as expressions as in the free product, m v . . . m v k k ∈ G ( M v ) where the superscript indicatesthat m i ∈ M v i . We often consider Green products of several copies of the same monoid,so this notation allows one to distguish elements coming from different components of theproduct, even if they happen to come from the same monoid. The intention and result ofthe imposed relations is that for an expression m v . . . m v k k of an element, if there is an i such that { v i , v i +1 } ∈ E , then we can rewrite the expression by replacing m v i i m v i +1 i +1 with m v i +1 i +1 m v i i . This move is called a shuffle , and two expressions are called shuffle equivalent if one can be obtained from the other by a sequence of shuffles. An expression m v . . . m v k k is reduced if whenever i < j and v i = v j , there exists l with i < l < j and { v i , v l } / ∈ E .If two reduced expressions are shuffle equivalent, they are clearly expressions of the sameelement. The converse is also true. Theorem 3.4 ([FK09], Thm. 1.1) . Every element of M is represented by a reduced ex-pression. Two reduced expressions represent the same element of M if and only if they areshuffle equivalent. In this section, we use a categorical description of Green products to define a similarconstruction in a more general context. The relevant property of
Mon that we need for thisgeneralization is that
Mon is a pointed category .Let C be a category. An object of C which is both initial and terminal is called a zeroobject . If C has such an object, C is called a pointed category [Qui67]. For any twoobjects A, B of a pointed category, there is a unique map 0 : A → B which is the compositeof the unique map from A to the zero object, and the unique map from the zero object37o B . If C is a pointed category with finite products, then for two objects A, B of C , theobjects admit canonical maps A → A × B . AA × BA B ∃ ! i A π A π B So we have the following maps
A BA × BA B i A i B π B π A satisfying the following properties. π A i A = 1 A π B i B = 1 B π B i A = 0 π A i B = 0This is suggestive of a biproduct, but in a general pointed category A × B is not necessarilyisomorphic to A + B .In Section 3.3, we use a generalized Green product to construct network models. Ageneralized Green product is a colimit of a diagram whose shape is derived from a givengraph. We describe the shapes of the diagrams here with directed multi-graphs. We referto them here as quivers to help distinguish them from other variants of graphs and the rolethey play in this chapter. A quiver is a pair of sets E , V , respectively called the set of edges and set of vertices , and a pair of functions s, t : E → V assigning to each edge its starting vertex and its terminating vertex respectively. A map of quivers is a pair of functions E E V V s t f E s t f V such that the s -square and the t -square both commute.We will use the word cospan to refer to the quiver with the following shape. • → • ← • Define a functor IC : SimpleGph → Quiv which replaces every edge with a cospan ( IC standsfor ‘insert cospan’). Specifically, given a simple graph ( V, E ) where E ⊆ (cid:0) V (cid:1) , define the38uiver Q ⇒ Q where Q = V ⊔ E and Q = { ( v, e ) ∈ V × E | v ∈ e } , then define the sourcemap s : Q → Q by projection onto the first component, and the target map t : Q → Q by projection onto the second component. For example, the simple graph1 234gives the quiver 1 { , } { , } { , } { , } G = ( V, E ) and G ′ = ( V ′ , E ′ ) be simple graphs, and f : G → G ′ a map of simplegraphs. Define a map of quivers ICf : IC ( G ) → IC ( G ′ ) by ICf = f V ⊔ f E and ICf ( v, e ) =( f V ( v ) , f E ( e )). IC ( G ) IC ( G ′ ) IC ( G ) IC ( G ′ ) s G t G ICf s G ′ t G ′ ICf This construction gives a coproduct preserving functor IC : SimpleGph → Quiv .Let F : Quiv → Cat denote the free category (or path category) functor [ML98]. Since F is a left adjoint, it preserves colimits. Notice that any quiver of the form IC ( G ) wouldnever have a path of length greater than 1. Thus the free path category on IC ( G ) simplyhas identity morphisms adjoined.The objects in the category F ( IC ( G )) come from two places. There is an object for eachvertex of G , and there is an object at the apex of the cospan for each edge in G . We callthese two subsets of objects vertex objects and edge objects . We abuse notation andrefer to the object given by the vertex u by the same name, and similar for edge objects.If { M v } v ∈ V is a family of monoids indexed by the set V , that means that there is afunctor M : V → Mon from the set V thought of as a discrete category. Notice that if G is asimple graph with vertex set V , then the discrete category V is a subcategory of F ( IC ( G )).We can then extend the functor M to D : F ( IC ( G )) → Mon in the following way. Obviously we let D ( u ) = M u for a vertex object u . If { u, v } is an edgein G , then D ( { u, v } ) = M u × M v . The morphism ( u, { u, v } ) is sent to the canonical map39 u → M u × M v . For example, for a family of monoids { M , . . . M } , we have the followingdiagram. M M × M M M × M M × M M × M M M Since there are no non-trivial pairs of composable morphisms in categories of the form F ( IC ( G )), nothing further needs to be checked to confirm D is a functor.Despite the way we are denoting these products, we are not considering them to beordered products. Alternatively, we could have used a more cumbersome notation thatdoes not suggest any order on the factors. Theorem 3.5.
Let V be a set, { M v } v ∈ V be a family of monoids indexed by V , and G =( V, E ) be a simple graph with vertex set V . The G Green product of M v is the colimit ofthe diagram D : F ( IC ( G )) → Mon defined as above. G ( M v ) ∼ = colim D. Proof.
We show that G ( M v ) satisfies the necessary universal property. The vertex objectsin the diagram have inclusion maps into the edge objects i u,v : M u → M u × M v , and all theobjects have inclusion maps into G ( M v ), j u : M u → G ( M v ) and j u,v : M u × M v → G ( M v )such that j u,v ◦ i u,v = j u . Note that due to the fact that we have unordered products forobjects, there is some redundancy in our notation, namely j u,v = j v,u . If we have a monoid Q and maps f u : M u → Q and f u,v : M u × M v → Q such that f u,v = f v,u f u,v ◦ i u,v = f u , then we define a map φ : G ( M v ) → Q by φ ( m v . . . m v k k ) = f v ( m ) . . . f v k ( m k ). Since thismap is defined via expressions of elements, Theorem 3.4 tells us that to check this map iswell-defined, we need only check that the values of two expressions that differ by a shuffleare the same. Let m v . . . m v k k be an expression, and i such that { v i , v i +1 } ∈ E . φ ( m v i i m v i +1 i +1 ) = f v i ( m i ) f v i +1 ( m i +1 )= f v i ,v i +1 ( m i , m i +1 )= f v i +1 ( m i +1 ) f v i ( m i )= φ ( m v i +1 i +1 m v i i )It is clear that φ ( m v . . . m v k k ) = φ ( m v . . . m v i − i − ) φ ( m v i i m v i +1 i +1 ) φ ( m v i +2 i +2 . . . m v k k ) ,
40o two shuffle equivalent expressions have the same value under φ , and φ is well-defined. Itis clearly a monoid homomorphism, and has the property φ ◦ j u = f u and φ ◦ j u,v = f u,v .To show this map is unique, assume there is another such map ψ : G ( M v ) → Q . Since ψ ◦ j u = f u , then ψ ( m u ) = f ( u ), and ψ ( m v . . . m v k k ) = ψ ( m v ) . . . ψ ( m v k k )= f v ( m ) . . . f v k ( m k )= φ ( m v . . . m v k k ) . This result makes it reasonable to generalize Green products in the following way.
Definition 3.6.
Let C be a pointed category with finite products and finite colimits, V aset, { A v } v ∈ V a family of objects of C indexed by V , and G a simple graph with vertex set V . Let D : F ( IC ( G )) → C be the diagram defined by v A v , { u, v } 7→ A u × A v , and themorphism ( u, { u, v } ) is mapped to the inclusion A u → A u × A v as above. The G Greenproduct of { A v } v ∈ V is the colimit of D in C , G C ( A v ) = colim D. If C = Mon , we denote the Green product simply as G ( A v ).In Section 3.3, we use this general notion of graph products in varieties of monoids to construct network models whose constituent monoids are in those varieties. Note thatsince F ◦ IC is a functor, the group Aut ( G ) of graph automorphisms of G naturally acts on G C ( A v ). We focus here on a special family of simple graphs known as the
Kneser graphs [Lov78].The
Kneser graph KG n,m has vertex set (cid:0) nm (cid:1) , the set of m -element subsets of an n -elementset, and an edge between two vertices if they are disjoint subsets. Since a simple graph isdefined as a collection of two-element subsets of an n -element set, the Kneser graph KG n, has a vertex for each edge in the complete graph on n , and has an edge between every pairof vertices which correspond to disjoint edges. So the Kneser graph KG n, can be thoughtof as describing the disjointness of edges in the complete graph on n . For instance, thecomplete graph on 5 is 41nd the corresponding Kneser graph KG , is the Petersen graph:For sets X, Y and a function f : X → Y , let f [ U ] = { f ( x ) | x ∈ U } for U ⊆ X . Let FinInj denote the category of finite sets and injective functions.
Lemma 3.7.
For k ∈ N , there is a functor (cid:0) − k (cid:1) : FinInj → FinInj which sends X to (cid:0) Xk (cid:1) theset of k -element subsets of X , and injections f : X → Y to the functions (cid:0) fk (cid:1) : (cid:0) Xk (cid:1) → (cid:0) Yk (cid:1) defined by (cid:0) fk (cid:1) ( U ) = f [ U ] . Note that this result holds for
Inj the category of sets and injective functions, but weonly require
FinInj for our purposes.
Proof. If f : X → Y is an injection, then | f [ U ] | = | U | for U ⊆ X . It then makes senseto restrict the induced map on power sets to subsets of a fixed cardinality. The map (cid:0) fk (cid:1) : (cid:0) mk (cid:1) → (cid:0) nk (cid:1) defined by (cid:0) fk (cid:1) ( U ) = f [ U ] is then well defined. If f [ U ] = f [ V ] and x ∈ U ,then f ( x ) ∈ f [ U ] = f [ V ], which implies there is a y ∈ V such that f ( y ) = f ( x ). Since f isinjective, then x = y ∈ V . Thus U = V by symmetry.Let i X and i Y denote the following inclusion maps. X YX + Y i X i Y Since these maps are injective, they induce maps (cid:0) i X k (cid:1) , (cid:0) i Y k (cid:1) , and we get a map Φ X,Y : (cid:0) Xk (cid:1) + (cid:0) Yk (cid:1) → (cid:0) X + Yk (cid:1) by the universal property in the following way. (cid:0) Xk (cid:1) (cid:0) Yk (cid:1)(cid:0) Xk (cid:1) + (cid:0) Yk (cid:1)(cid:0) X + Yk (cid:1) j X ( iXk ) j Y ( iYk ) ∃ !Φ X,Y emma 3.8. The functor (cid:0) − k (cid:1) is made lax symmetric monoidal ( (cid:18) − k (cid:19) , Φ , φ ) : ( FinInj , + , ∅ ) → ( FinInj , + , ∅ ) where the components of Φ are defined as above.Proof. The family of maps { Φ X,Y } is clearly a natural transformation. There is no choice forthe map φ : ∅ → (cid:0) ∅ k (cid:1) . The left and right unitor laws hold trivially. Checking the coherenceconditions for the associator and the symmetry are straightforward computations.For n, k ∈ N , the simple graph KG n,k has vertex set V = (cid:0) nk (cid:1) and edge set {{ u, v } ⊆ (cid:0) V (cid:1) | u ∩ v = ∅} . If f : m → n is injective, then we get a map (cid:0) fk (cid:1) between the vertex setsof KG m,k and KG n,k . Let { u, v } ∈ (cid:0) V (cid:1) be an edge in KG m,k . Then f [ u ] ∩ f [ v ] = ∅ byinjectivity, so { f [ u ] , f [ v ] } is an edge of KG n,k . An injection f then induces a map of graphs,denoted KG f,k : KG m,k → KG n,k . Since (cid:0) fk (cid:1) is injective, KG f,k is an embedding. Nothingabout this construction requires finiteness of the sets involved, but our applications onlycall for finite graphs. Proposition 3.9.
For k ∈ N , there is functor KG − ,k : FinInj → SimpleGph which sends n to KG n,k and f : m → n to KG f,k . Not only does KG m,k embed into KG n,k when m < n , but KG m,k + KG n,k embedsinto KG m + n,k . We construct the embedding KG m,k + KG n,k → KG m + n,k by using the laxstructure map from Lemma 3.8 for the vertex map, Φ m,n : (cid:0) mk (cid:1) + (cid:0) nk (cid:1) → (cid:0) m + nk (cid:1) . Restrictingthis map to either (cid:0) mk (cid:1) (resp. (cid:0) nk (cid:1) ) gives the map (cid:0) i m k (cid:1) (resp. (cid:0) i n k (cid:1) ) which we already knowinduces a map of graphs. Thus Φ m,n induces a map of graphs, which we call Ψ m,n . Proposition 3.10.
The functor KG − ,k is made lax (symmetric) monoidal ( KG − ,k , Ψ) : (
Inj , +) → ( SimpleGph , +) where the components of Ψ are defined as above.Proof. All the necessary properties for Ψ are inherited immediately from Φ.Let ( L, Λ) : (
Inj , +) → ( Cat , +) be the composite L = F ◦ IC ◦ KG − , with the obviouslaxator. Let M be a monoid. Then from the construction given in the previous subsection,for each n we get a diagram D n : L ( n ) → Mon which sends all vertex objects to M , all edgeobjects to M × M , and all nontrivial morphisms to inclusions M → M × M . Taking thecolimit of D n then gives the Green product KG n, ( M ). Note that we identify constituentmonoids with the corresponding submonoid of the graph product when this can be donewithout confusion. Proposition 3.11.
Let M p,q be a (cid:0) m + n (cid:1) family of monoids, and G and G be graphs with m and n vertices respectively. Let a ∈ M p ,q with p , q ≤ m and a ∈ M p ,q with p , q > m ,and let a , a be their values under the canonical inclusions M p,q ֒ → ( G ⊔ G )( M p,q ) . Then a a = a a in ( G ⊔ G )( M p,q ) .Proof. By definition, there is an edge in the Kneser graph KG m + n, between the vertices p , q and p , q . This imposes the desired commutativity relation.43 .2.3 Varieties of Monoids A finitary algebraic theory or Lawvere theory is a category T with finite productsin which every object is isomorphic to a finite cartesian power x n = Q n x of a distinguishedobject x [Law63, ALR03]. An algebra of a theory T , or T -algebra , is a product preservingfunctor T → Set . Let T Alg denote the category of T -algebras with natural transformationsfor morphisms. We are primarily concerned with monoids in this chapter. The theory ofmonoids T Mon has morphisms m : x × x → x and e : x → x , which makes the followingdiagrams commute. x x x × x x x × xx x x m × x x × m m x × e ≃ m e × x ≃ m A variety of T -algebras is a full subcategory of T Alg which is closed under products,subobjects, and homomorphic images. Birkhoff’s theorem implies that this is equivalent tothe category T ′ Alg of algebras of another theory T ′ which has the same morphisms, butsatisfies more commutative diagrams [BS81]. For example, commutative monoids are givenby algebras of the theory of commutative monoids T CMon , which has morphisms m, e asin T Mon , satisfies the same commutative diagrams as T Mon , but also satisfies the followingcommutative diagram x x x mb m where b : x → x is the braid isomorphism. We only use varieties of monoids in thischapter, so we give these “extra” conditions by equations, e.g. commutative monoids arethose which satisfy the equation ab = ba for all elements a, b . We call the extra equationsthe defining equations of the variety.A graphic monoid is a monoid which satisfies the graphic identity : aba = ab for allelements a, b . Graphic monoids are algebras of a theory T GMon . A semigroup obeying thisrelation is known as a left regular band [MSS20]. The term graphic monoid was introducedby Lawvere [Law89a]. Let M be a graphic monoid. If we let b be the unit of M , then thegraphic relation says that a = a . Every element of M is idempotent. If a, c ∈ M , then ca = c if c already has a as a factor.Graphic monoids are present when talking about types of information where a piece ofinformation cannot contain the same piece of information twice. A simple example can beseen in the powerset of a given set X , given the structure of a monoid by union. Of course,this example is overly simple because the operation is commutative idempotent, which isstronger than graphic. A more interesting example can be seen by considering the followingsimple graph. a b cx yWe will define a monoid structure on the set M = { , a, b, c, x, y } in the following way.First, 1 is a freely adjoined identity element. For p, q ∈ M \ { } , define pq as follows. Pick a44eneric point f in p and a generic point g in q . Then move a small distance along a straightline path from f to g . We define the product pq to be the component of the graph you landin. Here are some example computations: ab = x aa = abc = y xb = xac = x ca = y The last two demonstrate that this monoid is not commutative. More complicated examplescan be constructed by using the same idea for the operation, but applying it to differentspaces.The following fact is critical in Chapter 2. It follows immediately from the definitions.
Lemma 3.12.
Every variety of monoids is a pointed category and has finite colimits.
Our motivation for using graphic monoids is that we use the graphic relation to model“commitment” in the following way. Let M be a graphic monoid, where we think of anelement of M as a task or list of tasks. If we first commit to doing task x , and thencommit to doing task y , then we have the element xy as our task list, indicating that wecommitted to x before y . If we then try to commit to to doing x , the graphic relation savesus from recording this information twice. The relation also preserves the order in which wecommitted to x and y : if x is a task list of the form x = ab , and we have committed to xy ,and then try to commit to bc , we get ( xy )( bc ) = ( aby )( bc ) = a ( byb ) c = a ( by ) c = abyc = xyc .We want to construct a network model from a monoid in a variety V which has con-stituent monoids that are also in V . If M is a monoid in a variety V , then each constituentmonoid Γ M ( n ) is a product of several copies of M , and so is also in V by definition. Thusthe ordinary network model (given in Theorem 2.11) restricted to a variety gives a functor V →
NetMod V , where NetMod V denotes the category of V -valued network models.The free product of two monoids is a monoid, M + N an element of which is given bya list with entries in the set M ⊔ N such that if two consecutive entries of a list are eitherboth elements of M or both elements of N , then the list is identified with the list that is thesame everywhere except that those two entries are reduced to one entry occupied by theirproduct. Note that the empty list is identified with both the singleton list consisting of theidentity element of M , and the singleton list consisting of the identity element of N . Freeproducts of monoids gives the coproduct in the category of monoids Mon . Free productsof monoids are very similar to free products of groups, which can be found in most booksintroducing group theory [Hun74].If two monoids M and N are in a variety V , taking their free product will not necessarilyproduce a monoid in V , i.e. varieties are not necessarily closed under the coproduct of Mon .It is easy to find an example demonstrating this. Consider
IMon , the variety of idempotentmonoids, i.e. monoids satisfying the equation x = x for all elements x . The boolean monoid B is an object in IMon . The free product of B with itself B + B can be generated by elements a and b which correspond to the element 1 in each copy of B . The element ab ∈ B + B
45s not idempotent, as abab = ab . However, every variety V does have coproducts. Thecoproduct in a variety of monoids is the quotient of the free product by the congruencerelation generated by the variety’s defining equations. In Section 3.3 we give a construction V →
NetMod V which uses colimits in order to impose minimal relations.Lemma 3.12 tells us that it makes sense to talk about Green products in a variety, whichwe call varietal Green products . In the next section, we use varietal Green products withKneser graphs to construct network models. In this section, we state and prove the main result of this chapter. It says that given amonoid M in a variety V , we can construct a network model whose constituent monoids arealso in V , while avoiding to impose commutativity relations when possible. In the followingsection, we see how this construction resolves the dilemma presented in the question.Let M be a monoid in a variety V . Define Γ M, V ( n ) to be the KG n, Green product of (cid:0) n (cid:1) copies of M . Theorem 3.13.
For V a variety of monoids, Γ − , V : V →
NetMod V is a functor, as givenabove. The network model Γ M, V is called the V -varietal network model for M -weightedgraphs , or just the varietal M network model . In order to prove this, we must first show that a monoid M gives a network model, i.e. alax symmetric monoidal functor. The laxator for Γ M, V is canonically defined, but perhapsit is not as immediate as the one for the ordinary M network model. We treat this firstbefore returning to the proof of the main theorem.Let A and B be objects in a pointed category with finite products and coproducts. Let p A : A × B → A and p B : A × B → B denote the canonical projections, and i A : A → A + B and i B : B → A + B the canonical inclusions. The category CMon of commutative monoidsis such a category. Recall that the operation of a monoid is a monoid homomorphism if andonly if the monoid is commutative. We have A × BA ( A + B ) × ( A + B ) BA + B A + B A + B p A p B i A ∗ i B where ∗ denotes the operation in the commutative monoid A + B , and the dashed arrowis < i A p A , i B p B > given by universal property. The composite of the two maps goingdown the middle is the inverse to the canonical map A + B → A × B . The operation in anoncommutative monoid is not a monoid homomorphism, but all the above maps still exist as functions . Recall that we let ∪ denote the operation in the monoids Γ M, V ( n ). There isalways a homomorphism φ m,n : Γ M, V ( m ) + Γ M, V ( n ) → Γ M, V ( m + n ) by universal propertyof coproducts. Let γ : (Γ M, V ( m ) + Γ M, V ( n )) × (Γ M, V ( m ) + Γ M, V ( n )) → Γ M, V ( m ) + Γ M, V ( n )46enote the monoid operation of the coproduct. Γ M, V ( m ) × Γ M, V ( n )Γ M, V ( m ) (Γ M, V ( m ) + Γ M, V ( n )) × (Γ M, V ( m ) + Γ M, V ( n )) Γ M, V ( n )Γ M, V ( m ) + Γ M, V ( n ) Γ M, V ( m ) + Γ M, V ( n ) Γ M, V ( m ) + Γ M, V ( n )Γ M, V ( m + n ) p p i γ i φ The monoids Γ M, V ( n ) are constructed specifically so that φ ◦ γ ◦ < i ◦ p , i ◦ p > is amonoid homomorphism despite the fact that γ is not.In the proof of the following theorem, we utilize a string diagrammatic calculus suitedfor reasoning in a symmetric monoidal category. We refer the reader to Selinger’s thoroughexposition of such string diagramatic languages and their use in category theory [Sel11]. Lemma 3.14.
The function Γ M, V ( m ) × Γ M, V ( n ) → Γ M, V ( m + n ) given by φ ◦ ( i ◦ p ∪ i ◦ p ) is a monoid homomorphism. Moreover, the family of maps of this form gives a naturaltransformation, denoted ⊔ .Proof. We have the following actors in play: • the monoid operations ∪ k : Γ M, V ( k ) for k = m, n, m + n (we leave off the subscriptsbelow) • the monoid operation of the coproduct γ : (Γ M, V ( m ) + Γ M, V ( n )) × (Γ M, V ( m ) + Γ M, V ( n )) → Γ M, V ( m ) + Γ M, V ( n ) • the canonical inclusion maps i : Γ M, V ( m ) → Γ M, V ( m ) → Γ M, V ( n ) and i : Γ M, V ( n ) → Γ M, V ( m ) → Γ M, V ( n ) • the canonical map φ : Γ M, V ( m ) + Γ M, V ( n ) → Γ M, V ( m + n )We represent these string diagramatically (read from top to bottom) as follows. Notethat these are digrams in Set with its cartesian monoidal structure, because the monoidoperations ∪ k and γ are not necessarily monoid homomorphisms. ∪ , γ , i , i , φ
47e define ⊔ : Γ M, V ( m ) × Γ M, V ( n ) → Γ M, V ( m + n ) as follows. ⊔ = φ φ ∪ i i (3.1)Proposition 3.11 gives the following equation. φ φ ∪ i i = φ φ ∪ i i (3.2)Since φ is a homomorphism, we get the following equation. ∪ φ φ = γφ (3.3)Since i and i are homomorphisms, we get the following equations. γi j i j = γi j (3.4)48e want to show that ( g ⊔ h ) ∪ ( g ′ ⊔ h ′ ) = ( g ∪ g ′ ) ⊔ ( h ∪ h ′ ). We compute: ⊔ ⊔∪ =(3.1) ∪ ∪∪ i i i i φ φ φ φi i i i φ φ φ φ ∪∪ ∪ =(3.2)= i i i i φ φ φ φ ∪∪ ∪ = i i i i φ φ φ φ ∪ ∪∪ i i i i γ γφ φ ∪ =(3.3) =(3.4) ⊔∪ ∪ i φ i φ =(3.1) ⊔∪ ∪ Let σ ∈ S m and τ ∈ S n . ThenΓ M, V ( σ + τ )( g ⊔ h ) = Γ M, V ( σ + τ ) φ ( i ( g ) ∪ i ( h ))= Γ M, V ( σ ) φ ( i ( g )) ∪ Γ M, V ( τ ) φ ( i ( h ))= Γ M, V ( σ ( g )) ⊔ Γ M, V ( τ ( h )) ,
49o the following diagram commutes.Γ M, V ( m ) × Γ M, V ( n ) Γ M, V ( m + n )Γ M, V ( m ) × Γ M, V ( n ) Γ M, V ( m + n ) ⊔ Γ M, V ( σ ) × Γ M, V ( τ ) Γ M, V ( σ + τ ) ⊔ Thus ⊔ is a natural transformation. Proof of Theorem 3.13.
Checking the coherence conditions for ⊔ to be a laxator is a straight-forward computation. Let f : M → N . Then define the natural transformation f V : Γ M, V → Γ N, V with components ( f V ) n : Γ M, V ( n ) → Γ N, V ( n ) given by the universal property. Com-position is clearly preserved. Theorem 3.15.
The functor Γ − , V is left adjoint to E : NetMod V → V where E ( F ) = F ( ) for ( F, Φ) : ( S , +) → ( V , × ) a V -network model. Because of this, we call Γ M, V the free V -valued network model on the monoid M or the free V network model on M . Proof.
By construction, Γ M, V ( ) = M , so let the unit η = 1 V : 1 V → Γ − , V ( ).We use the universal property of Γ M, V to construct the counit. We define a map F ( ) → F ( n ) for each vertex in KG n, , and a map F ( ) × F ( ) → F ( n ) for each edge in KG n, .If i, j ≤ n , then F ((1 i )(2 j )) : F ( n ) → F ( n ). If e is the unit of the monoid F ( n − ),and m ∈ F ( ), then Φ ,n − ( m, e ) ∈ F ( n ). Define maps c i,j : F ( ) → F ( n ) by c i,j = F ((1 i )(2 j ))(Φ ,n − ( m, e )) . The intuition here is that m is a value on one edge of the graph, and e is a graph with n − m, e ) is the graph with n vertices, and just one m -valuededge between vertices 1 and 2. Then the permutation (1 i )(2 j ) permutes this one-edgegraph to put m between vertex i and vertex j . So the map c i,j places the one-edge monoid M at the i, j -position in the n -vertex monoid.Define maps c i,j,p,q : F ( ) × F ( ) → F ( n ) by c i,j,p,q ( m, m ′ ) = c i,j ( m ) c p,q ( m ′ ). The secondgives a monoid homomorphism precisely because ( F, Φ) is a network model.Then we get a map ( ǫ F ) n : Γ F ( ) , V ( n ) → F ( n ) by universal property, which gives amonoidal natural transformation automatically. That these maps form the components ofa natural transformation can be seen by a routine computation.Notice that ( ǫ Γ − , V ) M = ǫ Γ M, V = 1 Γ M, V , (Γ − , V η ) M = Γ M , V = 1 Γ M, V , ( Eǫ ) F = E ( ǫ F ) = ( ǫ F ) = 1 F (2) , ( ηE ) F = η F (2) = 1 F (2) . Thus, checking that the snake equations hold is routine.50 xample 3.16. In CMon , products and coproducts are isomorphic. In particular, for acommutative monoid M , Γ M, CMon ∼ = Γ M .Note that this does not indicate that varietal network models completely encompassordinary network models. If M is a noncommutative monoid,then Γ M, CMon is not defined,but Γ M is. The motivating example of network models in general is SG, the network model of simplegraphs. By Example 3.16, this network model is an example of the main construction of thischapter, SG = Γ B , CMon . The boolean monoid is not only an object in
CMon , it is also anobject in
GMon , the variety of graphic monoids. Then we can consider the network modelsΓ B , Mon and Γ B , GMon . Example 3.17.
Elements of the monoid Γ B , Mon ( n ) are words e p ,q . . . e p k ,q k . These wordsare interpreted as graphs with edges that look like they were built with popsicle sticks, andif two edges lie directly on top of each other, they are identified. Besides that relation, youcan stack edges as high as you want by placing them between different pairs of vertices, butsharing one vertex.There are networks one could imagine building with this popsicle stick intuition whichare not allowed by this formalism. For instance, consider a network with three nodes and anedge for each pair of nodes, each overlapping exactly one of its neighbors, forming an Escher-esque ever-ascending staircase. This sort of network is not allowed by the formalism, sincenetworks are actually equivalence classes of words, where letters have a definite positionrelative to each other. This is an important feature for this network model as it is necessaryto guarantee that the procedure in the following example is well-defined, giving an algebraof the related network operad. What this means in terms of popsicle stick intuition is thatallowed networks are built by placing popsicle sticks one at a time. Example 3.18.
Elements of the Γ B , GMon ( n ) are similar to those in the previous example,except that they must obey the graphic identity, xyx = xy for all x, y ∈ Γ B , GMon ( n ). Whatthis means in the graphical interpretation is that all edges can be identified with the lowestoccuring instance of an edge on the same vertex pair. This means that these networks inreduced form have at most as many edges as the complete simple graph with the samenumber of edges. Essentially these networks are simple graphs with a partial order on theedges which respects disjointness of edges.The networks in the previous example have exactly what we need in a network modelto realize networks of bounded degree as an algebra of a network operad. Example 3.19 ( Networks of bounded degree, revisited).
The degree of a vertex ina simple graph is the number of edges in the graph which contain that vertex. For k ∈ N ,we say that a simple graph is k -bounded if all vertices have degree less than or equal to k . Then we can consider the set B k ( n ) of k -bounded simple graphs. We can define anaction of Γ B , GMon ( n ) on B k ( n ) in the following way. Let g = e . . . e l ∈ Γ B , GMon ( n ) and51 ∈ B k ( n ). Choose a graph h ′ ∈ Γ B , GMon ( n ) which has the same edges as h . Define h = h ′ ,then define h i = h i − e i if that is k -bounded, else h i = h i − . Let hg denote h l , which is a k -bounded element of Γ B , GMon ( n ). Let Γ k B , GMon ( n ) denote the set of k -bounded elements ofΓ B , GMon ( n ). There is a function s : Γ k B , GMon ( n ) → B k ( n ). So we define hg to be s ( h l ). Thisis independent of the choice of h ′ and defines an action of Γ B , GMon on B k ( n ).The networks in the question in Section 3.1 can be represented by simple graphs withvertex degrees bounded by k . Then B k ( n ) gives an algebra of the operad O B , GMon . Thisresolves the conflict encountered in the question in Section 3.1. Ordinary network modelscould not record the order in which edges were added to a network, which was necessaryto define a systematic way of attempting to add new connections to a network which hasdegree limitations on each vertex. 52 hapter 4
Petri Nets
Petri nets are a widely studied formalism for describing collections of entities of differenttypes, and how they turn into other entities [GV13, Pet81]. In this chapter, we combinePetri nets with network models. This is worthwhile because while both formalisms involvenetworks, they serve different functions, and are in some sense complementary.A Petri net can be drawn as a bipartite directed graph with vertices of two kinds: places ,drawn as circles below, and transitions drawn as squares:In applications to chemistry, places are also called species . When we run a Petri net, westart by placing a finite number of tokens in each place: •••
This is called a marking . Then we repeatedly change the marking using the transitions. Forexample, the above marking can change to this: •• and then this: ••• types of entity, and the transitions describe ways thatone collection of entities of specified types can turn into another such collection.Network models serve a different function than Petri nets: they are a general tool forworking with networks of many kinds. A network model is a lax symmetric monoidalfunctor G : S ( C ) → Cat , where S ( C ) is the free strict symmetric monoidal category on a set C . Elements of C represent different kinds of “agents”. Unlike in a Petri net, we do notusually consider processes where these agents turn into other agents. Instead, we wish tostudy everything that can be done with a fixed collection of agents. Any object x ∈ S ( C ) isof the form c ⊗ · · · ⊗ c n for some c i ∈ C ; thus, it describes a collection of agents of variouskinds. The functor G maps this object to a category G ( x ) that describes everything thatcan be done with this collection of agents.In many examples considered so far, G ( x ) is a category whose morphisms are graphswhose nodes are agents of types c , . . . , c n . Composing these morphisms corresponds to overlaying graphs. Network models of this sort let us design networks where the nodes areagents and the edges are communication channels or shared commitments. In Chapter 2,the operation of overlaying graphs was always commutative. In Chapter 3 we introducedmore general noncommutative overlay operations. This lets us design networks where eachagent has a limit on how many communication channels or commitments it can handle;the noncommutativity allows us to take a first come, first served approach to resolvingconflicting commitments.Here we take a different tack: we instead take G ( x ) to be a category whose morphismsare processes that the given collection of agents, x , can carry out . Composition of morphismscorresponds to carrying out first one process and then another.This idea meshes well with Petri net theory, because any Petri net P determines asymmetric monoidal category F P whose morphisms are processes that can be carried outusing this Petri net. More precisely, the objects in
F P are markings of P , and the morphismsare sequences of ways to change these markings using transitions, e.g.: ••• •• → ••• Given a Petri net, then, how do we construct a network model G : S ( C ) → Cat , and inparticular, what is the set C ? In a network model the elements of C represent differentkinds of agents. In the simplest scenario, these agents persist in time. Thus, it is naturalto take C to be some set of “catalysts”. In chemistry, a reaction may require a catalyst toproceed, but it neither increases nor decrease the amount of this catalyst present. For aPetri net, catalysts are species that are neither increased nor decreased in number by anytransition. For example, species a is a catalyst in the following Petri net, so we outline it54n red: cb aτ τ but neither b nor c is a catalyst. The transition τ requires one token of type a as inputto proceed, but it also outputs one token of this type, so the total number of such tokensis unchanged. Similarly, the transition τ requires no tokens of type a as input to proceed,and it also outputs no tokens of this type, so the total number of such tokens is unchanged.In Theorem 4.9 we prove that given any Petri net P , and any subset C of the catalystsof P , there is a network model G : S ( C ) → Cat . An object x ∈ S ( C ) says how many tokensof each catalyst are present; G ( x ) is then the subcategory of F P where the objects aremarkings that have this specified amount of each catalyst, and morphisms are processesgoing between these.From the functor G : S ( C ) → Cat we can construct a category R G by the Grothendieckconstruction. Because G is symmetric monoidal we can make R G into a symmetric monoidalcategory by the monoidal Grothendieck construction of Chapter 5. The tensor product in R G describes doing processes in parallel. The category R G is similar to F P , but it isbetter suited to applications where agents each have their own individuality, because
F P is actually a commutative monoidal category, where permuting agents has no effect at all,while R G is not so degenerate. In Theorem 4.12 we make this precise by more concretelydescribing R G as a symmetric monoidal category, and clarifying its relation to F P .There are no morphisms between an object of G ( x ) and an object of G ( x ′ ) unless x ∼ = x ′ ,since no transitions can change the amount of catalysts present. The category F P is thus adisjoint union, or more precisely a coproduct, of subcategories
F P i where i , an element offree commutative monoid on C , specifies the amount of each catalyst present. The tensorproduct on F P has the property that tensoring an object in
F P i with one in F P j gives anobject in F P i + j , and similarly for morphisms.However, in Prop. 4.15 we show that each subcategory F P i also has its own tensorproduct, which describes doing one process and then another while reusing catalyst tokens.This tensor product makes F P i into a premonoidal category—an interesting generalizationof a monoidal category which we recall. Finally, in Theorem 4.17 we show that thesemonoidal structures define a lift of the functor G : S ( C ) → Cat to a functor ˆ G : S ( C ) → PreMonCat , where
PreMonCat is the category of strict premonoidal categories.
A Petri net generates a symmetric monoidal category whose objects are tensor prod-ucts of species and whose morphisms are built from the transitions by repeatedly takingcomposites and tensor products. There is a long line of work on this topic starting withthe papers of Meseguer–Montanari [MM90] and Engberg–Winskel [EW90], both dating toroughly 1990. It continues to this day, because the issues involved are surprisingly subtle55DMM89, Sas94, Sas95, Sas96, SS05, Mas20]. In particular, there are various kinds of sym-metric monoidal categories to choose from. Following the work of Master and Baez [BM20]we use ‘commutative’ monoidal categories. These are just commutative monoid objects in
Cat , so their associator: α a,b,c : ( a ⊗ b ) ⊗ c ∼ −→ a ⊗ ( b ⊗ c ) , their left and right unitor: λ a : I ⊗ a ∼ −→ a, ρ a : a ⊗ I ∼ −→ a, and even their braiding: σ a,b : a ⊗ b ∼ −→ b ⊗ a are all identity morphisms. While every symmetric monoidal category is equivalent to onewith trivial associator and unitors, this ceases to be true if we also require the braidingto be trivial. However, it seems that Petri nets most naturally serve to present symmetricmonoidal categories of this very strict sort. Thus, we shall describe a functor from thecategory of Petri nets to the category of commutative monoidal categories, which we call CMonCat : F : Petri → CMonCat . To begin, let
CMon be the category of commutative monoids and monoid homomor-phisms. There is a forgetful functor from
CMon to Set that sends commutative monoids totheir underlying sets and monoid homomorphisms to their underlying functions. It has aleft adjoint N : Set → CMon sending any set X to the free commutative monoid on X . Anelement a ∈ N [ X ] is formal linear combination of elements of X : a = X x ∈ X a x x, where the coefficients a x are natural numbers and all but finitely many are zero. The set X naturally includes in N [ X ], and for any function f : X → Y , N [ f ] : N [ X ] → N [ Y ] is theunique monoid homomorphism that extends f . We often abuse language and use N [ X ] tomean the underlying set of the free commutative monoid on X . Definition 4.1. A Petri net is a pair of functions of the following form: T N [ S ] . st We call T the set of transitions , S the set of places or species , s the source function,and t the target function. We call an element of N [ S ] a marking of the Petri net.For example, in this Petri net: τ τ ab c
56e have S = { a, b, c } , T = { τ , τ } , and s ( τ ) = a + b t ( τ ) = cs ( τ ) = c t ( τ ) = 2 b. The term ‘species’ is used in applications of Petri nets to chemistry. Since the concept of‘catalyst’ also arose in chemistry, we henceforth use the term ‘species’ rather than ‘places’.
Definition 4.2. A Petri net morphism from the Petri net P to the Petri net P ′ is a pairof functions ( f : T → T ′ , g : S → S ′ ) such that the following diagrams commute: T N [ S ] T ′ N [ S ′ ] sf N [ g ] s ′ T N [ S ] T ′ N [ S ′ ] tf N [ g ] t ′ Let
Petri denote the category of Petri nets and Petri net morphisms with compositiondefined by ( f, g ) ◦ ( f ′ , g ′ ) = ( f ◦ f ′ , g ◦ g ′ ) . Definition 4.3. A commutative monoidal category is a commutative monoid objectin ( Cat , × ). Let CMonCat denote the category of commutative monoid objects in (
Cat , × ).More concretely, a commutative monoidal category is a strict monoidal category forwhich a ⊗ b = b ⊗ a for all pairs of objects and all pairs of morphisms, and the braidisomorphism a ⊗ b → b ⊗ a is the identity map.Every Petri net P = ( s, t : T → N [ S ]) gives rise to a commutative monoidal category F P as follows. We take the commutative monoid of objects Ob(
F P ) to be the free commutativemonoid on S . We construct the commutative monoid of morphisms Mor ( F P ) as follows.First we generate morphisms recursively: • for every transition τ ∈ T we include a morphism τ : s ( τ ) → t ( τ ); • for any object a we include a morphism 1 a : a → a ; • for any morphisms f : a → b and g : a ′ → b ′ we include a morphism denoted f + g : a + a ′ → b + b ′ to serve as their tensor product; • for any morphisms f : a → b and g : b → c we include a morphism g ◦ f : a → c toserve as their composite.Then we quotient by an equivalence relation on morphisms that imposes the laws of acommutative monoidal category, obtaining the commutative monoid Mor ( F P ).Similarly, morphisms between Petri nets give morphisms between their commutativemonoidal categories. Given a Petri net morphism T N [ S ] T ′ N [ S ′ ] f N [ g ]
57e define the functor F ( f, g ) : F P → F P ′ to be N [ g ] on objects, and on morphisms to bethe unique map extending f that preserves identities, composition, and the tensor product.This functor is strict symmetric monoidal. Proposition 4.4.
There is a functor F : Petri → CMonCat defined as above.Proof.
This is straightforward; the proof that F is a left adjoint is harder [Mas20], but wedo not need this here. One thinks of a transition τ of a Petri net as a process that consumes the source species s ( τ ) and produces the target species t ( τ ). An example of something that can be representedby a Petri net is a chemical reaction network [BB18, BP17]. Indeed, this is why Carl Petrioriginally invented them. A ‘catalyst’ in a chemical reaction is a species that is necessaryfor the reaction to occur, or helps lower the activation energy for reaction, but is neitherincreased nor depleted by the reaction. We use a modest generalization of this notion,defining a catalyst in a Petri net to be a species that is neither increased nor depleted by any transition in the Petri net.Given a Petri net s, t : T → N [ S ], recall that for any marking a ∈ N [ S ] we have a = X x ∈ S a x x for certain coefficients a x ∈ N . Thus, for any transition τ of a Petri net, s ( τ ) x is thecoefficient of the place x in the source of τ , while t ( τ ) x is its coefficient in the target of τ . Definition 4.5.
A species x ∈ S in a Petri net P = ( s, t : T → N [ S ]) is called a catalyst if s ( τ ) x = t ( τ ) x for every transition τ ∈ T . Let S cat ⊆ S denote the set of catalysts in P . Definition 4.6. A Petri net with catalysts is a Petri net P = ( s, t : T → N [ S ]) with achosen subset C ⊆ S cat . We denote a Petri net P with catalysts C as ( P, C ).Suppose we have a Petri net with catalysts (
P, C ). Recall that the set of objects of
F P is the free commutative monoid N [ C ]. We have a natural isomorphism N [ S ] ∼ = N [ C ] × N [ S \ C ] . We write π C : N [ S ] → N [ C ]for the projection. Given any object a ∈ F P , π C ( a ) says how many catalysts of each speciesin C occur in a . Definition 4.7.
Given a Petri net with catalysts (
P, C ) and any i ∈ N [ C ], let F P i be thefull subcategory of F P whose objects are objects a ∈ F P with π C ( a ) = i .58orphisms in F P i describe processes that the Petri net can carry out with a specificfixed amount of every catalyst. Since no transition in P creates or destroys any catalyst, if f : a → b is a morphism in F P then π C ( a ) = π C ( b ) . Thus,
F P is the coproduct of all the subcategories
F P i : F P ∼ = a i ∈ N [ C ] F P i as categories. The subcategories F P i are not generally monoidal subcategories because if a, b ∈ F P and a + b is their tensor product then π C ( a + b ) = π C ( a ) + π C ( b )so for any i, j ∈ N [ C ] we have a ∈ F P i , b ∈ F P j ⇒ a + b ∈ F P i + j and similarly for morphisms. Thus, we can think of F P as a commutative monoidal category‘graded’ by N [ C ]. But note we are free to reinterpret any process as using a greater amountof various catalysts, by tensoring it with identity morphism on this additional amount ofcatalysts. That is, given any morphism in F P i , we can always tensor it with the identityon j to get a morphism in F P i + j .Since N [ C ] is a commutative monoid we can think of it as a commutative monoidalcategory with only identity morphisms, and we freely do this in what follows. Networkmodels rely on a similar but less trivial way of constructing a symmetric monoidal categoryfrom a set C . Namely, for any set C there is a category S ( C ) for which: • Objects are formal expressions of the form c ⊗ · · · ⊗ c n for n ∈ N and c , . . . , c n ∈ C . When n = 0 we write this expression as I . • There exist morphisms f : c ⊗ · · · ⊗ c m → c ′ ⊗ · · · ⊗ c ′ n only if m = n , and in that case a morphism is a permutation σ ∈ S n such that c ′ σ ( i ) = c i for all i = 1 , . . . , n . • Composition is the usual composition of permutations.In short, an object of S ( C ) is a list of catalysts, possibly empty, and allowing repetitions.A morphism is a permutation that maps one list to another list.As shown in Proposition 2.16, S ( C ) is the free strict symmetric monoidal category onthe set C . There is thus a strict symmetric monoidal functor p : S ( C ) → N [ C ]59ending each object c ⊗· · ·⊗ c n to the object c + · · · + c n , and sending every morphism to anidentity morphism. This can also be seen directly. In what follows, we use this functor p toconstruct a lax symmetric monoidal functor G : S ( C ) → Cat , where
Cat is made symmetricmonoidal using its cartesian product.
Proposition 4.8.
Given a Petri net with catalysts ( P, C ) , there exists a unique functor G : S ( C ) → Cat sending each object x ∈ S ( C ) to the category F P p ( x ) and each morphism in S ( C ) to an identity functor.Proof. The uniqueness is clear. For existence, note that since N [ C ] has only identity mor-phisms there is a functor H : N [ C ] → Cat sending each object x ∈ N [ C ] to the category F P p ( x ) . If we compose H with the functor p : S ( C ) → N [ C ] described above we obtain thefunctor G . Theorem 4.9.
The functor G : S ( C ) → Cat becomes lax symmetric monoidal with the laxstructure map Φ x,y : F P p ( x ) × F P p ( y ) → F P p ( x ⊗ y ) given by the tensor product in F P , and the map φ : 1 → F P sending the unique object of the terminal category ∈ Cat to the unit for the tensor productin
F P , which is the object ∈ F P .Proof. Recall that G is the composite of p : S ( C ) → N [ C ] and H : N [ C ] → Cat . The functor p is strict symmetric monoidal. The functor p is strict symmetric monoidal. One can checkthat the functor H becomes lax symmetric monoidal if we equip it with the lax structuremap F P i × F P j → F P i + j given by the tensor product in F P , and the map1 → F P sending the unique object of 1 ∈ Cat to the unit for the tensor product in
F P , namely0 ∈ N [ S ] = Ob( F P ). Composing the lax symmetric monoidal functor H and with the strictsymmetric monoidal functor p , we obtain the lax symmetric monoidal functor G describedin the theorem statement.We defined C -colored network model in Chapter 2 to be a lax symmetric monoidalfunctor from S ( C ) to Cat . Definition 4.10.
We call the C -colored network model G : S ( C ) → Cat of Theorem 4.9 the
Petri network model associated to the Petri net with catalysts (
P, C ).60 xample 4.11.
The following Petri net P has species S = { a, b, c, d, e } and transitions T = { τ , τ } : a bc d eτ τ Species a and b are catalysts, and the rest are not. We thus can take C = { a, b } and obtaina Petri net with catalysts ( P, C ), which in turn gives a Petri network model G : S ( C ) → Cat .We outline catalyst species in red, and also draw the edges connecting them to transitionsin red.Here is one possible interpretation of this Petri net. Tokens in c represent people at abase on land, tokens in d are people at the shore, and tokens in e are people on a nearbyisland. Tokens in a represent jeeps, each of which can carry two people at a time from thebase to the shore and then return to the base. Tokens in b represent boats that carry oneperson at a time from the shore to the island and then return.Let us examine the effect of the functor G : S ( C ) → Cat on various objects of S ( C ). Theobject a ∈ S ( C ) describes a situation where there is one jeep present but no boats. Thecategory G ( a ) is isomorphic to F X , where X is this Petri net: c d eτ That is, people can go from the base to the shore in pairs, but they cannot go to the island.Similarly, the object b describes a situation with one boat present but no jeeps, and thecategory G ( b ) is isomorphic to F Y , where Y is this Petri net: c d eτ Now people can only go from the shore to the island, one at a time.The object a ⊗ b ∈ S ( C ) describes a situation with one jeep and one boat. The category G ( a ⊗ b ) is isomorphic to F Z for this Petri net Z : c d eτ τ Now people can go from the base to the shore in pairs and also go from the shore to theisland one at a time.Surprisingly, an object x ∈ S ( C ) with additional jeeps and/or boats always produces acategory G ( x ) that is isomorphic to one of the three just shown: G ( a ) , G ( b ) and G ( a ⊗ b ).For example, consider the object b ⊗ b ∈ S ( C ), where there are two boats present but nojeeps. There is an isomorphism of categories − + b : G ( b ) → G ( b ⊗ b )defined as follows. Recall that G ( b ) = F P b and G ( b ⊗ b ) = F P b + b , where F P b and F P b + b are subcategories of F P . The functor − + b : F P b → F P b + b x ∈ F P b to the object x + b , and sends each morphism f : x → y in F P b to the morphism 1 b + f : b + x → b + y . That this defines a functor is clear; the surprisingpart is that it is an isomorphism. One might have thought that the presence of a secondboat would enable one to carry out a given task in more different ways.Indeed, while this is true in real life, the category F P is commutative monoidal, sotokens of the same species have no ‘individuality’: permuting them has no effect. There isthus, for example, no difference between the following two morphisms in F P b + b : • using one boat to transport one person from the base to shore and another boat totransport another person, and • using one boat to transport first one person and then another.It is useful to draw morphisms in F P as string diagrams, since such diagrams serve as ageneral notation for morphisms in monoidal categories [JS91]. For expository treatments,see [BS11, Sel11]. The rough idea is that objects of a monoidal category are drawn aslabelled wires, and a morphism f : x ⊗ · · · ⊗ x m → y ⊗ · · · ⊗ y n is drawn as a box with m wires coming in on top and n wires coming out at the bottom. Composites of morphisms aredrawn by attaching output wires of one morphism to input wires of another, while tensorproducts of morphisms are drawn by setting pictures side by side. In symmetric monoidalcategories, the braiding is drawn as a crossing of wires. The rules governing string diagramslet us manipulate them while not changing the morphisms they denote. In the case ofsymmetric monoidal categories, these rules are well known [JS91, Sel11]. For commutative monoidal categories there is one additional rule: x yy x = x yyx This says both that x ⊗ y = y ⊗ x and that the braiding σ x,y : x ⊗ y → y ⊗ x is the identity.Here is the string diagram notation for the equation we mentioned between two mor-phisms in F P : b b d dτ = τ b b e e = b b d dτ τ b b e e
62e draw the object b (standing for a boat) in red to emphasize that it serves as a catalyst.At left we are first using one boat to transport one person from the base to shore, andthen using another boat to transport another person. At right we are using the sameboat to transport first one person and then another, while another boat stands by anddoes nothing. These morphisms are equal because they differ only by the presence of thebraiding σ b,b : b + b → b + b in the left hand side, and this is an identity morphism.The above example illustrates an important point: in the commutative monoidal cate-gory F P , permuting catalyst tokens has no effect. Next we construct a symmetric monoidalcategory R G in which permuting such tokens has a nontrivial effect. One reason for want-ing this is that in applications, the catalyst tokens may represent agents with their ownindividuality. For example, when directing a boat to transport a person from base to shore,we need to say which boat should do this. For this we need a symmetric monoidal categorythat gives the catalyst tokens a nontrivial braiding.To create this category, we use the symmetric monoidal Grothendieck construction ofChapter 5. Given any symmetric monoidal category X and any lax symmetric monoidalfunctor F : X → Cat , this construction gives a symmetric monoidal category R F equippedwith a functor (indeed an opfibration) R F → X . In Chapter 2, we used this constructionto build an operad from any network model, whose operations are ways to assemble largernetworks from smaller ones. Now this construction has a new significance.Starting from a Petri network model G : S ( C ) → Cat , the symmetric monoidal Grothendieckconstruction gives a symmetric monoidal category R G in which: • an object is a pair ( x, a ) where x ∈ S ( C ) and a ∈ F P p ( x ) . • a morphism from ( x, a ) to ( x ′ , a ′ ) is a pair ( σ, f ) where σ : x → x ′ is a morphism in S ( C ) and f : a → a ′ is a morphism in F P . • morphisms are composed componentwise. • the tensor product is computed componentwise: in particular, the tensor product ofobjects ( x, a ) and ( x ′ , a ′ ) is ( x ⊗ x ′ , a + a ′ ). • the associators, unitors and braiding are also computed componentwise (and henceare trivial in the second component, since F P is a commutative monoidal category).The functor R G → S ( C ) simply sends each pair to its first component.This is simpler than one typically expects from the Grothendieck construction. Thereare two main reasons: first, G maps every morphism in S ( C ) to an identity morphism in Cat , and second, the lax structure map for G is simply the tensor product in F P . However,this construction still has an important effect: it makes the process of switching two tokensof the same catalyst species into a nontrivial morphism in R G . More formally, we have: Theorem 4.12. If G : S ( C ) → Cat is the Petri network model associated to the Petri netwith catalysts ( P, C ) , then R G is equivalent, as a symmetric monoidal category, to the fullsubcategory of S ( C ) × F P whose objects are those of the form ( x, a ) with x ∈ S ( C ) and a ∈ F P p ( x ) . roof. One can read this off from the description of R G given above.The difference between R G and F P is that the former category keeps track of pro-cesses where catalyst tokens are permuted, while the latter category treats them as identitymorphisms. In the terminology of Glabbeek and Plotkin, R G implements the ‘individualtoken philosophy’ on catalysts, in which permuting tokens of the same catalyst is regardedas having a nontrivial effect [vGP09]. By contrast, F P implements the ‘collective tokenphilosophy’, where all that matters is the number of tokens of each catalyst, and permutingthem has no effect.There is a map from R G to F P that forgets the individuality of the catalyst tokens. Amorphism in R G is a pair ( σ, f ) where σ : x → x ′ is a morphism in S ( C ) and f : a → a ′ isa morphism in F P with a ∈ G ( x ) , a ′ ∈ G ( x ′ ). There is a symmetric monoidal functor R G → F P that discards this extra information, mapping ( σ, f ) to f . The symmetric monoidal Grothendieckconstruction also gives a symmetric monoidal opfibration R G → S ( C )which maps ( σ, f ) to σ , by Chapter 5. Example 4.13.
Let (
P, C ) be the Petri net with catalysts in Ex. 4.11, and G : S ( C ) → Cat the resulting Petri network model. In R G the following two morphisms are not equal: b b d dτ τ b b e e = b b d dτ τ b b e e because the braiding of catalyst species in R G is nontrivial. This says that in R G weconsider these two processes as different: • using one boat to transport one person from the base to shore and another boat totransport another person, and • using one boat to transport first one person and then another.64n the other hand, in R G we have b b d dτ τ b b e e = b b d dτ τ b b e e because these morphisms differ only by two people on the shore switching place before theyboard the boats, and the braiding of non-catalyst species is the identity. In short, the R G construction implements the individual token philosophy only for catalyst tokens; tokens ofother species are governed by the collective token philosophy. We have seen that for a Petri net P , a choice of catalysts C lets us write the category F P as a coproduct of subcategories
F P i , one for each possible amount i ∈ N [ C ] of thecatalysts. The subcategory F P i is only a monoidal subcategory when i = 0. Indeed, only F P contains the monoidal unit of F P . However, we shall see that each subcategory
F P i can be given the structure of a premonoidal category, as defined by Power and Robinson[PR97]. We motivate our use of this structure by describing two failed attempts to make F P i into a monoidal category.Given two morphisms in F P i we typically cannot carry out these two processes simul-taneously, because of the limited availability of catalysts. But we can do first one and thenthe other. For example, imagine that two people are trying to walk through a doorway, butthe door is only wide enough for one person to walk through. The door is a resource thatis not depleted by its use, and thus a catalyst. Both people can use the door, but not atthe same time: they must make an arbitrary choice of who goes first.We can attempt to define a tensor product on F P i using this idea. Fix some amount ofcatalysts i ∈ N [ C ]. Objects of F P i are of the form i + a with a ∈ N [ S − C ]. On objects wedefine ( i + a ) ⊗ i ( i + a ′ ) = i + a + a ′ . The unit object for ⊗ i is therefore i + 0, or simply i . For morphisms f : i + a → i + bf ′ : i + a ′ → i + b ′ we define f ⊗ i f ′ = ( f + 1 b ′ ) ◦ (1 a + f ′ ) . f ⊗ i f ′ = ( f + 1 b ′ ) ◦ (1 a + f ′ ) of morphisms in F P i involves anarbitrary choice: namely, the choice to do f ′ first. This is perhaps clearer if we draw thismorphism as a string diagram in F P . i a a ′ f f ′ b b ′ i If instead we choose to do f first, we can define a tensor product i ⊗ which is the same onobjects but given on morphisms by f i ⊗ f ′ = (1 b + f ′ ) ◦ ( f + 1 a ′ ) . It looks like this: i a a ′ f f ′ b b ′ i Unfortunately, neither of these tensor products makes
F P i into a monoidal category! Eachmakes the set of objects Ob( F P i ) and the set of morphisms Mor ( F P i ) into a monoid in sucha way that the source and target maps s, t : Mor ( F P i ) → Ob(
F P i ), as well as the identity-assigning map i : Ob( F P i ) → Mor ( F P i ), are monoid homomorphisms. The problem is thatneither obeys the interchange law, so neither of these tensor products defines a functor from F P i × F P i to F P i . For example,(1 ⊗ i f ′ ) ◦ ( f ⊗ i = ( f ⊗ i ◦ (1 ⊗ i f ′ ) . The other tensor product suffers from the same problem.What is going on here? It turns out that
F P i is a ‘strict premonoidal category’. Whilethese structures first arose in computer science [PR97], they are also mathematically natural,for the following reason. There are only two symmetric monoidal closed structures on Cat ,66p to isomorphism [FKL80]. One is the the cartesian product. The other is the ‘funnytensor product’ [Web13]. A monoid in
Cat with its cartesian product is a strict monoidalcategory, but a monoid in
Cat with its funny tensor product is a strict premonoidal category.The funny tensor product C (cid:3) D of categories C and D is defined as the following pushoutin Cat : C × D C × DC × D C (cid:3) D i × × j Here C is the subcategory of C consisting of all the objects and only identity morphisms, i : C → C is the inclusion, and similarly for j : D → D . Thus, given morphisms f : x → y in C and f ′ : x ′ → y ′ in C , the category C (cid:3) D in contains a square of the form x (cid:3) x ′ x (cid:3) y ′ x ′ (cid:3) y x ′ (cid:3) y ′ , f (cid:3) (cid:3) f ′ f (cid:3) (cid:3) f ′ but in general this square does not commute, unlike the corresponding square in C × D . Definition 4.14. A strict premonoidal category is a category C equipped with a functor ⊠ : C (cid:3) C → C that obeys the associative law and an object I ∈ C that serves as a left andright unit for ⊠ .Given two morphisms f : x → y , f ′ : x ′ → y ′ in a strict premonoidal category C weobtain a square x ⊠ x ′ x ⊠ y ′ x ′ ⊠ y x ′ ⊠ y ′ , f ⊠ ⊠ f ′ f ⊠ ⊠ f ′ but this square may not commute. There are thus two candidates for a morphism from x ⊠ x ′ to y ⊠ y ′ . When these always agree, we can make C monoidal by setting f ⊠ f ′ equal to either (and thus both) of these candidates. We shall give F P i a strict premonoidalstructure where these two candidates do not agree: one is f ⊗ i f ′ while the other is f i ⊗ f ′ .This explains the meaning of these two failed attempts to give F P i a monoidal structure.Thanks to the description of C (cid:3) C as a pushout, to know the tensor product ⊠ in a strictpremonoidal category C it suffices to know x ⊠ y , x ⊠ f and f ⊠ y for all objects x, y andmorphisms f of C . (Here we find it useful to write x ⊠ f for 1 x ⊠ f and f ⊠ y for f ⊠ y .)In the case at hand, we define ⊠ i : F P i (cid:3) F P i → F P i on objects by setting ( i + a ) ⊠ i ( i + a ′ ) = i + a + a ′ a, a ′ ∈ N [ S − C ], while for morphisms f : i + a → i + bf ′ : i + a ′ → i + b ′ we set a ⊠ f ′ = f ′ + 1 a , f ⊠ a ′ = f + 1 a ′ . Proposition 4.15.
The tensor product ⊠ i makes F P i into a strict premonoidal category.Proof. This can be checked directly, but this is also a special case of a construction inPower and Robinson’s paper on premonoidal categories [PR97, Ex. 3.4]. They describe aconstruction, sometimes called ‘linear state passing’ [MS14], that takes any object i in anysymmetric monoidal category C and yields a premonoidal category C i where objects are ofthe form i ⊗ c for c ∈ C and morphisms are morphisms in C of the form f : i ⊗ c → i ⊗ c ′ .We are considering the special case where C = F P , and because
F P is commutativemonoidal the resulting premonoidal category is strict: all the coherence isomorphisms areidentities.Finally, we show that the tensor products ⊠ i on the categories F P i let us lift our networkmodel G from Cat to the category of strict premonoidal categories.
Definition 4.16.
Let
PreMonCat be the category of strict premonoidal categories and strict premonoidal functors , meaning functors between strict premonoidal categoriesthat strictly preserve the tensor product. Let U : PreMonCat → Cat denote the forgetfulfunctor which sends a strict premonoidal category to its underlying category.
Theorem 4.17.
The network model G : S ( C ) → Cat lifts to a functor ˆ G : S ( C ) → PreMonCat : PreMonCatS ( C ) Cat UG ˆ G where ˆ G ( x ) = F P p ( x ) with the strict premonoidal structure described in Prop. 4.15.Proof. Since G sends each morphism in S ( C ) to an identity functor, so must ˆ G .68 hapter 5 Monoidal GrothendieckConstruction
The Grothendieck construction [Gro71] exhibits one of the most fundamental relationsin category theory, namely the equivalence between contravariant pseudofunctors into
Cat and fibrations. In previous chapters, we have described how the to construct a total cate-gory , denoted R F , from a functor of the form F : X op → Cat . Actually, we really could havebeen using pseudofunctors, since
Cat is more naturally thought of as a 2-category. We referto pseudofunctors of the form F : X op → Cat as indexed categories . The construction of R F from a given indexed category essentially forgets the distinction between the categories F x for x ∈ X , and incorporates the functors F f : F y → F x as maps between the objects of
F y and
F x . The distinction between these categories could be remembered via a fibration ,a special sort of functor P : R F → X , which tells you how to take preimage categoriesof the objects, P − ( x ), and turn certain maps in R F into functors between the preimagecategories. For a general fibration P : A → X , the category X is called the base category and the category A is called the total category . For an object x ∈ X , the preimage category P − ( x ) is called the fibre of P over x . A fibration is precisely what is needed to reconstructall the data in the indexed category from its total category. Indeed, the Grothendieck con-struction gives an equivalence between the 2-categories ICat of indexed categories, and
Fib offibrations. This equivalence allows us to move between the worlds of indexed categories andfibred categories, providing access to tools and results from both. We recall the basic theoryof fibrations, indexed categories, and the Grothendieck construction in Appendix C.1.Due to the importance of the Grothendieck construction, it is only natural that onewould be interested in extra structure these objects may have, and how the correspondenceextends. In particular, a version which handles monoidal structures on the various categoriesin play could potentially be very useful, as monoidal categories are of central interest inboth pure and applied category theory. There are several categories to consider as equippedwith monoidal structures in this scenario: fibers P − ( x ) of a fibration P : A → X , itsbase category X , its total category A , the indexing category X of an indexed category F : X op → Cat , and the categories
F x indexed by F . Of course these options are not69eally all distinct. The base of the fibrations correspond to the indexing category under theequivalence, and the fibres correspond to the individual categories selected by the indexedcategory. This boils our options down to two monoidal variants: fibre-wise, and global.In the first variant—the fibre-wise approach—the fibres are equipped with a monoidalstructure, and the reindexing functors are equipped with a strict monoidal structure. Theadditional structure this gives to the corresponding indexed categories turns them intopseudofunctors into MonCat , which were called indexed monoidal categories by Hofstraand de Marchi [HM06]. In the second variant—the global approach—the total categoryand the base category of the fibration are each equipped with the structure of a monoidalcategory, and the fibration is equipped with a strict monoidal structure. The correspondingstructure equipped to the related indexed category is a little less obvious. The indexingcategory is equipped with a monoidal structure as in the fibration side of the picture, andthe pseudofunctor is now equipped with the structure of a lax monoidal structure into
Cat with its cartesian structure. We call these monoidally indexed categories or just monoidalindexed categories .Both of these variants can be seen as special cases of a much more general phenomenon.Pseudomonoids are a categorification of monoid objects internal to a monoidal category.It would be reasonable to call it “monoidal category internal to a monoidal 2-category”.We can see both of the monoidal variants of both fibrations and indexed categories de-scribed above as examples of pseudomonoids in certain 2-categories of fibrations or indexedcategories.The 2-category
ICat ( X ) of indexed categories over a fixed base category has finite prod-ucts, and thus a cartesian monoidal structure. Pseudomonoids taken with respect to thismonoidal structure are precisely pseudofunctors X op → MonCat , i.e. the fibre-wise monoidalindexed categories described above. Similarly, the 2-category
Fib ( X ) of fibrations over afixed base category has a cartesian monoidal structure, for which pseudomonoids are pre-cisely the fibre-wise monoidal fibrations described above.The 2-category ICat of indexed categories over different base categories has finite prod-ucts, and thus a cartesian monoidal structure. Pseudomonoids taken with respect to thismonoidal structure are precisely lax monoidal pseudofunctors ( X op , ⊗ ) → ( Cat , × ), i.e.the global monoidal indexed categories described above. Similarly, the 2-category Fib offibrations over different base categories has a cartesian monoidal structure, for which pseu-domonoids are precisely the global monoidal fibrations described above.An immediate consequence of this perspective on these objects is that the Grothendieckconstruction lifts naturally into both settings. The 2-category of fibre-wise monoidal fi-brations is equivalent to the 2-category of fibre-wise monoidal indexed categories, c.f. The-orem 5.8. Similarly, the 2-category of global monoidal fibrations is equivalent to the 2-category of global monoidal indexed categories, c.f. Theorem 5.11.When X is cartesian monoidal, a global monoidal structure can be constructed fromfibre-wise monoidal data, and vice versa, c.f. Theorem 5.13. We use our high-level perspec-tive to give a new proof of the result of Shulman giving an equivalence between fibre-wisemonoidal indexed categories and global monoidal fibrations over cartesian base categories[Shu08].The fact that the monoidal Grothendieck construction naturally arises in diverse settings70s what motivated the theoretical clarification presented here. We gather a few examples inthe last section of the chapter to exhibit the various constructions concretely, and we areconvinced that many more exist and would benefit from such a viewpoint. The examplesinclude standard (op)fibrations such as the (co)domain (op)fibration and families classi-fied in their monoidal contexts, as well as certain special algebraic cases of interest suchas monoid-(co)algebras as objects in monoidal Grothendieck categories. Moreover, globalcategories of (co)modules for (co)monoids in any monoidal category, as well as (co)modulesfor (co)monads in monoidal double categories also naturally fit in this context. Finally,certain categorical approaches to systems theory employ algebras for monoidal categories,namely monoidal indexed categories, as their basic compositional tool for nesting of sys-tems; clearly these also fall into place, giving rise to total monoidal categories of systemswith new potential to be explored.In Section 5.2 and Section 5.3, we give the fibre-wise and global monoidal versionsof fibrations and indexed categories. In the first version, the fibers are equipped witha monoidal structure. In the second, the base and total categories are monoidal, and thefibration is (strict) monoidal. In Section 5.4, we lift the Grothendieck construction into thesemonoidal settings as well. In Section 5.5, we give a detailed description of the monoidalstructures given by the correspondences. We begin by describing the two monoidal variants of fibrations. This requires familiaritywith notions such as monoidal 2-categories, pseudomonoids, and the 2-categories
Fib and
Fib ( X ). The 2-categories Fib and
OpFib of (op)fibrations over arbitrary bases, explained inAppendix C.1, have a cartesian monoidal structure inherited from
Cat . For two fibrations P and Q , their product in Cat P × Q : A × B → X × Y (5.1)is also a fibration, where a cartesian lifting is a pair consisting of a P -lifting and a Q -lifting;similarly for opfibrations. The monoidal unit is the trivial (op)fibration 1 : → . Sincethe monoidal structure is cartesian, they are both symmetric monoidal 2-categories. Werefer to a pseudomonoid in ( Fib , × , ) as a monoidal fibration . By the following result,this aligns with the common notion of monoidal fibration [Shu08]. Proposition 5.1.
A monoidal fibration P : A → X is a fibration for which both the total A and base category X are monoidal, P is a strict monoidal functor and the tensor product ⊗ A of A preserves cartesian liftings.Proof. The multiplication and unit are fibred 1-cells µ = ( ⊗ A , ⊗ X ) : P × P → P and ǫ = ( I A , I X ) : → P displayed as follows. A × A A and AX × X X X ⊗ A P × P P I A P ⊗ X I X (5.2)71 morphism ( φ , φ ) in A × A is P × P -cartesian if and only if φ and φ are both P -cartesian. The condition of ( ⊗ A , ⊗ X ) forming a fibred 1-cell tells us precisely that φ ⊗ A φ is P -cartesian. The pieces of associativity and unitality 2-cells corresponding to A and to X give precisely the associativity and unitality structures for each to be given the structureof a monoidal category. The functor P is strict with respect to these monoidal structureson A and X due to the fact that the diagrams above commute.A monoidal fibred 1-cell between two monoidal fibrations P : A → X and Q : B → Y is a (strong) morphism of pseudomonoids between them, as defined in Appendix B.
Proposition 5.2.
A monoidal fibred 1-cell between two monoidal fibrations P and Q is afibred 1-cell ( H, F ) where both functors are monoidal, ( H, φ, φ ) and ( F, ψ, ψ ) , such that Q ( φ a,b ) = ψ P a,P b and Qφ = ψ .Proof. A monoidal fibred 1-cell amounts to a fibred 1-cell, i.e. a commutative square
A BX Y
HP QF (5.3)where H preserves cartesian liftings, along with invertible 2-cells Equation (B.7) in Fib satisfying Equation (B.8). By Equation (C.6), these are fibred 2-cells
B × BA × A BA Y × YX × X YX ⊗ B QφP × P H × H ⊗ A QH ⊗ Y ψF × F ⊗ X FP BA YX I B I A QHφ I Y I X FP ψ where φ and ψ are natural isomorphisms with components φ a,b : Ha ⊗ Hb ∼ −→ H ( a ⊗ b ) , ψ x,y : F x ⊗ F y ∼ −→ F ( x ⊗ y )72uch that φ is above ψ , i.e. the following diagram commutes: Q ( Ha ⊗ Hb ) QH ( a ⊗ b ) QHa ⊗ QHb F P ( a ⊗ b ) F P a ⊗ F P b F ( P a ⊗ P b ) Qφ a,b Equation (5.2)
Equation (5.3)
Equation (5.3)
Equation (5.2) ψ Pa,Pb
Similarly, φ and ψ have single components φ : I B ∼ −→ H ( I A ) and ψ : I Y ∼ −→ F ( I X ) suchthat Q ( φ ) = ψ . These two conditions in fact say that the identity transformation, a.k.a.commutative square Equation (5.3) is a monoidal one, as expressed in [Shu08, 12.5]. Therelevant axioms dictate that ( φ, φ ) and ( ψ, ψ ) give H and F the structure of strongmonoidal functors.For lax or oplax morphisms of pseudomonoids in Fib , we obtain appropriate notions ofmonoidal fibred 1-cells, where the top and bottom functors of Equation (5.3) are lax oroplax monoidal respectively.Finally, a monoidal fibred 2-cell is a 2-cell between morphisms (
H, F ) and (
K, G ) ofpseudomonoids P , Q in Fib . Proposition 5.3.
A monoidal fibred 2-cell between two monoidal fibred 1-cells is an ordi-nary fibred 2-cell ( α, β ) where both natural transformations are monoidal.Proof. Unpacking the definition, we see that a monoidal fibred 2-cell is a fibred 2-cell asdescribed in Appendix C.1
A BX Y
P HK ⇓ β QFG ⇓ α satisfying the axioms Equation (B.9). These amount to the fact that both β and α aremonoidal natural transformations between the respective lax monoidal functors.We denote by PsMon ( Fib ) =
MonFib the 2-category of monoidal fibrations, monoidalfibred 1-cells and monoidal fibred 2-cells. By changing the notion of morphisms betweenpseudomonoids to lax or oplax, we obtain 2-categories
MonFib lax and
MonFib opl . Thereare also 2-categories
BrMonFib and
SymMonFib of braided (resp. symmetric ) monoidalfibrations , braided (resp. symmetric ) monoidal fibred 1-cells and monoidal fibred2-cells, defined to be BrPsMon ( Fib ) and
SymPsMon ( Fib ) respectively; see Proposition B.3.Dually, we have appropriate 2-categories of monoidal opfibrations , monoidal op-fibred 1-cells and monoidal opfibred 2-cells and their braided and symmetric vari-ations, MonOpFib , BrMonOpFib and
SymMonOpFib . All the structures are constructed73ually, where a monoidal opfibration, namely a pseudomonoid in the cartesian monoidal(
OpFib , × , ), is a strict monoidal functor such that the tensor product of the total categorypreserves cocartesian liftings.All the above 2-categories have sub-2-categories of monoidal (op)fibrations over a fixedmonoidal base ( X , ⊗ , I ), e.g. MonFib ( X ) and MonOpFib ( X ). The morphisms are monoid-al (op)fibred functors , i.e. fibred 1-cells of the form ( H, X ) with H monoidal, and the2-cells are monoidal (op)fibred natural transformations , i.e. fibred 2-cells of the form( β, X ) with β monoidal. These 2-categories correspond to the ‘global’ monoidal part ofthe story.Moreover, the above constructions can be adjusted accordingly to the context of splitfibrations. Explicitly, the 2-category PsMon ( Fib s ) = MonFib s has as objects monoidal splitfibrations , namely split fibrations P : A → X between monoidal categories which are strictmonoidal functors and ⊗ A strictly preserves cartesian liftings (compare to Proposition 5.1).Furthermore, the hom-categories MonFib s ( P, Q ) between monoidal split fibrations are fullsubcategories of
MonFib ( P, Q ) spanned by the monoidal fibred 1-cells which are split asfibred 1-cells, namely (
H, F ) as in Proposition 5.2 where H strictly preserves cartesianliftings.We end this section by considering a different monoidal object in the context of (op)fibra-tions, starting over from the usual 2-categories of (op)fibrations over a fixed base X ,(op)fibred functor and (op)fibred natural transformations Fib ( X ) and OpFib ( X ). Noticethat contrary to the earlier devopment, there is no monoidal structure on X . Both these2-categories are also cartesian monoidal, but in a different manner than Fib and
OpFib , dueto the cartesian monoidal structure of
Cat / X ; see for example [Jac99, 1.7.4]. Explicitly, forfibrations P : A → X and Q : B → X , their tensor product P ⊠ Q is given by any of the twoequal functors to X from the following pullback A × X B AB X y P ⊠ Q PQ (5.4)since fibrations are closed under pullbacks and of course composition. The monoidal unitis 1 X : X → X .A pseudomonoid in (
Fib ( X ) , ⊠ , X ) is an ordinary fibration P : A → X equipped withtwo fibred functors ( µ, X ) : P ⊠ P → P and ( ǫ, X ) : 1 X → P displayed as A × X A AX P ⊠ P µ P
X AX ǫ X P (5.5)along with invertible fibred 2-cells satisfying the usual axioms. In more detail, the pullback A × X A consists of pairs of objects of A which are in the same fibre of P , and P ⊠ P sends such a pair to their underlying object defining their fibre. The functor µ maps any( a, b ) ∈ A x to some m ( a, b ) := a ⊗ x b ∈ A x and the map ǫ sends an object x ∈ X to a chosen74ne, I x , in its fibre. The invertible 2-cells and the axioms guarantee that these maps definea monoidal structure on each fibre A x , providing the associativity, left and right unitors.The fact that µ and ǫ preserve cartesian liftings translate into a strong monoidal structureon the reindexing functors: for any f : x → y and a, b ∈ A y , f ∗ a ⊗ x f ∗ b ∼ = f ∗ ( a ⊗ y b ) and I y ∼ = f ∗ ( I x ).A (lax) morphism between two such fibrations is a fibred functor Equation (C.4) suchthat the induced functors H x : A x → B x between the fibres as in Equation (C.5) are (lax)monoidal, whereas a 2-cell between them is a fibred natural transformation β : H ⇒ K Equation (C.7) which is monoidal when restricted to the fibers, β x | A x : H x ⇒ K x . Inthis way, we obtain the 2-category PsMon ( Fib ( X )) and dually PsMon ( OpFib ( X )). These2-categories correspond to the ‘fibrewise’ monoidal part of the story.Finally, taking pseudomonoids in the 2-category of split fibrations over a fixed base,we obtain the 2-category PsMon ( Fib s ( X )) with objects split fibrations equipped with afibrewise tensor product and unit as above, but now the reindexing functors strictly preservethat monoidal structure since the top functors of Equation (5.5) strictly preserve cartesianliftings: f ∗ a ⊗ x f ∗ b = f ∗ ( a ⊗ y b ) and I y = f ∗ ( I x ). Moreover, PsMon ( Fib s ( X ))( P, Q ) is the fullsubcategory of
PsMon ( Fib ( X ))( P, Q ) spanned by split fibred functors, namely H : A → B which strictly preserve cartesian liftings but still H x are monoidal functors between themonoidal fibres as before.As is evident from the above descriptions, the 2-categories MonFib ( X ) and PsMon ( Fib ( X ))are different in general. A monoidal fibration over X is a strict monoidal functor, whereas apseudomonoid in fixed-base fibrations is a fibration with monoidal fibres in a coherent way:none of the base or the total category need to be monoidal. The 2-categories of indexed and opindexed categories
ICat and
OpICat , explained inAppendix C.1, are both monoidal. Explicitly, given two indexed categories M : X op → Cat and N : Y op → Cat , their tensor product
M ⊗ N : (
X × Y ) op → Cat is the composite(
X × Y ) op ∼ = X op × Y op M×N −−−−→
Cat × Cat × −→ Cat (5.6)i.e. (
M ⊗ N )( x, y ) = M ( x ) × N ( y ) using the cartesian monoidal structure of Cat . Themonoidal unit is the indexed category ∆ : op → Cat that picks out the terminal category in Cat , and similarly for opindexed categories. Notice that this monoidal 2-structure,formed pointwise in
Cat , is also cartesian.We call a pseudomonoid in (
ICat , ⊗ , ∆ ) a monoidal indexed category . Proposition 5.4.
A monoidal indexed category is a lax monoidal pseudofunctor ( M , µ, µ ) : ( X op , ⊗ op , I ) → ( Cat , × , ) , where ( X , ⊗ , I ) is an (ordinary) monoidal category.Proof. Unpacking the definition, we see that a monoidal indexed category is an indexed cat-egory M : X op → Cat equipped with multiplication and unit indexed 1-cells ( ⊗ X , µ ) : M ⊗ → M , ( η, µ ) : ∆ → M which by Equation (C.8) are as follows. X op × X op Cat X op M⊗M⊗ op ⇓ µ M op Cat X op ∆ I op ⇓ µ M These come equipped with invertible indexed 2-cells as in Equation (B.6); the axioms thisdata is required to satisfy, on the one hand, render X a monoidal category with ⊗ : X ×X →X its tensor product functor and I : → X its unit. On the other hand, the resulting axiomsfor the components µ x,y : M x × M y → M ( x ⊗ y ) , µ : → M ( I ) (5.7)of the above pseudonatural transformations precisely give M the structure of a lax monoidalpseudofunctor, recalled in Appendix B.We then define a monoidal indexed 1-cell to be a (strong) morphism between pseu-domonoids in ( ICat , ⊗ , ∆ ). Proposition 5.5.
A monoidal indexed 1-cell between two monoidal indexed categories M and N is an indexed 1-cell ( F, τ ) , where the functor F is (strong) monoidal and the pseudo-natural transformation τ is monoidal.Proof. Unpacking the definition, we see that a monoidal indexed 1-cell is an indexed 1-cell(
F, τ ) :
M → N X op Cat Y op M F op ⇓ τ N between two monoidal indexed categories ( M , µ, µ ) and ( N , ν, ν ) equipped with two in-vertible indexed 2-cells ( ψ, m ) and ( ψ , m ) as in Equation (B.7), which explicitly consistof natural isomorphisms ψ , ψ and invertible modifications X op × X op X op × X op X op Y op × Y op Cat X op Cat Y op Y op Y op ⊗ op F op × F op M⊗M ⇓ τ × τ M⊗M⊗ op ⇓ µF op ⇓ ψ N ⊗N⊗ op ⇓ ν m ⇛ M F op ⇓ τ id N N op op X op Cat X op Cat Y op Y op Y op I op ∆ ⇓ ν I op ∆ I op ⇓ µ F op ⇓ ψ m ⇛ M F op ⇓ τ id N N as dictated by the general form Equation (C.10) of indexed 2-cells. The natural isomor-phisms ψ and ψ have components ψ x,z : F x ⊗ F y ∼ −→ F ( x ⊗ y ) , ψ : I ∼ −→ F ( I ) in Y op whereas the modifications m and m are given by families of invertible natural transforma-tions N F x × N
F y N ( F x ⊗ F y ) M x ×M y N F ( x ⊗ y ) M ( x ⊗ y ) ν F x,F y N ψ x,y τ x × τ y µ x,y ⇓ m x,y τ x ⊗ y N ( I ) N ( F I ) M ( I ) N ψ ν µ ⇓ m τ I The appropriate coherence axioms ensure that the functor F : X → Y has a strong monoidalstructure (
F, ψ, ψ ), and that the pseudonatural transformation τ : M ⇒ N ◦ F op is monoidalwith m x,y , m as in Equation (B.4). Notice that F op being monoidal makes F monoidalwith inverse structure isomorphisms.Finally, a monoidal indexed 2-cell is a 2-cell between morphisms of pseudomonoidsin ( ICat , ⊗ , ∆ ). Proposition 5.6.
A monoidal indexed 2-cell between two monoidal indexed 1-cells ( F, τ ) and ( G, σ ) is an indexed 2-cell ( α, m ) such that α is an ordinary monoidal natural trans-formation and m is a monoidal modification.Proof. Following the definition of Appendix B, an indexed 2-cell ( a, m ) : (
F, τ ) ⇒ ( G, σ ) :
M →N as in Equation (C.10), which consists of a natural transformation α : F ⇒ G and a mod-ification m with components M x N F x N Gx σ x τ x ⇓ m x N α x is monoidal, exactly when α : F ⇒ G is compatible with the strong monoidal structures of F and G , and the modification m : τ ⇛ N α op ◦ σ satisfies Equation (B.5) for the inducedmonoidal structures on its domain and target pseudonatural transformations.77e write PsMon ( ICat ) =
MonICat , the 2-category of monoidal indexed categories, monoidalindexed 1-cells and monoidal indexed 2-cells. Moreover, their braided and symmetric coun-terparts form
BrMonICat and
SymMonICat respectively, as the 2-categories of braided andsymmetric pseudomonoids in (
ICat , ⊗ , ∆ ) formally discussed in Appendix B. Similarly, wehave 2-categories of (braided or symmetric) monoidal opindexed categories, 1-cellsand 2-cells MonOpICat , BrMonOpICat and
SymMonOpICat .All these 2-categories have sub-2-categories of monoidal (op)indexed categories with afixed monoidal domain ( X , ⊗ , I ), and specifically MonICat ( X ) = Mon2Cat ps ( X op , Cat ) (5.8)
MonOpICat ( X ) = Mon2Cat ps ( X , Cat )the functor 2-categories of lax monoidal pseudofunctors, monoidal pseudonatural transfor-mations and monoidal modifications.Moreover, we can consider pseudomonoids in the strict context. Explicitly, the 2-category
PsMon ( ICat s ) = MonICat s has as objects monoidal strict indexed categories namely (2-)functors M : X op → Cat from an ordinary monoidal category X which are laxmonoidal as before, but the laxator and unitor Equation (5.7) are strictly natural ratherthan pseudonatural transformations. The hom-categories PsMon ( ICat s )( M , N ) betweenmonoidal strict indexed categories are full subcategories of MonICat ( M , N ) spanned bystrict natural transformations—which are however still lax monoidal, i.e. equipped withisomorphisms Equation (B.4).Similarly to the previous Section 5.2 on fibrations, we end this section with the study ofpseudomonoids in a different but related monoidal 2-category, namely ICat ( X ) = ps ( X op , Cat )of indexed categories with a fixed domain X . Working in this 2-category, or in OpICat ( X ),there is no assumed monoidal structure on X . Their monoidal structure is again cartesian:for two X -indexed categories M , N : X op → Cat , their product is M ⊠ N : X op ∆ −→ X op × X op M×N −−−−→
Cat × Cat × −→ Cat (5.9)with pointwise components ( M ⊠ N )( x ) = M ( x ) × N ( x ) in Cat . The monoidal unit is just X op ! −→ ∆ −−→ Cat , which we will also call ∆ .A pseudomonoid in ( ICat ( X ) , ⊠ , ∆ ) is a pseudofunctor M : X op → Cat equipped withindexed functors Equation (C.9) µ : M ⊠ M → M and ǫ : ∆ → M namely X op × X op Cat × Cat X op Cat X op Cat
M×M × ∆ ∆ ⇓ µ M ! M ⇓ ǫ with components µ x : M x × M x → M x and ǫ x : → M x which are pseudonatural via M x × M x M y × M y M x M y µ x M f ×M f ∼ = µ y M f M x M y = ǫ x ∼ = ǫ y M f (5.10)78f we denote µ x = ⊗ x and ǫ x = I x , the pseudomonoid invertible 2-cells Equation (B.6)and the axioms these data satisfy make each M x into a monoidal category ( M x, ⊗ x , I x ),and each M f into a strong monoidal functor: the above isomorphisms have components M f ( a ) ⊗ y M f ( b ) ∼ = M f ( a ⊗ x b ) and I y ∼ = M f ( I x ) for any a, b ∈ M x .Such a structure, namely a pseudofunctor M : X op → MonCat into the 2-category ofmonoidal categories, strong monoidal functors and monoidal natural transformations, wasdirectly defined as an indexed strong monoidal category in [HM06], and as indexed monoidalcategory in [PS12]. We will avoid this notation in order to not create confusion with theterm monoidal indexed categories .A strong morphism of pseudomonoids Equation (B.7) in (
ICat ( X ) , ⊠ , ∆ ) ends up beinga pseudonatural trasformation τ : M ⇒ N : X op → Cat (indexed functor) whose com-ponents τ x : M x → N x are strong monoidal functors, whereas a 2-cell between strongmorphisms of pseudomonoids is an ordinary modification X op Cat MN m ⇛ τ σ whose components m x : τ x ⇒ σ x are monoidal natural transformations.We thus obtain the 2-categories PsMon ( ICat ( X )) as well as PsMon ( OpICat ( X )); fromthe above descriptions, it is clear that PsMon ( ICat ( X )) = ps ( X op , MonCat ) (5.11)
PsMon ( OpICat ( X )) = ps ( X , MonCat )which will also be rediscovered by Proposition 5.15.Finally, taking pseudomonoids in strict X -indexed categories ICat s ( X ) = [ X op , Cat ] pro-duces the 2-category
PsMon ( ICat s ( X )) with objects functors M : X op → MonCat st intomonoidal categories with strict monoidal functors: the isomorphisms Equation (5.10) arenow equalities due to strict naturality of the multiplication and unit. Then the hom-categories PsMon ( ICat s ( X ))( M , N ) are full subcategories of PsMon ( ICat ( X ))( M , N ) spannedby strictly natural transformations τ : M ⇒ N , still with strong monoidal components τ x .For example, it would not be correct to write PsMon ( ICat s ( X )) = [ X op , MonCat (st) ].It is evident that
MonICat ( X ) and PsMon ( ICat ( X )) are in principle different. A monoidalindexed category with base X is a lax monoidal pseudofunctor into Cat (and X is required tobe monoidal already), whereas a pseudomonoid in X -indexed categories is a pseudofunctorfrom an ordinary category X into MonCat . In Appendix C.1, we recalled the standard equivalence between fibrations and indexedcategories via the Grothendieck construction. We will now lift this correspondence to theirmonoidal versions studied in Sections 5.2 and 5.3, using general results about pseudomonoidsin arbitrary monoidal 2-categories described in Appendix B.79ince both
Fib and
ICat are cartesian monoidal 2-categories, via Equation (5.1) andEquation (5.6) respectively, our first task is to ensure that they are monoidally equivalent.
Lemma 5.7.
The 2-equivalence
Fib ≃ ICat between the cartesian monoidal 2-categories offibrations and indexed categories is (symmetric) monoidal.Proof.
Since they form an equivalence, both 2-functors from Theorem C.1 preserve limits,therefore are monoidal 2-functors. Moreover, it can be verified that the natural isomor-phisms with components
F ∼ = F P F and P ∼ = P F P are monoidal with respect to the cartesianstructure, due to universal properties of products. Theorem 5.8.
There are 2-equivalences
MonFib ≃ MonICatBrMonFib ≃ BrMonICatSymMonFib ≃ SymMonICat between the 2-categories of monoidal fibrations and monoidal indexed categories, as wellas their braided and symmetric versions. Dually, there is a 2-equivalence
MonOpFib ≃ MonOpICat between the 2-categories of monoidal opfibrations and monoidal opindexed cat-egories, as well as their braided and symmetric versions.Proof.
Since
MonFib = PsMon ( Fib ) and
MonICat = PsMon ( ICat ), we obtain the equivalenceas a special case of Proposition B.5; similar for
OpFib ≃ OpICat . Corollary 5.9.
The above 2-equivalences restrict to the sub-2-categories of fixed bases ordomains, which by Equation (5.8) are
MonFib ( X ) ≃ Mon2Cat ps ( X op , Cat ) MonOpFib ( X ) ≃ Mon2Cat ps ( X op , Cat )These results correspond to the global monoidal structure of fibrations and indexedcategories. Even though they were directly derived via abstract reasoning, for expositionpurposes we briefly describe this equivalence on the level of objects; some relevant detailscan also be found in [BFMP20, Sec 6]. Independently and much earlier, in his thesis [Shu09]Shulman explores such a fixed-base equivalence on the level of double categories (of monoidalfibrations and monoidal pseudofunctors over the same base).Suppose that ( M , µ, µ ) : ( X op , ⊗ , I ) → ( Cat , × , ) is a monoidal indexed category, i.e.a lax monoidal pseudofunctor with structure maps Equation (5.7). The induced monoidalproduct ⊗ µ : R M × R M → R M on the Grothendieck category is defined on objects by( x, a ) ⊗ µ ( y, b ) = ( x ⊗ y, µ x,y ( a, b )) (5.12)and I µ = ( I, µ ( ∗ )) is the unit object. Clearly, the induced fibration R M → X which mapseach pair to the underlying X -object strictly preserves the monoidal structure. Moreover,pseudonaturality of µ implies that ⊗ µ preserves cartesian liftings, so all clauses of Proposi-tion 5.1 are satisfied. For a more detailed exposition of the structure, as well as the braidedand symmetric version, we refer the reader to the Section 5.5.1.We can also restrict to the context of split fibrations and strict indexed categories.80 heorem 5.10. There are 2-equivalences
MonFib s ≃ MonICat s MonOpFib s ≃ MonOpICat s between monoidal split (op)fibrations and monoidal strict (op)indexed categories, as well asfor the fixed-base case.Proof. Again by applying
PsMon (-) to the 2-equivalence
ICat s ≃ Fib s , we obtain equivalencesbetween the respective structures discussed in Sections 5.2 and 5.3, as the strict counter-parts of Theorem 5.8 and Corollary 5.9. Recall that a monoidal strict indexed categoryis a lax monoidal 2-functor X op → Cat whose structure maps ( φ, φ ) are strictly naturaltransformations, and corresponds to a split fibration which is monoidal like before, only thetensor product of the total category strictly preserves cartesian liftings.We close this section in a similar manner to Sections 5.2 and 5.3, namely by working inthe cartesian monoidal 2-categories ( Fib ( X ) , ⊠ , X ) and ( ICat ( X ) , ⊠ , ∆ ) of fibrations andindexed categories with a fixed base category. Theorem 5.11.
There are 2-equivalences between (op)fibrations with monoidal fibres andstrong monoidal reindexing functors, and pseudofunctors into
MonCatPsMon ( Fib ( X )) ≃ ps ( X op , MonCat ) PsMon ( OpFib ( X )) ≃ ps ( X op , MonCat ) Moreover, these restrict to 2-equivalences between split (op)fibrations with monoidal fibresand strict monoidal reindexing functors, and ordinary functors into
MonCat st .Proof. Since
Fib ( X ) ≃ ICat ( X ) is also a monoidal 2-equivalence, Proposition B.5 appliesonce more – recall Equation (5.11).These equivalences correspond to the fibrewise monoidal structure on fibrations andindexed categories. In more detail, a pseudofunctor M : X op → MonCat maps every object x to a monoidal category M x and every morphism f : x → y to a strong monoidal functor M f : M y → M x ; under the usual Grothendieck construction, these are precisely the fibrecategories and the reindexing functors between them for the induced fibration, as describedat the end of Section 5.2. Notice how, in particular, X is not a monoidal category, as wasthe case in Corollary 5.9.A very similar, relaxed version of the fibrewise monoidal correspondence seems to con-nect the concepts of an indexed monoidal category , defined in [HM06] as a pseudofunctor M : X op → MonCat lax , and that of of a lax monoidal fibration , defined in [Zaw11]. Noticethat these terms are misleading with respect to ours: an indexed monoidal category is not a monoidal indexed category, and also a lax monoidal fibration is not a functor with a laxmonoidal stucture.Briefly, there is a full sub-2-category
Fib opl ( X ) ⊆ Cat / X of fibrations, namely fibred 1-cells Equation (C.3) which are not required to have a cartesian functor on top. As discussed81n [Shu08, Prop.3.6], this is 2-equivalent to ps,opl ( X op , Cat ), the 2-category of pseud-ofunctors, oplax natural transformations and modifications. Describing pseudomonoidstherein appears to give rise to a fibration with monoidal fibres and lax monoidal reindexingfunctors between them, or equivalently a pseudofunctor into
MonCat lax . We omit the detailsso as to not digress from our main development.
The bulk of this chapter is dedicated to proving various monoidal variations of theequivalence between fibrations and indexed categories, using general results in monoidal2-category theory. In this section, we detail the descriptions of the (braided/symmetric)monoidal structures on the total category of the Grothendieck construction, assuming theappropriate data is present. We also provide a hands-on correspondence that underlies theproof of Theorem 5.13 regarding the transfer of monoidal structure from a functor to itstarget and vice versa. We hope this section can serve as a quick and clear reference on somefundamental constructions of this chapter.
As sketched under Corollary 5.9, let ( X , ⊗ , I ) be a monoidal category, and( M , µ, µ ) : ( X op , ⊗ op , I ) → ( Cat , × , )a monoidal indexed category, a.k.a. lax monoidal pseudofunctor. Recall that µ is pseudo-natural transformation consisting of functors µ x,y : M x × M y → M ( x ⊗ y ) for any objects x and y of X , and natural isomorphisms M z × M w M x × M y M ( z ⊗ w ) M ( x ⊗ y ) M f ×M gµ z,w µ x,yµf,g ∼ = M ( f ⊗ g ) for any arrows f : x → z and g : y → w in X . Also the unique component of µ is thefunctor µ : → M ( I ).The induced tensor product functor on the total category, denoted as ⊗ µ : R M× R M → R M , is given on objects by ( x, a ) ⊗ µ ( y, b ) = ( x ⊗ y, µ x,y ( a, b ))On morphisms ( f : x → z, k : a → ( M f ) c ) and ( g : y → w, ℓ : b → ( M g ) d ), we get( f, k ) ⊗ µ ( g, ℓ ) = ( x ⊗ y f ⊗ g −−→ z ⊗ w, µ f,g ( µ x,y ( k, ℓ )))where the latter is the composite morphism µ x,y ( a, b ) µ x,y ( k,ℓ ) −−−−−→ µ x,y (( M f )( c ) , ( M g )( d )) ∼ −→ M ( f ⊗ g )( µ z.w ( c, d )) in M ( x ⊗ y ) . I µ = ( I, µ ).If a x,y,z : ( x ⊗ y ) ⊗ z → x ⊗ ( y ⊗ z ) denotes the associator in X , the associator for( R M , ⊗ µ , I µ ) is given by α ( x,b ) , ( y,c ) , ( z,d ) = ( α x,y,z , ω x,y,z ( b, c, d ))where ω is the invertible modification Equation (B.2).If l x : I ⊗ x → x and r x : x ⊗ I → x are the left and right unitors in X , the unitors in R M are defined as λ x = ( l x , ξ -1 x ( a )) : ( I, µ ) ⊗ µ ( x, a ) → ( x, a ) ρ x = ( r x , ζ x ( a )) : ( x, a ) ⊗ µ ( I, µ ) → ( x, a )where ζ and ξ are invertible modifications as in Equation (B.2).We now turn to the correspondence between 1-cells of Theorem 5.8: given a monoidalindexed 1-cell ( X , ⊗ , I ) op ( Cat , × , )( Y , ⊗ , I ) op ( M ,µ,µ )( F,ψ,ψ ) op ⇓ τ ( N ,ν,ν ) where M and N are lax monoidal pseudofunctors and F is a monoidal functor, as inProposition 5.5, we first of all obtain an ordinary fibred 1-cell ( P τ , F ) : P M → P N asexplained above Equation (C.12) R M R NX Y P τ P M P N F with P τ ( x, a ) = ( F x, τ x ( a )). The functor F is already monoidal, and P τ obtains a monoidalstructure too: for example, there are isomorphisms P τ ( x, a ) ⊗ ν P τ ( y, b ) ∼ −→ P τ (( x, a ) ⊗ µ ( y, b )) in R N between the objects P τ ( x, a ) ⊗ ν P τ ( y, b ) = ( F x, τ x ( a )) ⊗ ν ( F y, τ y ( b ) = ( F x ⊗ F y, ν
F x,F y ( τ x ( a ) , τ y ( b )) P τ (( x, a ) ⊗ µ ( y, b )) = P τ ( x ⊗ y, µ x,y ( a, b )) = ( F ( x ⊗ y ) , τ x ⊗ y ( µ x,y ( a, b )))given by ψ x,y : F x ⊗ F y ∼ −→ F ( x ⊗ y ) and by ν F x,F y ( τ x ( a ) , τ y ( b )) ∼ = N ( ψ x,y )( τ x ⊗ y ( µ x,y ( a, b )))83ssentially given by the monoidal pseudonatural isomorphism Equation (B.4) for τ : M ⇒N F op . As a result, ( P τ , F ) is indeed a monoidal fibred 1-cell as in Proposition 5.2.Finally, it can be verified that starting with a monoidal indexed 2-cell as in Proposi-tion 5.6, the induced fibred 2-cell Equation (C.13) is monoidal, i.e. P m satisfies the conditionsof a monoidal natural transformation.Regarding the induced braided and symmetric monoidal structures, suppose that ( X , ⊗ , I )is a braided monoidal category, with braiding b with components β x,y : x ⊗ y ∼ −→ y ⊗ x ;then X op is braided monoidal with the inverse braiding, namely ( X op , ⊗ op , I, β − ). Nowif ( M , µ, µ ) : X op → Cat is a braided lax monoidal pseudofunctor, i.e. a braided monoidalindexed category, by Theorem 5.8 we have an induced braided monoidal structure on( R M , ⊗ µ , I µ ), namely B ( x,a ) , ( y,b ) : ( x, a ) ⊗ µ ( y, b ) = ( x ⊗ y, µ x,y ( a, b )) → ( y, b ) ⊗ µ ( x, a ) = ( y ⊗ x, µ y,x ( b, a ))are given by β x,y : x ⊗ y ∼ = y ⊗ x in X and ( v x,y ) ( a,b ) : µ x,y ( a, b ) ∼ = M ( β − x,y )( µ y,x ( b, a )), where v is as in Equation (B.11).If M is a symmetric lax monoidal pseudofunctor, it can be verified that B ( y,b ) , ( x,a ) ◦ B ( x,a ) , ( y,b ) = 1 ( x,a ) ⊗ µ ( y,b ) therefore R M is also symmetric monoidal, as is the monoidal fibration P M : R M → X . Here we detail the correspondence between monoidal opindexed categories and a pseud-ofunctors into
MonCat when the domain is a cocartesian monoidal category, as estab-lished by Theorem 5.13; the one for indexed categories is of course similar. We denoteby ∇ x : x + x → x the induced natural components due to the universal property of coprod-uct, and ι x : x → x + y the inclusion into a coproduct.Start with a lax monoidal pseudofunctor M : ( X , + , → ( Cat , × , ) equipped with µ x,y : M ( x ) × M ( y ) → M ( x + y ) and µ : → M (0), which gives the global monoidal struc-ture Equation (5.12) of the corresponding opfibration. There exists an induced monoidalstructure on each fibre M ( x ) as follows: ⊗ x : M ( x ) × M ( x ) µ x,x −−→ M ( x + x ) M ( ∇ ) −−−−→ M ( x ) (5.13) I x : µ −→ M (0) M (!) −−−→ M ( x )Moreover, each M f : M x → M y is a strong monoidal functor, with φ a,b : ( M f )( a ) ⊗ y ( M f )( b ) ∼ −→ M f ( a ⊗ x b ) and φ : I y ∼ −→ ( M f ) I x essentially given by the following isomor-84hisms M x × M x M y × M y M ( x + x ) M ( y + y ) M x M y M f ×M fµ x,x µf,f ∼ = µ y,y M ( ∇ x ) M ( f + f ) ∼ = M ( ∇ y ) M f M (0) M (0) M x M y µ µ ∼ = M (!) M (!) M f (5.14)since ∇ and ! are natural and M is a pseudofunctor.In the opposite direction, take an ordinary pseudofunctor M : X →
MonCat into the2-category of monoidal categories, strong monoidal functors and monoidal natural trans-formations, with ⊗ x : M ( x ) × M ( x ) → M ( x ) and I x the fibrewise monoidal structures inevery M x . We can use those to endow M with a lax monoidal structure via µ x,y : M ( x ) × M ( y ) M ( ι x ) ×M ( ι y ) −−−−−−−−−→ M ( x + y ) × M ( x + y ) ⊗ x + y −−−→ M ( x + y ) µ : I −→ M (0)The fact that all M f are strong monoidal imply that the above components form pseudo-natural transformations, and all appropriate conditions are satisfied.In the strict context, a lax monoidal 2-functor M : ( X , + , → ( Cat , × , ) with naturallaxator and unitor bijectively corresponds to a functor X →
MonCat st since Equation (5.14)are in fact strictly commutative, by naturality of µ, µ and functoriality of M .In the even more special case of an ordinary lax monoidal functor M : ( X , + , → ( Cat , × , ), the fibres M ( x ) turn out to be strict monoidal. For example, strict associativityof the tensor is established by M x × M x × M x M x × M x M x × M ( x + x ) M ( x + x ) M ( x + x + x ) M x M ( x + x ) × M x M ( x + x ) M x × M x × M x M x × M x ×⊗ x × µ x,x ⊗ x µ x,x ( ∗ ) ×M ( ∇ ) µ x,x + x M∇M (1+ ∇ ) M ( ∇ +1) µ x + x,x M ( ∇ ) × M∇ µ x,x × ⊗ x × µ x ⊗ x where the three diamond-shaped diagrams on the right commute due to naturality of µ aswell as associativity of ∇ and functoriality of M already in the monoidal strict opindexedcase, whereas ( ∗ ) is in general ω from Equation (B.2) which in this case is an identity, andthe four triangular diagrams commute due to Equation (5.13).85 .5.3 Comparison with Higher-Dimensional Grothendieck Constructions Monoidal categories are precisely bicategories with one object. As recalled in Ap-pendix C.1, there is a theory of fibred bicategories and indexed bicategories, and a cor-responding Grothendieck construction. It is natural to consider the possibility that themonoidal Grothendieck constructions presented here are special cases of this bicategoricalversion. However, it is easy to see that this cannot be the case. When one restricts theirview to just the objects, the bicategorical Grothendieck construction is just taking the dis-joint union of the object sets of the fibres. If you consider an indexed monoidal category asa special case of an indexed bicategory, where each fibre has one object, then generally youwould not expect the total bicategory to have one object. It would have as many objectsas the base category. Thus, the result would not be a monoidal category. The constructiongiven here always produces a monoidal category.
In the previous section, we obtain two different equivalences between fixed-base fibrationsand fixed-domain indexed categories of monoidal flavor: Corollary 5.9 where both total andbase categories are monoidal, and Theorem 5.11 where only the fibres are monoidal. Clearly,neither of these two cases implies the other in general. The global monoidal structure asdefined in Equation (5.12) sends two objects in arbitrary fibres to a new object lying in thefibre of the tensor of their underlying objects in the base, whereas having a fibre-wise tensorproducts does not give a way of multiplying objects in different fibres of the total category.In [Shu08], Shulman introduces monoidal fibrations (Proposition 5.1) as a building blockfor fibrant double categories. Due to the nature of the examples, the results restrict tothe case where the base of the monoidal fibration P : A → X is equipped with specifi-cally a cartesian or cocartesian monoidal structure; the main idea is that these fibrationsform a “parameterized family of monoidal categories”. Formally, a central result thereinlifts the Grothendieck construction to the monoidal setting, by showing an equivalence be-tween monoidal fibrations over a fixed (co)cartesian base and ordinary pseudofunctors into
MonCat . Theorem 5.12 ([Shu08]) . If X is cartesian monoidal, MonFib ( X ) ≃ ps ( X op , MonCat ) (5.15)
Dually, if X is cocartesian monoidal, MonOpFib ( X ) ≃ ps ( X , MonCat ) . Bringing all these structures together, we obtain the following.
Theorem 5.13. If X is a cartesian monoidal category, MonFib ( X ) Mon2Cat ps ( X op , Cat ) PsMon ( Fib ( X )) ps ( X op , MonCat ) ≃ ≃ ≃ ≃ ually, if X is a cocartesian monoidal category, MonOpFib ( X ) Mon2Cat ps ( X , Cat ) PsMon ( OpFib ( X )) ps ( X , MonCat ) ≃ ≃ ≃ ≃ In the strict context, the restricted equivalences give a correspondence between monoidal splitfibrations over X and functors X op → MonCat st , and between monoidal split opfibrationsover X and functors X →
MonCat st . The original proof of Theorem 5.12 is an explicit, piece-by-piece construction of anequivalence, and employs the reindexing functors ∆ ∗ and π ∗ induced by the diagonal andprojections in order to move between the appropriate fibres and build the required struc-tures. The global monoidal structure is therein called external and the fibre-wise internal .Here we present a different argument that does not focus on the fibrations side. Theequivalence between lax monoidal pseudofunctors X op → Cat and ordinary pseudofunctors X op → MonCat , which essentially provides a way of transferring the monoidal structurefrom the target category to the functor itself and vice versa, brings a new perspective onthe behavior of such objects.
Lemma 5.14.
For any two monoidal 2-categories K and L , the following are true.1. For an arbitrary 2-category A , ps ( A , Mon2Cat ps ( K , L )) ≃ Mon2Cat ps ( K , ps ( A , L )) (5.16)
2. For a cocartesian 2-category A , ps ( A , Mon2Cat ps ( K , L )) ≃ Mon2Cat ps ( A × K , L ) (5.17) Proof.
First of all, recall [Str80, 1.34] that there are equivalences ps ( A , ps ( K , L )) ≃ ps ( A × K , L ) ≃ ps ( K , ps ( A , L ))which underlie Equation (5.16) and Equation (5.17) for the respective pseudofunctors; sothe only part needed is the correspondence between the respective monoidal structures.Notice that A × K is a monoidal 2-category since both A and K are, and also ps ( A , L )is monoidal since L is: define ⊗ [] and I [] by ( F ⊗ [] G )( a ) = F a ⊗ L G a (similarly to Equa-tion (5.9)) and I [] : A ! −→ I L −→ L .First, we prove 1. Take a pseudofunctor F : A →
Mon2Cat ps ( K , L ). For every a ∈ A ,its image pseudofunctor F a is lax monoidal, i.e. comes equipped with maps in L : φ ax,y : ( F a )( x ) ⊗ L ( F a )( y ) → ( F a )( x ⊗ K y ) , φ a : I L → ( F a ) I K (5.18)for every x, y ∈ K , satisfying coherence axioms.87ow define the pseudofunctor F : K → ps ( A , L ), with ( F x )( a ) := ( F a )( x ). It has alax monoidal structure, given by pseudonatural transformations F x ⊗ [] F y ⇒ F ( x ⊗ K y ) , I [] ⇒ F ( I K )whose components evaluated on some a ∈ A are defined to be Equation (5.18). Pseudonat-urality and lax monoidal axioms follow, and in a similar way we can establish the oppositedirection and verify the equivalence.Now, we prove 2. If A is a cocartesian monoidal 2-category, a lax monoidal pseudofunctor F : A →
Mon2Cat ps ( K , L ) induces a pseudofunctor ˜ F : A × K → L by ˜ F ( a, x ) := ( F a )( x ).Its lax monoidal structure is given by the composite˜ F ( a, x ) ⊗ L ˜ F ( b, y ) ˜ F ( a + b, x ⊗ K y )( F a )( x ) ⊗ L ( F b )( y ) ( F ( a + b ))( x ⊗ K y )( F ( a + b ))( x ) ⊗ L ( F ( a + b ))( y ) ψ ( a,x ) , ( b,y ) ( F ι a ) x ⊗ ( F ι b ) y φ a + bx,y where a ι a −→ a + b ι b ←− b are the inclusions, and ψ : I L φ −→ ˜ F (0 , I K ); the respective axiomsfollow.In the opposite direction, starting with some pseudofunctor G : A × K → L equippedwith a lax monoidal structure ψ ( a,x ) , ( b,y ) and ψ , we can build ˆ G : A →
Mon2Cat ps ( K , L ) forwhich every ˆ G a is a lax monoidal pseudofunctor, via( ˆ G a )( x ) ⊗ L ( ˆ G b )( y ) ( ˆ G a )( x ⊗ K y ) G ( a, x ) ⊗ L G ( a, y ) G ( a + a, x ⊗ K y ) G ( a, x ⊗ K y ) φ a ( x,y ) ψ ( a,x ) , ( a,y ) G ( ∇ , φ a : I L ψ −→ G (0 , I K ) G (! , −−−→ G ( a, I K )The equivalence follows, using the universal properties of coproducts and initial object. Proof of Theorem 5.13.
The top and bottom right 2-categories of the first square are equiv-alent as follows, where X op is cocartesian. ps ( X op , MonCat ) ≃ ps ( X op , PsMon ( Cat )) Equation (B.10) ≃ ps ( X op , Mon2Cat ps ( , Cat )) Equation (5.17) ≃ Mon2Cat ps ( X op × , Cat ) ≃ Mon2Cat ps ( X op , Cat )The strict context equivalence can be explicitly verified as a special case of the above, wherethe corresponding 1-cells and 2-cells are as described in Section 5.2 and Section 5.3.88he decisive step in the above proof is the much broader Lemma 5.14; for a groundedexplanation of the correspondence of the relevant structures, see Section 5.5.2. In simplerwords, a lax monoidal structure of a pseudofunctor F : ( A , + , → ( Cat , × , ) gives apseudofunctor F : A →
MonCat and vice versa: in a sense, ‘monoidality’ can move betweenthe functor and its target.As another corollary of Lemma 5.14, we can formally deduce that pseudomonoids in(
ICat ( X ) , ⊠ , ∆ ) are functors into MonCat , as described at the end of Section 5.3.
Proposition 5.15.
For any X , PsMon ( ICat ( X )) ≃ ps ( X op , MonCat ) .Proof. There are equivalences
PsMon ( ICat ( X )) = PsMon ( ps ( X op , Cat )) ≃ Mon2Cat ps ( , ps ( X op , Cat )) Equation (5.16) ≃ ps ( X op , Mon2Cat ps ( , Cat )) Equation (B.10) ≃ ps ( X op , PsMon ( Cat )) ≃ ps ( X op , MonCat )as desired.As a first and meaningful example of Theorem 5.13, recall that the categories
Fib and
ICat are themselves fibred over
Cat , with fibres
Fib ( X ) and ICat ( X ) respectively. The basecategory in both cases is the cartesian monoidal category ( Cat , × , Fib , ICat and
Fib ( X ), ICat ( X ), instrumental for the study of global and fibre-wise monoidal structures,follow the very same abstract pattern. Proposition 5.16.
The fibrations
Fib → Cat and
ICat → Cat are monoidal, and more-over their fibres
Fib ( X ) and ICat ( X ) are monoidal and the reindexing functors are strongmonoidal.Proof. The pseudofunctors inducing
Fib → Cat and
ICat → Cat are
Cat op CAT Cat op CAT X Fib ( X ) X ICat ( X ) Y Fib ( Y ) Y ICat ( Y ) F FF ∗ −◦ F op where CAT is the 2-category of possibly large categories, F ∗ takes pullbacks along F and − ◦ F op precomposes with the opposite of F . These are both lax monoidal, with therespective structures essentially being Equation (5.1) and Equation (5.6) giving the globalmonoidal structure on the fibrations.Since the base of both monoidal fibrations is cartesian, the global monoidal structure isequivalent to a fibre-wise monoidal structure, as per the theme of this whole section. Theinduced monoidal structure on each Fib ( X ) is given by Equation (5.4) and on each ICat ( X )by Equation (5.9), and F ∗ , − ◦ F op are strong monoidal functors accordingly.89he above essentially lifts the global and fibre-wise monoidal structure developmentone level up, exhibiting fibrations and indexed categories as examples of the monoidalGrothendieck construction themselves.Concluding this investigation on monoidal structures of fibrations and indexed cate-gories, we consider the (co)cartesian monoidal (op)fibration case; for example, a monoidalfibration P : ( A , × , → ( X , × ,
1) as in Proposition 5.1 where P preserves products (orcoproducts for opfibrations) on the nose. As remarked in [Shu08, 12.9], the equivalenceEquation (5.15) restricts to one between pseudofunctors which land to cartesian monoidalcategories, and monoidal fibrations where the total category is cartesian monoidal. Withthe appropriate 1-cells and 2-cells that preserve the structure, we can write the respectiveequivalences as ps ( X op , Cart ) ≃ cMonFib ( X ) for cartesian X (5.19) ps ( X , Cocart ) ≃ cocMonOpFib ( X ) for cocartesian X where the prefixes c and coc correspond to the respective (co)cartesian structures. Explic-itly, in order for the total category to specifically be endowed with (co)cartesian monoidalstructure, it is required not only that the base category is but also the fibres are and thereindexing functors preserve finite (co)products.This special case of the monoidal Grothendieck construction that connects the existenceof (co)products and initial/terminal object in the fibres and in the total category, is remi-niscent (and also an example of) the general theory of fibred limits originated from [Gra66].Explicitly, [Her99, Cor. 4.9] deduces that if the base of a fibration P : A → X has J -limitsfor any small category J , then the fibres have and the reindexing functors preserve J -limitsif and only if A has J -limits and P strictly preserves them, and dually for opfibrations andcolimits. Hence for finite (co)products in (op)fibrations, Equation (5.19) re-discovers thatresult using the monoidal Grothendieck correspondence.Moreover, since the squares of Theorem 5.13 reduce to their (co)cartesian variants, wewould like to identify the conditions that the corresponding lax monoidal pseudofunctorinto Cat needs to satisfy in order to give rise to a (co)cartesian monoidal (op)fibration.We employ Proposition A.17 to tackle the opfibration case: if, in a symmetric monoidalcategory X , there exist monoidal natural transformations with components ∇ x : x ⊗ x → x, u x : I → x satisfying the commutativity of I ⊗ x x ⊗ x x ⊗ I x ⊗ xx x u x ⊗ ∼ ℓ x ∇ x ∼ r x ⊗ u x ∇ x (5.20)then X is cocartesian monoidal. In fact, it is the case that a symmetric monoidal categoryis cocartesian if and only if Mon ( X ) ∼ = X .Suppose ( M , µ, µ ) : X →
Cat is a (symmetric) lax monoidal pseudofunctor, such thatthe corresponding Grothendieck category ( R M , ⊗ µ , I µ ) described in Section 5.4 is cocarte-sian monoidal. This means there are monoidal natural transformations with components ∇ ( x,a ) : ( x, a ) ⊗ µ ( x, a ) → ( x, a ) and u ( x,a ) : ( I, µ ( ∗ )) → ( x, a )90aking the diagrams Equation (5.20) commute. Explicitly, by Equation (5.12), ∇ ( x,a ) consists of morphisms f x : x ⊗ x → x in X and κ a : ( M f x )( µ x,x ( a, a )) → a in M x , whereas u ( x,a ) consists of i x : I → x in X and λ a : ( M i x ) µ → a in M x .The conditions Equation (5.20) say that the composites( I, µ ) ⊗ µ ( x, a ) u ( x,a ) ⊗ µ ( x,a ) −−−−−−−−−→ ( x, a ) ⊗ µ ( x, a ) ∇ ( x,a ) −−−−→ ( x, a )( x, a ) ⊗ µ ( I, µ ) ( x,a ) ⊗ µ u ( x,a ) −−−−−−−−−→ ( x, a ) ⊗ µ ( x, a ) ∇ ( x,a ) −−−−→ ( x, a )are equal to the left and right unitor on x , where all respective structures are detailedin Section 5.5.1. Using the composition inside R M analogously to Equation (C.11), theseconditions translate, on the one hand, to the base being cocartesian monoidal ( X , + ,
0) with f x = ∇ x and i x = u x . On the other hand, κ a and λ a form natural transformations M x × M x M ( x + x ) M x M x µ x,x M ( ∇ x )∆ 1 ⇓ κ x M (0) M x M x µ M ( u x )! 1 ⇓ λ x (5.21)satisfying the commutativity of M ( ∇ x ◦ ( u x + 1))( µ ,x ( µ ( ∗ ) , a )) ( M ( ∇ x ) ◦ M ( u x + 1))(( µ ,x ( µ ( ∗ ) , a )) M ( ∇ x )( µ x,x ( M ( u x )( µ ( ∗ ) , a ))) M ( ∇ x )( µ x,x ( a, a )) M ( ℓ x )( µ ,x ( µ ( ∗ ) , a )) a id ∼ δ ∼ M ( ∇ x )( µ ux, ) M ( ∇ x )( µ x,x ( λ xa ,γ )) κ xa ξ ∼ and a similar one with µ on second arguments. The above greatly simplifies if M is justa lax monoidal functor: the first condition becomes 1 a ∼ = κ xa ◦ M ( ∇ x )( µ x,x ( λ xa , a ∼ = κ xa ◦ M ( ∇ x )( µ x,x (1 a , λ xa )). Corollary 5.17.
A lax monoidal pseudofunctor M : ( X , + , → ( Cat , × , ) equipped withnatural transformations κ and λ as in Equation (5.21) corresponds to an ordinary pseud-ofunctor M : X →
Cocart , or equivalently Equation (5.19) to a cocartesian monoidal opfi-bration.
In this section, we explore certain settings where the equivalence between monoidalfibrations and monoidal indexed categories naturally arises. Instead of going into details thatwould result in a much longer text, we mostly sketch the appropriate example cases up tothe point of exhibition of the monoidal Grothendieck correspondence, providing indicationsof further work and references for the interested reader.91 .7.1 Fundamental Bifibration
For any category X , the codomain or fundamental opfibration is the usual functor fromits arrow category cod : X −→ X mapping every morphism to its codomain and every commutative square to its right-handside leg. It uniquely corresponds to the strict opindexed category, i.e. mere functor X Cat x X /xy X /y f f ! (5.22)that maps an object to the slice category over it and a morphism to the post-compositionfunctor f ! = f ◦ − induced by it.If the category has a monoidal structure ( X , ⊗ , I ), this (2-)functor naturally becomeslax monoidal with structure maps X /x × X /y ⊗ −→ X / ( x ⊗ y ) , I −→ X /I. (5.23)These components form strictly natural transformations, and for example the invertiblemodification ω Equation (B.2) has components the evident isomorphisms, for ( f, g, h ) ∈X /x × X /y × X /z , between a ⊗ ( b ⊗ c ) f ⊗ ( g ⊗ h ) −−−−−→ x ⊗ ( y ⊗ z ) ∼ = ( x ⊗ y ) ⊗ z (5.24)( a ⊗ b ) ⊗ c ( f ⊗ g ) ⊗ h −−−−−→ ( x ⊗ y ) ⊗ z By Theorem 5.10, this monoidal strict opindexed category correspondes to a monoidal splitfibration, i.e. ( X , ⊗ , I ) is monoidal and cod strict monoidal, where ⊗ X strictly preservescartesian liftings via f ! k ⊗ g ! ℓ = ( f ⊗ g ) ! ( k ⊗ ℓ ) – which can of course be independentlyverified. However in general, the slice categories X /x do not inherit the monoidal structure:there is no way to restrict the global monoidal structure to a fibrewise one.According to Theorem 5.13, there is an induced monoidal structure on the categories X /x and a strict monoidal structure on all f ! only when the monoidal structure on X isgiven by binary coproducts and an initial object (i.e. cocartesian). In that case, for each k : a → x and ℓ : b → x in the same fibre X /x , their tensor product in X /x is given by a + b k + ℓ −−−→ x + x ∇ x −−→ x as a simple example of Equation (5.13). In fact, this is precisely the coproduct of two objectsin X /x , and 0 ! −→ x the initial object, due to the way colimits in the slice categories areconstructed. Therefore this falls under the cocartesian-fibres special case Equation (5.19),bijectively corresponding to the cocartesian structure on X inherited from X .92ow suppose an ordinary category X has pullbacks. This endows the codomain functoralso with a fibration structure, corresponding to the indexed category X op Cat x X /xy X /y f f ∗ with the same mapping on objects as Equation (5.22) but by taking pullbacks rather thanpost-composing along morphisms, a pseudofunctorial assignment. This gives cod : X → X a bifibration structure, also by that classic fact that f ! ⊣ f ∗ .In this case, if X has a general monoidal structure, there is no naturally induced laxmonoidal structure of that pseudofunctor as before: there is no reason for the pullback of atensor to be isomorphic to the tensor of two pullbacks. However, if X is cartesian monoidal(hence has all finite limits), the components X /x × X /y × −→ X / ( x × y ) , ∆ ! −→ X / k : a → x and ℓ : b → x in X /x , their induced product is given by • a × bx x × x δ ∗ ( k × ℓ ) y k × ℓδ and 1 x : x → x is the unit of each slice X /x , this indexed monoidal category also describedin [HM06]. The monoidal fibration structure on cod : ( X , × , ) → ( X, × ,
1) is the evi-dent one, so it again falls in the special case Equation (5.19) now for cartesian fibres, byconstruction of products in slice categories.As a final remark, analogous constructions hold for the domain functor which is againa bifibration: its fibration structure comes from pre-composing along morphisms, whereasits opfibration structure comes from taking pushouts along morphisms.
Recall that for any category C , the standard family fibration is induced by the (strict)functor [ − , C ] : Set op → Cat (5.25)which maps every discrete category X to the functor category [ X, C ] and every function f : X → Y to the functor f ∗ = [ f, f . The total category of the93nduced fibration Fam ( C ) → C has as objects pairs ( X, M : X → C ) essentially given by afamily of X -indexed objects in C , written { M x } x ∈ X , whereas the morphisms are X C Y Mf ⇓ αN namely a function f : X → Y together with families of morphisms α x : M x → N fx in C .Notice the similarity of this description with Equation (C.8), which for the strict indexedcategories case looks like a non-discrete version of the family fibration, for C = Cat . More-over, it is a folklore fact that
Fam ( C ) is the free coproduct cocompletion on the category C . On the other hand, we could consider the opfibration induced by the very same functorEquation (5.25), denoted by Maf ( C ) → Set op . The objects of Maf ( C ) are the same as Fam ( C ), but morphisms { M x } x ∈ X → { N y } y ∈ Y between them are functions g : Y → X (i.e. X → Y in Set op ) together with families of arrows β y : M gy → N y in C . Notice that these arenow indexed over the set Y rather than X like before, and in fact Maf ( X ) = Fam ( X op ) op .In the case that the category is monoidal ( C , ⊗ , I ), the (2-)functor [ − , C ] has a canonicallax monoidal structure. Explicitly, by taking its domain Set op to be cocartesian by theusual cartesian monoidal structure ( Set , × , φ X,Y : [ X, C ] × [ Y, C ] → [ X × Y, C ] , φ : I C −→ [ , C ] ∼ = C where φ X,Y corresponds, under the tensor-hom adjunction in
Cat , to[ X, C ] × [ Y, C ] × X × Y ∼ −→ [ X, C ] × X × [ Y, C ] × Y ev X × ev Y −−−−−−→ C × C ⊗ −→ C . These are again natural components, and for example Equation (B.2) has componentsthe natural isomorphisms between the assignments
M x ⊗ ( N y ⊗ U z ) and (
M x ⊗ N y ) ⊗ U z . By Theorem 5.10, this monoidal strict indexed category endows the correspondingsplit fibration
Fam ( X ) → Set with a monoidal structure via { M x } ⊗ { N y } := { M x ⊗ N y } X × Y . On the other hand, we could use the dual part of the same theorem, and insteadconsider the induced monoidal split opfibration Maf ( X ) → Set op corresponding to the same([ − , C ] , φ, φ ).Moreover, since Set is cartesian, Theorem 5.13 also applies in both cases, giving amonoidal structure to the fibres as well: for M : X → C and N : X → C , their fibrewisetensor product and unit are given by X ∆ −→ X × X M × N −−−−→ C × C ⊗ −→ C , X ! −→ I −→ C which are precisely constructed as in Equation (5.13). Once again, notice the direct similarywith Equation (5.9), the fibrewise monoidal structure on ICat ( X ).As an interesting example, consider C = Mod R for a commutative ring R , with its usualtensor product ⊗ R . In [CDL06], the authors introduce a category T of Turaev R -modules, as94ell as a category Z of Zunino R -modules, which serve as symmetric monoidal categorieswhere group-(co)algebras and Hopf group-(co)algebras, [Tur00], live as (co)monoids andHopf monoids respectively.In more detail, the objects of both T and Z are defined to be pairs ( X, M ) where X is a set and { M x } x ∈ X is an X -indexed family of R -modules, and their morphisms arerespectively( T ) ( s : M g ( y ) → N y in Mod R g : Y → X in Set ( Z ) ( t : M x → N f ( x ) in Mod R f : X → Y in Set
There is a symmetric pointwise monoidal structure, { M x ⊗ R N y } X × Y , and there are strictmonoidal forgetful functors T →
Set op , Z →
Set . It is therein shown that comonoids in T are monoid-coalgebras and monoids in Z are monoid-algebras , i.e. families of R -modulesindexed over a monoid, together with respective families of linear maps( T ) C g ∗ h → C g ⊗ C h ( Z ) A g ⊗ A h → A g ∗ h C e → R R → A e satisfying appropriate axioms. Based on the above, it is clear that T = Maf ( Mod R ) and Z = Fam ( Mod R ), which clarifies the origin of these categories and can be directly used tofurther generalize the notions of Hopf group-(co)monoids in arbitrary monoidal categories. For any monoidal category V , there exist global categories of modules and comodules,denoted by Mod and
Comod [Vas14, 6.2]. Their objects are all (co)modules over (co)monoidsin V , whereas a morphism between an A -module M and a B -module N is given by a monoidmap f : A → B together with a morphism k : M → N in V satisfying the commutativity of A ⊗ M MA ⊗ N B ⊗ N N µ ⊗ k kf ⊗ µ where µ denotes the respective action, and dually for comodules. Both these categoriesarise as the total categories induced by the Grothendieck construction on the functors Mon ( V ) op Cat Comon ( V ) Cat A Mod V ( A ) C Comod V ( C ) B Mod V ( B ) D Comod V ( D ) f g g ! f ∗ (5.26)where f ∗ and g ! are (co)restriction of scalars: if M is a B -module, f ∗ ( M ) is an A -modulevia the action A ⊗ M f ⊗ −−→ B ⊗ M µ −→ M. Mod → Mon ( V ) and Comod → Comon ( V ), mapa (co)module to its respective (co)monoid.Recall that when ( V , ⊗ , I, σ ) is braided monoidal, its categories of monoids and como-noids inherit the monoidal structure: if A and B are monoids, then A ⊗ B has also a monoidstructure via A ⊗ B ⊗ A ⊗ B ⊗ σ ⊗ −−−−→ A ⊗ A ⊗ B ⊗ B m ⊗ m −−−→ A ⊗ B, I ∼ = I ⊗ I j ⊗ j −−→ A ⊗ B where m and j give the respective monoid structures. In that case, the induced splitfibration and opfibration are both monoidal. This can be deduced by directly checking theconditions of Proposition 5.1, as was the case in the relevant references, or in our setting byusing Theorem 5.10 since both (2-)functors Equation (5.26) are lax monoidal. For example,for any A, B ∈ Mon ( V ) there are natural maps φ A,B : Mod V ( A ) × Mod V ( B ) → Mod V ( A ⊗ B ) φ : → Mod V ( I )with φ A,B ( M, N ) = M ⊗ N , with the A ⊗ B -module structure being A ⊗ B ⊗ M ⊗ N ⊗ σ ⊗ −−−−→ A ⊗ M ⊗ B ⊗ N µ ⊗ µ −−−→ M ⊗ N and φ ( ∗ ) = I , which are pseudoassociative and pseudounital in the sense that e.g. for any M, N, P ∈ Mod V ( A ) × Mod V ( B ) × Mod V ( C ), M ⊗ ( N ⊗ P ) is only isomorphic to ( M ⊗ N ) ⊗ P as ( A ⊗ B ) ⊗ C -modules.Notice that in general, the monoidal bases Mon ( V ) and Comon ( V ) are not (co)cartesian,since they have the same tensor as ( V , ⊗ , I, σ ). Therefore this case does not fall underTheorem 5.13, hence the fibre categories are not monoidal. For example in ( Vect k , ⊗ k , k ),the k -tensor product of two A -modules for a k -algebra A is not an A -module as well.We remark that the induced monoidal opfibration Comod → Comon ( V ) in fact servesas the monoidal base of an enriched fibration structure on Mod → Mon ( V ) as explainedin [Vas18], built upon an enrichment between the monoidal bases Mon ( V ) in Comon ( V )established in [HLFV17]. Moreover, analogous monoidal structures are induced on the(op)fibrations of monads and comonads in any fibrant monoidal double category, see [Vas19,Prop. 3.18]. In [SSV20] as well as in earlier works e.g. [VSL15], the authors investigate a categoricalframework for modeling systems of systems using algebras for a monoidal category. In moredetail, systems in a broad sense are perceived as lax monoidal pseudofunctors W C → Cat where W C is the monoidal category of C - labeled boxes and wiring diagrams with types in afinite product category C . Briefly, the objects in W C are pairs X = ( X in , X out ) of finite setsequipped with functions to ob C , thought of as boxes X a ...a m b ...b n X in = { a , . . . , a m } are the input ports, X out = { b , . . . , b n } the output ones and allwires are associated to a C -object expressing the type of information that can go throughthem. A morphism φ : X → Y in this category consists of a pair of functions (cid:26) φ in : X in → X out + Y in φ out : Y out → X out that respect the C -types, which roughly express which port is ‘fed information’ by which.Graphically, we can picture it as YX... ...φ : X → Y (5.27)Composition of morphisms can be thought of a zoomed-in picture of three boxes, and themonoidal structure amounts to parallel placement of boxes as in X X ... ...... ... There is a close connection between the definition of W C and that of Dialectica categoriesas well as lenses ; such considerations are the topic of work in progress [FHJ + W C to Cat that essentially receive a general picture such as X X X X X Y (which really takes place in the underlying operad of W C ) and assign systems of a certainkind to all inner boxes; the lax monoidal and pseudofunctorial structure of this assignmentformally produce a system of the same kind for the outer box.Examples of such systems are discrete dynamical systems (Moore machines in the finitecase), continuous dynamical systems but also more general systems with deterministic ortotal conditions; details can be found in the provided references. Since all these systemsare lax monoidal pseudofunctors from the non-cocartesian monoidal category of wiringdiagrams to Cat , i.e. monoidal indexed categories, the monoidal Grothendieck constructionTheorem 5.8 induces a corresponding monoidal fibration in each system case, and this globalstructure does not reduce to a fibrewise one.97or example, the algebra for discrete dynamical systems [SSV20, Sec. 2.3]DDS : W Set → Cat (5.28)assigns to each box X = ( X in , X out ) the category of all discrete dynamical systems with fixedinput and output sets being Q x ∈ X in x and Q y ∈ X out y respectively. There exist morphismsbetween systems of the same input and output set, but not between those with different ones.To each morphism, i.e. wiring diagram as in Equation (5.27), DDS produces a functor thatmaps an inner discrete dynamical system to a new outer one, with changed input and outputsets accordingly. (Pseudo)functoriality of this assignment allows the coherent zoom-in andzoom-out on dynamical systems built out of smaller dynamical systems, and monoidalityallows the creation of new dynamical systems on parallel boxes.Being a monoidal indexed category, Equation (5.28) gives rise to a monoidal opfibrationover W Set . Its total category R DDS has objects all dynamical systems with arbitrary inputand output sets, morphisms that can now go between systems of different inputs/outputs,and also a natural tensor product inherited from that in W Set and the laxator of R DDS. Ina sense, this category has all the required flexibility for the direct communication (via mor-phisms in the total category) between any discrete dynamical system, or any composite ofsystems or parallel placement of them, whereas the wiring diagram algebra Equation (5.28)focuses on the machinery of building new discrete dynamical systems systems from old.This classic change of point of view also transfers over to maps of algebras, i.e. indexedmonoidal 1-cells. As an example, see [SSV20, Sec. 5.1], discrete dynamical systems cannaturally be viewed as general total and deterministic machines denoted by Mch td , via amonoidal pseudonatural transformation W Set
Cat W ] Int N DDS ⇓ Mch td which also changes the type of input and output wires from sets to discrete interval sheaves ] Int N . This gives rise to a monoidal opfibred 1-cell R DDS R Mch td W Set W f Int N which provides a direct functorial translation between the one sort of system to the otherin a way compatible with the monoidal structure.As a final note, this method of modeling certain objects as algebras for a monoidal cat-egory (a.k.a. strict or general monoidal indexed categories) carries over to further contextsthan systems and the wiring diagram category. Examples include hypergraph categories98s algebras on cospans [FS19] and traced monoidal categories as algebras on cobordisms[SSR17]. In all these cases, the monoidal Grothendieck construction gives a potentiallyfruitful change of perspective that should be further investigated. As we show in Appendix C.4.2, the category of (directed, multi) graphs, is bifibred overset, where the bifibration V : Grph → Set is given by sending a graph to its vertex set.Since V : Grph → Set preserves products, then it can be given the structure of a strictmonoidal monoidal functor with respect to the cartesian monoidal structures on
Grph and
Set . Since the cartesian morphisms are those that form pullback squares, and productsin
Grph are given pointwise, then the monoidal structure in
Grph preserves cartesian mor-phisms. We can then apply Corollary 5.9 to obtain a symmetric lax monoidal structure forthe pseudofunctor
Grph ∗ : Set op → Cat . The lax structure map γ X,Y : Grph X × Grph Y → Grph X × Y is given by taking the product of the two graphs within Grph . Notice the producthas vertex set given by X × Y . Since the base category is cartesian monoidal, we can applyTheorem 5.13, granting a symmetric monoidal structure to the fibres Grph X . The monoidalproduct is given by the following composite. Grph X × Grph
X γ
X,X −−−→
Grph X × X ∆ ∗ −−→ Grph X Simply put, this operation is given by taking the product of the two graphs on X , and thenrestricting to the vertices on the diagonal. Indeed, this is the cartesian monoidal structureon Grph X .Since the category Set also has all finite colimits, we obtain a symmetric lax monoidalstructure for the pseudofunctor
Grph ∗ : Set op → Cat . The lax structure map φ X,Y : Grph X × Grph Y → Grph X + Y is given by taking the disjoint union of the two graphs. Notice thedisjoint union has vertex set given by X + Y . Since the base category is cocartesian monoidal,we can apply Theorem 5.13, granting a symmetric monoidal structure to the fibres Grph X .The monoidal product is given by the following composite. Grph X × Grph
X φ
X,X −−−→
Grph X + X ∆ ∗ −−→ Grph X Simply put, this operation is given by taking the disjoint union of the edges. Indeed, thisis the cocartesian monoidal structure on
Grph X . This is also the overlay operation for thenetwork model of directed multi graphs. 99 ppendix A Monoidal Categories
Monoidal categories lie at the center of applied category theory. This section is includedmainly to establish notation and terminology used throughout this thesis. Some standardreferences are [ML98] and [EGNO15].
A.1 Definitions
A.1.1 Monoidal, Braided, and Symmetric Categories
Definition A.1. A monoidal category ( C , ⊗ , I, α, λ, ρ ) consists of • a category C• a functor ⊗ : C × C → C called the tensor • a functor I : 1 → C called the unit • a natural transformation α with components of the form α x,y,z : ( x ⊗ y ) ⊗ z → x ⊗ ( y ⊗ z )called the associator • a natural transformation λ with components of the form λ x : I ⊗ x → x called the leftunitor • a natural transformation ρ with components of the form ρ x : x ⊗ I → x called the right unitor such that the following diagrams commute. 100 entagon identity: ( w ⊗ x ) ⊗ ( y ⊗ z )(( w ⊗ x ) ⊗ y ) ⊗ z w ⊗ ( x ⊗ ( y ⊗ z ))( w ⊗ ( x ⊗ y )) ⊗ z w ⊗ (( x ⊗ y ) ⊗ z ) α w,x,y ⊗ z α w ⊗ x,y,z α w,x,y ⊗ z α w,x ⊗ y,z w ⊗ α x,y,z (A.1) Triangle identity: ( x ⊗ I ) ⊗ y x ⊗ ( I ⊗ y ) x ⊗ y α x,I,y ρ x ⊗ y x ⊗ λ y (A.2)A strict monoidal category is one where the associator, left unitor, and right unitor areall identity.A braided monoidal category [JS93] is a monoidal category equipped with a naturaltransformation β called the braiding with components β x,y : x ⊗ y → y ⊗ x , such that thefollowing diagrams commute.( x ⊗ y ) ⊗ z x ⊗ ( y ⊗ z )( y ⊗ x ) ⊗ z ( y ⊗ z ) ⊗ xy ⊗ ( x ⊗ z ) y ⊗ ( z ⊗ x ) α x,y,z β x,y ⊗ z β x,y ⊗ z α y,x,z α y,z,x y ⊗ β x,z x ⊗ ( y ⊗ z ) ( x ⊗ y ) ⊗ zx ⊗ ( z ⊗ y ) z ⊗ ( x ⊗ y )( x ⊗ z ) ⊗ y ( z ⊗ x ) ⊗ y α − x,y,z x ⊗ β y,z β x ⊗ y,z α − x,z,y α − z,x,y β x,z ⊗ y (A.3)A symmetric monoidal category is a braided monoidal category where the braidingsatisfies the equation β y,x ◦ β x,y = 1 x ⊗ y for all objects x, y ∈ C . A commutative monoidalcategory is a symmetric monoidal category where the braiding is identity.For general (braided/symmetric) monoidal categories, we write C , D , or E . A.1.2 Monoidal, Braided, and Symmetric Functors
Definition A.2.
Let ( C , ⊗ C , I C , α C , λ C , ρ C ) and ( D , ⊗ D , I D , α D , λ D , ρ D ) be monoidal cate-gories. A lax monoidal functor from C to D consists of101 a functor F : C → D• a natural transformation with components φ x,y : F x ⊗ D F y → F ( x ⊗ C y ) called the laxator • a natural transformation with unique component φ : I D → F I C called the unit lax-ator such that the following diagrams commute.( F x ⊗ D F y ) ⊗ D F z F x ⊗ D ( F y ⊗ D F z ) F ( x ⊗ C y ) ⊗ D F z F x ⊗ D F ( y ⊗ C z ) F (( x ⊗ C y ) ⊗ C z ) F ( x ⊗ C ( y ⊗ C z )) α D F x,F y,F z φ x,y ⊗ D F z F x ⊗ D φ y,z φ x ⊗C y,z φ x,y ⊗C z F ( α C x,y,z ) (A.4) I D ⊗ D F x F xF I C ⊗ D F x F ( I C ⊗ C x ) λ D F x φ ⊗ D F x φ I C ,x F ( λ C x ) F x ⊗ D I D F xF x ⊗ D F I C F ( x ⊗ C I C ) ρ D x F x ⊗ D φ φ x,I C F ( ρ C x ) (A.5)We say that F is simply a monoidal functor when φ and φ are natural isomorphisms. Itis worth noting that there exists a notion of “oplax” monoidal functors, where the structuremap is reversed: φ x,y : F ( x ⊗ y ) → F x ⊗ F y . However, oplax monoidal functors do notappear in this thesis, so we spend no further time on them.A lax braided monoidal functor is a lax monoidal functor (
F, φ, φ ) : ( C , ⊗ C , I C ) → ( D , ⊗ D , I D ) where C and D are braided monoidal categories, with β C and β D being therespective braidings, such that the following diagram commutes. F x ⊗ D F y F y ⊗ D F xF ( x ⊗ C y ) F ( y ⊗ C x ) β D F x,F y φ x,y φ y,x F β C x,y (A.6)A (lax) braided monoidal functor between symmetric monoidal categories is called a (lax)symmetric monoidal functor with no further requirements. Lemma A.3.
Composition of lax monoidal functors is strictly associative.
We get categories
MonCat ℓ , MonCat , BrMonCat ℓ , BrMonCat , SymMonCat ℓ , and SymMonCat where the objects are monoidal categories, the functors are monoidal categories, the prefix Br (resp. Sym ) indicates the objects and morphisms are braided (resp. symmetric), and thesubscript ℓ indicated the morphisms are lax monoidal.102 .1.3 Monoidal Natural Transformations Definition A.4.
Let (
F, φ, φ ) and ( G, γ, γ ) be lax monoidal functors. A monoidalnatural transformation is a natural transformation θ : F ⇒ G such that the followingdiagrams commute. F x ⊗ D F y Gx ⊗ D GyF ( x ⊗ C y ) G ( x ⊗ C y ) θ x ⊗ D θ y φ x,y γ x,y θ x ⊗ Cy I D F I C GI C φ γ θ I C (A.7)There are no new laws which can be imposed on a monoidal natural transformation betweenbraided or symmetric monoidal functors. So we do not specialize this concept any further. A.2 Examples
Example A.5.
Let ( M, · , e ) be a monoid. If we can consider M as a discrete category,then it can be given a strict monoidal structure where the tensor is given by · and the unitis e . The functor Mon ֒ → MonCat which realizes a monoid as a discrete monoidal categoryis full and faithful. If we think of this as “forgetting discreteness”, then discreteness is aproperty.
Example A.6.
Given a monoidal category ( C , ⊗ , I ), we can define ⊗ rev : C × C → C by C × C CC × C ⊗ rev β Cat C , C ⊗ This defines an idempotent automorphism on
MonCat . Example A.7.
Given a monoidal category ( C , ⊗ , I ), the category C can be equipped witha monoidal structure given by ⊗ op : C op × C op → C op and the same unit object. This definesan idempotent automorphism on MonCat . Example A.8.
Any category C with finite products can be equipped with a symmetricmonoidal structure as follows. For every pair of objects c, d , choose some object satisfyingthe universal property of the product of c and d , call it c × d . Given a pair of morphisms f : a → b and g : c → d , the universal property gives a morphism f × g : a × b → c × d asfollows. a × ba bc × dc d π a π b ∃ ! f gπ c π d × : C × C → C . Consider a pair of morphisms( f , f ) : ( a , a ) → ( b , b ) and ( g , g ) : ( b , b ) → ( c , c ). Since ( g ◦ f ) × ( g ◦ g ) and( g × g ) ◦ ( f × f ) both make the following diagram commute, they must be equal. a × a a a b c × c b c c f f g g Identity maps are preserved because the identity map on a × b makes the diagram belowcommute. a × ba ba × ba b π a π b a b π a π b We define the unit object to be some chosen terminal object, call it 1. The associator,unitors, pentagon, hexagon, braiding, hexagon law, and symmetric law can all be derivedfrom the universal property of products. This gives C the structure of a symmetric monoidalcategory. This is called the cartesian monoidal structure , and ( C , × ,
1) is called a cartesian monoidal category . Example A.9.
Any category with finite coproducts can be equipped with a symmetricmonoidal structure by Example A.8 and Example A.7.
Example A.10. If C is monoidal and D is a category, the functor category C D can begiven a pointwise monoidal structure as follows. Define ⊗ pt : C D × C D → C D by ⊗ pt = ⊗ ( F × G ) ◦ ∆. The unit object 1 → C D is given by currying the composite D ! −→ I −→ C . Therest of the structures and the necessary properties all carry over from their counterpartsin C . Similarly, if C is braided or symmetric, then C D can be given a pointwise braided orsymmetric monoidal structure respectively. Example A.11.
Let C be a small monoidal category. Then the Day convolution tensorproduct [Day70] ⊗ Day : Set C op × Set C op → Set C op is the following left Kan extension. C op × C op Set C op ( X,Y ) ⊗ X ⊗ Day Y X ⊗ Day Y : c Z c ,c ∈ C C ( c ⊗ c , c ) × X ( c ) × Y ( c )Similarly, we can define the unit via left Kan extension.1 Set C op ∆1 I I
Day
Day convolution gives the functor category
Set C op a monoidal structure. Many nice proper-ties of this structure can be found in the literature, e.g. [Lor19]. However, these propertiesare not heavily used in this thesis, so we choose to leave them out. A.3 Monoid Objects
A monoidal structure is exactly what a category needs to have if we want to definemonoid objects in this category.
Definition A.12.
Let ( C , ⊗ , I ) be a monoidal category. A monoid object internal to C consist of • an object x ∈ C• a morphism µ : x ⊗ x → x • a morphism ε : I → x such that the following diagrams commute.( x ⊗ x ) ⊗ x x ⊗ ( x ⊗ x ) x ⊗ x x ⊗ xx α x,x,x µ ⊗ x x ⊗ µµ µ (A.8) I ⊗ x x ⊗ x x ⊗ Ix ε ⊗ xλ x µ x ⊗ ερ x (A.9)Alternatively, we can express these structures with string diagrams as follows. • multiplication 105 neutral elementsuch that = (A.10)= = (A.11)Let ( x, µ, ε ) and ( y, ν, δ ) be monoids in C . A morphism f : x → y is called a monoidhomomorphism if the following diagrams commute. x ⊗ x y ⊗ yx y f ⊗ fµ νf x y ε δf In strings, these equations are depicted as follows. f f = f f =Let Mon ( C , ⊗ ) denote the category of monoid objects in C and their homomorphisms. If C is Set with its cartesian monoidal structure, we simply denote the category of monoids by
Mon . Definition A.13.
Let ( C , ⊗ , I ) be a braided monoidal category. A commutative monoid in C is a monoid object in C where the following equation holds.= (A.12)Let CMon ( C , ⊗ ) denote the category of commutative monoid objects in C and their homo-morphisms. If C is Set with its cartesian monoidal structure, we simply denote the categoryof commutative monoids by
CMon . 106 .4 The Eckmann–Hilton Argument
Theorem A.14.
Let C be a braided monoidal category, and let x be an object equippedwith two distinct monoid structures ( x, µ, ε ) and ( x, ν, η ) such that µ and ν are related bythe following equation. µ ◦ ( ν ⊗ ν ) = ν ( µ ⊗ µ ) ◦ (1 ⊗ β ⊗
1) (A.13)
Then ε = η , µ = ν , and ( x, µ, ε ) is commutative. It is important to note that if C is Set or some other concrete category, and the operationsare instead denoted by ◦ and ⋆ , Equation (A.13) becomes ( a ◦ b ) ⋆ ( c ◦ d ) = ( a⋆c ) ◦ ( b⋆d ). Dueto this formulation, this relation as it appears in many contexts is called the middle-fourinterchange law . Proof.
We prove it using string diagrams, just for fun. Let the following string diagramcomponents represent ε , µ , η , and ν , respectively., , ,Then we can draw Equation (A.13) as follows.=First, we show that the units coincide.= = = = =Since they are equal, we denote the unit with a black circle in the remainder of the proof.Next, we show in one calculation that the two operations are equal and commutative.= = = = = = Corollary A.15.
We have the following equivalences of categories.
Mon ( Mon , × ) ∼ = Mon ( CMon , × ) ∼ = CMon ( Mon , × ) ∼ = CMon ( CMon , × ) ∼ = CMon .5 Characterizing (co)cartesian monoidal categories
In the previous section, we saw that a category with finite products can be equippedwith a canonical symmetric monoidal structure, and dually so can a category with finitecoproducts. In this section, we give conditions under which a symmetric monoidal categoryis monoidally equivalent to one given by a (co)cartesian structure [Fox76].
Example A.16.
Let C be a category with finite coproducts. By Example A.9, C can beequipped with a cocartesian monoidal structure, with tensor denoted by +, and the unit(which is an initial object) denoted by 0. Let x be any object in C . Universal property ofcoproducts gives a map ∇ : x + x → xx xx + xx i x i x ∇ x We draw string diagrams with respect to the cocartesian monoidal structure on C . Thenthe map ∇ x is depicted as follows.Also, the universal property of an initial object gives a map ! x : 0 → x , depicted as follows.We show that this gives x the structure of a commutative monoid. We begin by findinga formula for the left unitor of the cocartesian monoidal structure on C . Notice that theleft unitor makes the following diagram commute (by definition) x x + 0 x i x ∃ ! and thus does so uniquely. Compare this to the diagram x x x + 0 xx + xx i x ∇ x ∇ x ◦ (1 x +! x ) ◦ i x = 1 x andsimilarly ∇ x ◦ (! x + 1 x ) ◦ i ′ x = 1 x , which we draw as follows.= =To show ∇ x is associative, we want to show that the following equation holds.=We have three inclusion maps i , i , i : x → x + x + x , which are given in strings below., ,The universal property of coproducts says that if the composites of the morphisms on the leftand right side of the associativity equation above with any of the three inclusions is alwaysthe identity morphism on x , then those two morphisms must be equal. So we compute:= = = == == = = == =Thus we have that ∇ x is associative.Recall that the braiding in C is derived from the universal property in the following way. x xx + xx + x i i i i σ x xx x + x xx + xx i i i σ i ∇ has precisely the frame for the universal construction of ∇ x . Thus ∇ x ◦ σ = ∇ x , displayedas follows. = Proposition A.17.
A symmetric monoidal category is cocartesian if and only if each objecthas a natural commutative monoid structure.Proof.
Given an object x in a cocartesian monoidal category C , we constructed a commu-tative monoid structure on x in Example A.16.We have to show that the multiplication maps ∇ x : x + x → x form the components of a natural transformation ∇ : + ◦ ∆ ⇒ C . For a given morphism f : x → y , the naturality square is x + x y + yx y f + f ∇ x ∇ y f Recall that the map f + f is derived from the universal property of coproducts by thefollowing diagram. x xy x + x yy + y f i i fi ′ f + f i ′ ∇ y ◦ ( f + f ) = f ◦ ∇ x . x xy x + x yy + yy f i i fi ′ y f + f i ′ ′ y ∇ y x xx x + x xxy x i i x x f ∇ x x ff The frames of the above diagrams are identical, and they are equal to the frame whichproduces h f, f i , the copairing of f with itself. So by universal property, they are equal.The naturality square of the units collapses into the triangle below.0 x y ! x ! y f which commutes by initiality of 0.We have shown one direction: that if C is cocartesian monoidal, then each object hasa natural commutative monoid structure. Now we must show the converse. Assume that( C , ⊗ , I ) is a symmetric monoidal category such that each object has a natural commutativemonoid structure m : ⊗ ◦ ∆ ⇒ C and ε : ∆ I ⇒ C . We represent the components of thesestructures with string diagrams as follows. ,The unit object is initial by naturality of ε . I Ix y ε x I ε y f Now we must show that the monoidal structure on C is cocartesian, i.e. that the unitobject is initial and tensor is coproduct. To show that x ⊗ y is actually the coproductof x and y , we first must provide inclusions, and then show that this cone satisfies theappropriate universal property. We propose that the inclusion maps i x : x → x ⊗ y and i y : y → x ⊗ y are given in string diagrams as follows.,111et q be an object of C , and f : x → q and g : y → q be maps in C . Define the map h : x ⊗ y → q to be the following composite. f g Then we show the diagram x yx ⊗ yq i x f i y gh commutes by the following calculations. f g = f = ff g = g = g Let k : x ⊗ y → q be a map which makes that diagram commute. Then h = f g = k k = k = k Thus h is the unique such map. This demonstrates x ⊗ y as the coproduct of x and y .There is a dual statement which characterizes cartesian monoidal categories, but in orderto state it, we must first define comonoid . Definition A.18. A comonoid object in a monoidal category C is monoid in C op . Equiv-alently, a comonoid is an object x ∈ C equipped with a comultiplication map µ : x → x ⊗ x and a counit map ε : x → I , which we express as,112atisfying the following equations. == =Let Comon ( C , ⊗ ) denote the category of comonoid objects in C and their homomorphisms. Definition A.19. A cocommutative comonoid is a comonoid for which the followingequation holds. =Let CoComon ( C , ⊗ ) denote the category of cocommutative comonoids in C and their homo-morphisms. Proposition A.20.
A symmetric monoidal category is cartesian if and only if each objecthas a natural cocommutative comonoid structure.Proof.
This is dual to Proposition A.17.
Corollary A.21.
Let ( C , ⊗ , I ) be a symmetric monoidal category. Then CMon ( C , ⊗ ) hasa cocartesian monoidal structure given by ⊗ , and CoComon ( C , ⊗ ) has a cartesian monoidalstructure given by ⊗ . ppendix B Monoidal 2-Categories andPseudomonoids
There are many sources for the basic theory of 2-categories and bicategories [B´en67,KS74, Lac10, JY21]. Below we sketch some basic definitions and constructions regardingmonoidal 2-categories, necessary for what follows; relevant references where explicit axiomscan be found are [Car95, GPS95, DS97, McC00].
B.1 Monoidal 2-Categories A monoidal 2-category K is a 2-category equipped with a pseudofunctor ⊗ : K ×K → K and a unit object I : 1 → K which are associative and unital up to coherentequivalence. A lax monoidal pseudofunctor F : K → L between monoidal 2-categoriesis a pseudofunctor equipped with pseudonatural transformations
K × K L × LK L
F×F⊗ K ⊗ L F µ K L I K I L F µ (B.1)with components µ a,b : F a ⊗ F b → F ( a ⊗ b ), µ : I → F I , and invertible modifications L L L L K K L K L LK K K K⇓ µ × ⊗ L × ⇓ µ ⊗ L ⇓ × µ ⊗ L × ×⊗ L ∼ = ⊗ L F×F×F⊗ K × ×⊗ K F×F⊗ K ω ⇛ F×F×F ×⊗ K ⇓ µ ⊗ L ∼ = ⊗ K F F×F⊗ K F (B.2)114 L × L LK × K K × I F× I ∼ = F∼ = ⊗ L × µ ⊗ K F×F ⇓ µ F ζ ⇛ K L FF ⇓ ξ ⇛ K L × L LK × K K I × I ×F ∼ = F∼ = ⊗ L µ × ⊗ K F×F ⇓ µ F subject to coherence conditions which can be found in Definition 2 in [DS97]. A monoidalpseudonatural transformation τ : F ⇒ G between two lax monoidal pseudofunctors( F , µ, µ ) and ( G , ν, ν ) is a pseudonatural transformation equipped with two invertiblemodifications K × K L × L K × K L × L u ⇛ K L K L
F×FG×G⊗ ⇓ ν ⇓ τ × τ ⊗ F×F⊗ ⇓ µ ⊗G FG ⇓ τ (B.3) L L u ⇛ K K⇓ ν I L K ⇓ µ I L I K G F G ⇓ τ that consist of natural isomorphisms with components u a,b : ν a,b ◦ ( τ a ⊗ τ b ) ∼ −→ τ a ⊗ b ◦ µ a,b , u : ν ∼ −→ τ I ◦ µ (B.4)satisfying coherence conditions which can be found in [GPS95, Section 3.3].The above notions of course generalize those of an ordinary monoidal category, laxmonoidal functor and monoidal natural transformation. However, in our higher dimensionalsetting, there is now room for a structure not present for monoidal 1-categories.A monoidal modification between two monoidal pseudonatural transformations ( τ, u, u )and ( σ, v, v ) is a modification K L FG m ⇛ τ σ which consists of pseudonatural transformations m a : τ a ⇒ σ a compatible with the monoidal115tructures, in the sense that G a ⊗ G b G a ⊗ G b F a ⊗ F b G ( x ⊗ y ) F a ⊗ F b G ( a ⊗ b ) F ( a ⊗ b ) F ( a ⊗ b ) ν a,b ν a,b σ a ⊗ σ b µ a,b ⇓ v a,b = µ a,b σ a ⊗ σ b ⇓ u a,b τ a ⊗ τ b ⇓ m x ⊗ m y τ a ⊗ b σ a ⊗ b ⇓ m a ⊗ b τ a ⊗ b (B.5) I G ( I ) I G ( I ) F ( I ) F ( I ) µ ν ⇓ v = ⇓ u µ ν τ I σ I ⇓ m I τ I For any monoidal 2-categories K , L there are 2-categories Mon2Cat ps ( K , L ) denotedby WMonHom ( K , L ) in [DS97] for bicategories. If we take lax monoidal 2-functors andmonoidal 2-transformations, the corresponding sub-2-category is denoted by Mon2Cat ( K , L ). B.2 Pseudomonoids A pseudomonoid in a monoidal 2-category ( K , ⊗ , I ) is an object a equipped withmultiplication m : a ⊗ a → a , unit j : I → a , and invertible 2-cells a ⊗ a ⊗ a a ⊗ a a ⊗ I a ⊗ a I ⊗ aa ⊗ a a a ⊗ mm ⊗ α ∼ = mm ⊗ j ∼ m λ ∼ = ρ ∼ = j ⊗ ∼ m (B.6)expressing associativity and unitality up to isomorphism, that satisfy appropriate coherenceconditions. A lax morphism between pseudomonoids a, b is a 1-cell f : a → b equippedwith 2-cells a ⊗ a b ⊗ ba b m f ⊗ f mfφ Ia b j jfφ (B.7)such that the following conditions hold: b ⊗ b ⊗ b b ⊗ b ⊗ aa ⊗ a ⊗ a a ⊗ a ba ⊗ a a m ⊗ ⇓ φ ⊗ f mm ⊗ ⊗ mf ⊗ f ⊗ f mf ⊗ f ⇓ φα ∼ = m f = b ⊗ b ⊗ b b ⊗ ba ⊗ a ⊗ a b ⊗ b ba ⊗ a b ⊗ b m ⊗ ⊗ m ⇓ f ⊗ φ mα ∼ = f ⊗ f ⊗ f ⊗ m m ⇓ φf ⊗ fm f (B.8)116 ∼ = a ⊗ I b ⊗ b ba ⊗ a a ⊗ j f ⊗ j a λ ∼ = f f ⊗ λ ∼ = m f ⊗ φ mf ⊗ f ⇓ φ f = a b ff ⇓ f = a ∼ = I ⊗ a b ⊗ b ba ⊗ a a j ⊗ j ⊗ a ρ ∼ = fρ ⊗ f ∼ = mφ ⊗ f mf ⊗ f ⇓ φ f If ( f, φ, φ ) and ( g, ψ, ψ ) are two lax morphisms between pseudomonoids a and b , a between them σ : f ⇒ g in K which is compatible with multiplications and units, inthe sense that b ⊗ ba ⊗ a ba mf ⊗ f g ⊗ gm ⇓ σ ⊗ σ ψ ⇓ g = b ⊗ ba ⊗ a ba mf ⊗ fm φ ⇓ ⇓ σf g (B.9) I ba jjφ ⇓ f g ⇓ σ = I ba j j ⇓ ψ g We obtain a 2-category
PsMon lax ( K ) for any monoidal 2-category K , which is sometimesdenoted by Mon ( K ) [CLS10]. By changing the direction of the 2-cells in Equation (B.7) andthe rest of the axioms appropriately, or asking for them to be invertible, we have 2-categories PsMon opl ( K ) and PsMon ( K ) of pseudomonoids with oplax or (strong) morphisms be-tween them. Example B.1.
The prototypical example is that of the monoidal 2-category K = ( Cat , × , )of categories, functors, and natural transformations with the cartesian product of categoriesand the unit category with a unique object and arrow. A pseudomonoid in ( Cat , × , ) is amonoidal category, a lax (resp. oplax, strong) morphism between two of these is precisely alax (resp. oplax, strong) monoidal functor, and a 2-cell is a monoidal natural transformation.Therefore we obtain the well-known 2-categories MonCat lax , MonCat opl and
MonCat .There is an evident similarity between the structures defined above, e.g. Equation (B.1)and Equation (B.7), or Equation (B.3) and Equation (B.9). This is due to the fact thatmonoidal 2-categories, lax monoidal pseudofunctors and monoidal pseudonatural transfor-mations are themselves appropriate pseudomonoid-related notions in a higher level; we donot get into such details, as they are not pertinent to the present work.For our purposes, we are interested in a different observation: any pseudomonoid a in amonoidal 2-category K can in fact be expressed as a lax monoidal normal pseudofunctor A : → K with A ( ∗ ) = a , namely one where A (1 ∗ ) is equal to 1 a . Moreover, a monoidal117seudonatural transformation τ : A ⇒ B : → K bijectively corresponds to a strong mor-phism between the pseudomonoids a and b , and similarly for monoidal modifications and2-cells. Since every pseudofunctor is equivalent to a normal one, the 2-category of pseu-domonoids PsMon ( K ) can be equivalently viewed as Mon2Cat ps ( , K ), the 2-category of laxmonoidal pseudofunctors → K , monoidal pseudonatural transformations and monoidalmodifications.As was already shown in [DS97, Prop. 5], any lax monoidal 2-functor F : K → L takespseudomonoids to pseudomonoids, and in fact [McC00] there is a functor
PsMon ( F ) thatcommutes with the respective forgetful functors PsMon ( K ) PsMon ( L ) K L . PsMon ( F ) F Based on the above, and since every pseudofunctor from into a 2-category trivially pre-serves composition on the nose and every pseudonatural transformation is really 2-natural,we can define a hom-2-functor that clarifies these assignments. Proposition B.2.
There is a 2-functor
PsMon ( − ) ≃ Mon2Cat ps ( , − ) : Mon2Cat → (B.10) which maps a monoidal 2-category to its 2-category of pseudomonoids, strong morphismsand 2-cells between them. The theory in [DS97, McC00] extends the above definitions to the case of braided and symmetric pseudomonoids in braided and symmetric monoidal 2-categories . Briefly recallthat a braiding for ( K , ⊗ , I ) is a pseudonatural equivalence with components β a,b : a ⊗ b → b ⊗ a and invertible modifications, whereas a syllepsis is an invertible modification a ⊗ b −→ a ⊗ b ⇛ a ⊗ b β a,b −−→ b ⊗ a β b,a −−→ a ⊗ b which is called symmetry if it satisfies extra axioms. With the appropriate notions of braided and symmetric lax monoidal pseudofunctors and monoidal pseudonatural trans-formations (and usual monoidal modifications), we have 3-categories BrMon2Cat ps and SymMon2Cat ps . Indicatively, a lax monoidal pseudofunctor comes equipped an invertiblemodification with components F a ⊗ F b F b ⊗ F a F b × F a F ( b ⊗ a ) µ a,b β F a, F b ⇓ v a,b F ( β a,b ) µ b,a (B.11)As earlier, there exist 2-categories of braided and symmetric pseudomonoids with strongmorphisms between them, expressed as BrPsMon ( K ) = BrMon2Cat (ps) ( , K )118nd SymPsMon ( K ) = SymMon2Cat (ps) ( , K ) . Proposition B.3.
There are 2-functors
BrPsMon : BrMon2Cat → , SymPsMon : SymMon2Cat → which map a braided or symmetric monoidal 2-category to its 2-category of braided or sym-metric pseudomonoids. Finally, recall the notion of a monoidal 2-equivalence arising as the equivalence internalto the 2-category
Mon2Cat . Definition B.4. A monoidal 2-equivalence is a 2-equivalence F : K ≃ L : G where both2-functors are lax monoidal, and the 2-natural isomorphisms 1 K ∼ = F G , GF ∼ = 1 L aremonoidal. Similarly for braided and symmetric monoidal 2-equivalences.As is the case for any 2-functor between 2-categories, PsMon as well as
BrPsMon and
SymPsMon map equivalences to equivalences.
Proposition B.5.
Any monoidal 2-equivalence
K ≃ L induces a 2-equivalence between therespective 2-categories of pseudomonoids
PsMon ( K ) ≃ PsMon ( L ) . Similarly any braided orsymmetric monoidal 2-equivalence induces BrPsMon ( K ) ≃ BrPsMon ( L ) or SymPsMon ( K ) ≃ SymPsMon ( L ) . ppendix C Fibrations and Indexed Categories
We recall some basic facts and constructions from the theory of fibrations and indexedcategories, as well as the equivalence between them via the Grothendieck construction.Several indicative references for the general theory are [Gra66, B´en85, Her94, Bor94, Jac99,Joh02, Vis05, Str20, JY21].
C.1 Fibrations
Consider a functor P : A → X . A morphism φ : a → b in A over a morphism f = P ( φ ) : x → y in X is called cartesian if and only if, for all g : x ′ → x in X and θ : a ′ → b in A with P θ = f ◦ g , there exists a unique arrow ψ : a ′ → a such that P ψ = g and θ = φ ◦ ψ : a ′ a b in A x ′ x y in X θ ∃ ! ψ φf ◦ g = P θg f = P φ (C.1)For x ∈ Ob X , the fibre of P over x written A x , is the subcategory of A which consists ofobjects a such that P ( a ) = x and morphisms φ with P ( φ ) = 1 x , called vertical morphisms.The functor P : A → X is called a fibration if and only if, for all f : x → y in X and b ∈ A Y , there is a cartesian morphism φ with codomain b above f ; it is called a cartesianlifting of f to b . The category X is then called the base of the fibration, and A its totalcategory .Dually, the functor U : C → X is an opfibration if U op is a fibration, i.e. for every c ∈ C x and h : x → y in X , there is a cocartesian morphism with domain c above h , the120 ocartesian lifting of h to c with the dual universal property: d ′ c d in C y ′ x y in X γβ ∃ ! δk ◦ h = Uγh = Uβ k A bifibration is a functor which is both a fibration and opfibration.If P : A → X is a fibration, assuming the axiom of choice we may select a cartesianarrow over each f : x → y in X and b ∈ A y , denoted by Cart ( f, b ) : f ∗ ( b ) → b . Such a choiceof cartesian liftings is called a cleavage for P , which is then called a cloven fibration ; anyfibration is henceforth assumed to be cloven. Dually, if U is an opfibration, for any c ∈ C x and h : x → y in X we can choose a cocartesian lifting of h to c , Cocart ( h, c ) : c → h ! ( c ). Thechoice of (co)cartesian liftings in an (op)fibration induces a so-called reindexing functor between the fibre categories f ∗ : A y → A x and h ! : C x → C y (C.2)respectively, for each morphism f : x → y and h : x → y in the base category. It can beverified by the (co)cartesian universal property that 1 A x ∼ = (1 x ) ∗ and that for composablemorphism in the base category, g ∗ ◦ f ∗ ∼ = ( g ◦ f ) ∗ , as well as (1 x ) ! ∼ = 1 C x and ( k ◦ h ) ! ∼ = k ! ◦ h ! .If these isomorphisms are equalities, we have the notion of a split (op)fibration.A fibred 1-cell ( H, F ) : P → Q between fibrations P : A → X and Q : B → Y is givenby a commutative square of functors and categories
A BX Y
HP QF (C.3)where the top H preserves cartesian liftings, meaning that if φ is P -cartesian, then Hφ is Q -cartesian. In particular, when P and Q are fibrations over the same base category, wemay consider fibred 1-cells of the form ( H, X ) displayed as A BX
HP Q (C.4)121nd H is then called a fibred functor . Dually, we have the notion of an opfibred 1-cell and opfibred functor . Notice that any such (op)fibred 1-cell induces functors between thefibres, by commutativity of Equation (C.3): H x : A x −→ B F x (C.5)A fibred 2-cell between fibred 1-cells (
H, F ) and (
K, G ) is a pair of natural transfor-mations ( β : H ⇒ K, α : F ⇒ G ) with β above α , i.e. Q ( β a ) = α P a for all a ∈ A , displayedas A BX Y H ⇓ β KP QF ⇓ α G (C.6)A fibred natural transformation is of the form ( β, X ) : ( H, X ) ⇒ ( K, X ) A BX H ⇓ β KP Q (C.7)Dually, we have the notion of an opfibred 2-cell and opfibred natural transformation between opfibred 1-cells and functors respectively.We thus obtain a 2-category
Fib of fibrations over arbitrary base categories, fibred 1-cells and fibred 2-cells. There is also a 2-category
Fib ( X ) of fibrations over a fixed basecategory X , fibred functors and fibred natural transformations. Dually, we have the 2-categories OpFib and
OpFib ( X ). Moreover, we also have 2-categories Fib sp and OpFib sp ofsplit (op)fibrations, and (op)fibred 1-cells that preserve the cartesian liftings ‘on the nose’.Notice that Fib and
OpFib are both sub-2-categories of
Cat = [ , Cat ], the arrow 2-category of
Cat . Similarly,
Fib ( X ) and OpFib ( X ) are sub-2-categories of Cat / X , the slice2-category of functors into X . Due to that, both these (1-)categories form fibrations them-selves. Explicitly, the functor cod : Fib → Cat which maps a fibration to its base is afibration, with fibres
Fib ( X ) and cartesian liftings pullbacks along fibrations. In fact, it isa [Her99, Buc14]. C.2 Indexed Categories
We now turn to the world of indexed categories. Given an ordinary category X , an X - indexed category is a pseudofunctor M : X op → Cat X is viewed as a 2-category with trivial 2-cells; it comes with natural isomorphisms δ g,f : ( M g ) ◦ ( M f ) ∼ −→ M ( g ◦ f ) and γ x : 1 M x ∼ −→ M (1 x ) for every x ∈ X and composablemorphisms f and g , satisfying coherence axioms. Dually, an X - opindexed category is an X op -indexed category, i.e. a pseudofunctor X →
Cat . If an (op)indexed category strictlypreserves composition, i.e. is a (2-)functor, then it is called strict .An indexed 1-cell ( F, τ ) :
M → N between indexed categories M : X op → Cat and N : Y op → Cat consists of an ordinary functor F : X → Y along with a pseudonaturaltransformation τ : M ⇒ N ◦ F op X op Cat Y op M F op ⇓ τ N (C.8)with components functors τ x : M x → N F x , equipped with coherent natural isomorphisms τ f : ( N F f ) ◦ τ x ∼ −→ τ y ◦ ( M f ) for any f : x → y in X . For indexed categories with the samebase, we may consider indexed 1-cells of the form (1 X , τ ) X op Cat MN ⇓ τ (C.9)which are called indexed functors . Dually, we have the notion of an opindexed 1-cell and opindexed functor .An indexed 2-cell ( α, m ) between indexed 1-cells ( F, τ ) and (
G, σ ), pictured as X op Cat Y op M F op G op α op ⇐ N στ m ⇛ consists of an ordinary natural transformation α : F ⇒ G and a modification m X op Cat X op Cat Y op Y op M F op ⇓ τ m ⇛ M G op F op ⇓ σ ⇓ α op N N (C.10)given by a family of natural transformations m x : τ x ⇒ N α x ◦ σ x . Notice that takingopposites is a 2-functor ( − ) op : Cat → Cat co , on which the above diagrams rely. An indexed atural transformation between two indexed functors is an indexed 2-cell of the form(1 X , m ). Dually, we have the notion of an opindexed 2-cell and opindexed naturaltransformation between opindexed 1-cells and functors respectively.Notice that an indexed 2-cell ( α, m ) is invertible if and only if both α is a naturalisomorphism and the modification m is invertible, due to the way vertical composition isformed.We obtain a 2-category ICat of indexed categories over arbitrary bases, indexed 1-cellsand indexed 2-cells. In particular, there is a 2-category
ICat ( X ) of indexed categories withfixed domain X , indexed functors and indexed natural transformations, which coincideswith the functor 2-category ps ( X op , Cat ).Dually, we have the 2-categories
OpICat and
OpICat ( X ) = ps ( X , Cat ). Notice thatdue to the absence of opposites in the world of opindexed categories, opindexed 2-cells havea different form than Equation (C.10), namely X Cat X Cat
Y Y M FG ⇓ τ ⇓ α m ⇛ M G ⇓ σ N N
Moreover, we have 2-categories of strict (op)indexed categories and (op)indexed 1-cells thatconsist of strict natural transformations τ Equation (C.8), i.e.
ICat ( X ) = [ X op , Cat ] and
OpICat sp ( X ) = [ X , Cat ] the usual functor 2-categories.Notice that these categories also form fibrations over
Cat , this time essentially usingthe family fibration also seen in Section 5.7.2. The functor
ICat → Cat sends an indexedcategory to its domain and an indexed 1-cell to its first component. It is a split fibration,with fibres
ICat ( X ) and cartesian liftings pre-composition with functors. In fact, it is alsoa 2-fibration as explained in [Buc14, 2.3.2]. C.3 The Grothendieck Construction
In the first volume of the
S´eminaire de G´eom´etrie Alg´ebrique du Bois Marie [Gro71],Grothendieck introduced a construction for a fibration P M : R M → X from a given in-dexed category M : X op → Cat as follows. If δ and γ are the structure pseudonaturaltransformations of the pseudofunctor M , the total category R M has • objects ( x, a ) with x ∈ X and a ∈ M x ; • morphisms ( f, k ) : ( x, a ) → ( y, b ) with f : x → y a morphism in X , and k : a → ( M f )( b ) a morphism in M x ; • composition ( g, ℓ ) ◦ ( f, k ) : ( x, a ) → ( y, b ) → ( z, c ) is given by g ◦ f : a → b → c in X and a k −→ ( M f )( b ) ( M g )( ℓ ) −−−−−→ ( M g ◦ M f )( c ) ( δ f,g ) c −−−−→ M ( g ◦ f )( c ) in M x ; (C.11)124 unit 1 ( x,a ) : ( x, a ) → ( x, a ) is given by 1 x : x → x in X and a = 1 M x a ( γ x ) a −−−→ ( M x )( a ) in M x. The fibration P M : R M → X is given by ( x, a ) x on objects and ( f, k ) f onmorphisms, and the cartesian lifting of any ( y, b ) in R M along f : x → y in X is precisely( f, ( M f ) b ). Its fibres are precisely M x and the reindexing functors between them are M f .In the other direction, given a (cloven) fibration P : A → X , we can define an indexedcategory M P : X op → Cat that sends each object x of X to its fibre category A x , andeach morphism f : x → y to the corresponding reindexing functor f ∗ : A y → A x as inEquation (C.2). The isomorphisms of cartesian liftings f ∗ ◦ g ∗ ∼ = ( g ◦ f ) ∗ and 1 A x ∼ = 1 ∗ x render this assignment pseudofunctorial.Details of the above, as well as the correspondence between 1-cells and 2-cells, can befound in the provided references. Briefly, given a pseudonatural transformation τ : M →N ◦ F op Equation (C.8) with components τ x : M x → N F x , define a functor P τ : R M → R N mapping ( x ∈ X , a ∈ M x ) to the pair ( F x ∈ Y , τ x ( a ) ∈ N F x ) and accordingly for arrows.This makes the square R M R NX Y P τ P M P N F (C.12)commute, and moreover P τ preserves cartesian liftings due to pseudonaturality of τ . More-over, given an indexed 2-cell ( α, m ) : ( F, τ ) ⇒ ( G, σ ) as in Equation (C.10), we can form afibred 2-cell R M R NX Y P τ P σ ⇓ P m P M P N FG ⇓ α (C.13)where α : F ⇒ G is piece of the given structure, whereas P m is given by components( P m ) ( x,a ) : P τ ( x, a ) = ( F x, τ x a ) → P σ ( x, a ) = ( Gx, σ x a ) in R N explicitly formed by α x : F x → Gx in Y and ( m x ) a : τ x a → ( N α x ) σ x a in N F x .The following theorem summarizes these standard results.
Theorem C.1.
1. Every fibration P : A → X gives rise to a pseudofunctor M P : X op → Cat .2. Every indexed category M : X op → Cat gives rise to a fibration P M : R M → X . . The above correspondences yield an equivalence of 2-categories ICat ( X ) ≃ Fib ( X ) so that M P M ≃ M and P M P ≃ P .4. The above 2-equivalence extends to one between 2-categories of arbitrary-base fibrationsand arbitrary-domain indexed categories ICat ≃ Fib (C.14)If we combine the above with the fact that the 2-categories
Fib and
ICat are fibredover
Cat with fibres
Fib ( X ) and ICat ( X ) respectively, we obtain the following Cat -fibredequivalence
ICat FibCat ≃ (C.15)There is an analogous story for opindexed categories and opfibrations that results into a2-equivalences OpICat ( X ) ≃ OpFib ( X ) and OpICat ≃ OpFib , as well as for the split versionsof (op)indexed and (op)fibred categories.
C.4 Examples
C.4.1 Fundamental Fibration
Let 2 denote the category with two objects, and one non-identity morphism ⋆ → • . Fora category X , the functor category X then consists of the arrows of X as objects, andcommuting squares between them as the morphisms.For any category X , the codomain or fundamental opfibration is the usual functor fromits arrow category cod : X −→ X mapping every morphism to its codomain and every commutative square to its right-handside leg. It uniquely corresponds to the strict opindexed category, i.e. functor X Cat x X /xy X /y f f ! (C.16)that maps an object to the slice category over it and a morphism to the post-compositionfunctor f ! = f ◦ − induced by it. 126 .4.2 Graphs Consider (directed, multi) graphs, i.e. presheaves on the category G = V E ts .For a presheaf g : G op → Set , the set g V is the set of vertices of the graph, the set g E isthe set of edges of the graph, and the maps g s , g t : g E → g V assign to an edge its startingand terminating vertex respectively. Let Grph denoted the category of graphs,
Set G op .Sometimes it is helpful to think of a graph as a single map of the form ( g s , g T ) : g E → g V × g V .When convenient, we will abuse notation by simply referring to this map as g : g E → g V .Consider the inclusion of a terminal category 1 into G which selects the object V . Thisinduces a functor V : Grph → Set by precomposing, which sends a graph g to its vertex set g V . As we show below, this functor is in fact a bifibration. The idea here is that if youhave a function f : x → y , you can pull a graph on y back to a graph on x , and you canalso push a graph on x forward to a graph on y . Proposition C.2.
A morphism φ : g → h in Grph is V -cartesian if and only if the square g E h E g V h V ( g s ,g t ) φ E ( h s ,h t ) φ V is a pullback in Set .Proof.
A simple manipulation shows that the universal property of φ forming a pullbacksquare is the same as the universal property for it to be V -cartesian. Proposition C.3.
The functor V : Grph → Set is a fibration.Proof.
Let f : x → y be a function, and g ∈ Grph with g V = y . Then we can take a pullbackof the following diagram. x f −→ y = h V ( h s ,h t ) ←−−−− h E By Proposition C.2, this map is a cartesian lift of f .By the Grothendieck correspondence, there is a indexed category Set op → Cat .Thispseudofunctor assigns to a set X the category Grph X of graphs which have vertex set X ,and graph morphisms which fix the vertices. Given a function f : X → Y , this pseudofunctorgives a functor f ∗ : Grph Y → Grph X which sends a graph g over Y to the pullback, as in theproof of Proposition C.3. Since there is also an opindexed category with the same fibres,we denote this by Grph ∗ , referring to the action on morphisms.To show that V is also an opfibration, it is actually easier to construct an explicitsplitting. We can derive a characterization of the cocartesian maps from there. Proposition C.4.
The functor V : Grph → Set is an opfibration. roof.
Let g ∈ Grph , y ∈ Set , and f : g V → y a function. Then we can obtain a graph withvertex set y by taking the following composite. g E g −→ g V = g V f −→ y We claim the induced map of graphs displayed below is in fact a cocartesian lift of f . g E g E g V y g f ◦ gf Let h be a graph, φ : g → h a map of graphs, and φ : y → h V . h E h V g E g E g V y hφ V φ E g f ◦ gf The only map which may take the place of the dashed arrow is φ E . Corollary C.5.
A morphism φ : g → h in Grph is V -cocartesian if and only if it is bijectiveon edges. By the Grothendieck correspondence, there is a corresponding opindexed category
Grph ∗ : Set → Cat , again referring to the action on morphisms. This must have the same fibres
Grph X asthe indexed category Grph ∗ above. Given a function f : X → Y , this pseudofunctor givesa functor f ∗ : Grph X → Grph Y which sends a graph g over X to the composite, as in theproof of Corollary C.5. Corollary C.6.
The functor V : Grph → Set is a bifibration.
C.4.3 Ring Modules
For a ring R , denote by Mod R the category of R -modules and their homomorphisms.Given a ring homomorphism f : R → S , and an S -module N , we can give the underlyingabelian group of N the structure of an R -module, denoted f ∗ N , by the formula r.x := f ( r ) .x where r ∈ R and x ∈ N . This pullback construction is functorial: f ∗ : Mod S → Mod R f ◦ g ) ∗ = g ∗ f ∗ Indeed, the above defines a functor
Mod − : Ring op → Cat , a (strict) indexed category. Note,one could choose to be persnickety about size here, but we do not. We can then apply theGrothendieck construction, resulting in a category
Mod := R Mod − where • an object is a pair ( R ∈ Ring , M ∈ Mod R ) • a morphism is a pair ( f, φ ) where f : R → S is a ring homomorphism, and φ : M → f ∗ N is an S -module homomorphism.The category Mod admits a fibration
Mod → Ring which forgets the module in a ring-modulepair. 129 ppendix D
Species and Operads
D.1 Combinatorial Species
Combinatorial species were introduced by Joyal [Joy81]. A standard reference for thecombinatorial perspective is [BLL98]. In the previous section, we noted that the category
Set can be given both a cartesian and cocartesian monoidal structure. Moreover, thesesatisfy a distributive law reminiscent of rings. A × ( B + C ) ∼ = A × B + A × C Consider the subcategory
FinBij consisting of finite sets and bijections. This subcategoryis closed under both finite sums and products, and thus inherits both monoidal structures.However,
FinBij lacks the maps that would be the projections and inclusions necessary forthese structures to be cartesian or cocartesian themselves. By abuse of notation, we willcontinue to denote them by + and × . Definition D.1. A combinatorial species or simply species is a functor F : FinBij → Set .The category of species is the functor category
Set
FinBij .There are several operations which have been defined on species. These operations makeup the building blocks of a calculus for counting families of combinatorial gadgets.
Definition D.2.
Being a presheaf category,
Set
FinBij has colimits given pointwise, thusgiving it a cocartesian structure. We refer to this operation simply as addition . Onobjects, this operation is given by ( F + G )( U ) = F ( U ) + G ( U ). Definition D.3.
If we apply Day convolution as in Example A.11 to the + monoidalstructure on
FinBij , we get the operation which we refer to as multiplication of species.On objects, this operation is given by the following formula.( F · G )( U ) = X V ⊆ U F ( V ) × G ( U \ V ) Definition D.4.
Being a presheaf category,
Set
FinBij has products which are given pointwise,thus giving a cartesian monoidal structure. This is called the
Hadamard product . Onobjects, it is given by ( F × G )( U ) = F ( U ) × G ( U ). This tends to be less useful thanmultiplication, but it certainly has its purposes.130 efinition D.5. We define the
Dirichlet product on species to be the Day convolution(as in Example A.11) of the × monoidal structure on FinBij .To define differentiation of species, we will need to make use of the shift operator ,denoted +1, on
FinBij , which is defined as the composite
FinBij FinBijFinBij × FinBij × FinBij +1 ∼ FinBij × ∆1 + where the map ∆1 : 1 → FinBij is the monoidal unit with respect to × . Definition D.6.
The differentiation operator on species is given by D = +1 ∗ : Set
FinBij → Set
FinBij . In other words, for a given species F , the derivative of F is given by F ′ = F ◦ +1,or F ′ ( U ) = F ( U + 1) on objects. The motivation for calling this operation differentiationis that it actually corresponds to taking the formal derivative of its generation series. Definition D.7.
The composition product or substitution product is given by thefollowing formula. ( F ◦ G )( U ) = X π partition of U F ( π ) × Y p ∈ π G ( p ) ! The species which acts as unit for this product is the singleton indicator functor, i.e. I ◦ ( U )is a singleton is U is, and is empty otherwise. This monoidal structure is not symmetric. D.2 Operads
An operad is a generalization of category which incorporates the notion of an arrowhaving multiple inputs. A category is an arrow-like compositional system, consisting of acollection of directed arrows, a collection of labels for the endpoints called objects, and arule for turning a path of such arrows into a single arrow which is associative and unital. Anoperad is also a compositional system, but now tree-like . An operad consists of a collectionof directed “short” trees . . .a collection of labels for the endpoints called objects, and a rule for turning a big tree ofshort trees 131nto a single short treewhich is associative and unital. There are several good references for the theory of operads[MSS02, Yau16, M´en15, BD98, Kel05, Lei04]. Here, we follow the treatment given by Yauin [Yau20].
D.2.1 Definition of Operad
Let C be a non-empty set, whose elements we call colors . Recall that we denote thefree symmetric monoidal category on C by S ( C ). Below, we define operads to be monoidsin the presheaf category Set S ( C ) × C with respect to a certain monoidal structure. First wemust define this monoidal structure, which is somewhat involved.For this section, we denote objects of S ( C ) by either c or ( c , . . . , c n ) depending oncontext. We denote the monoidal structure on S ( C ) by +. For an object X ∈ Set S ( C ) op × C ,we denote the set assigned to ( c, d ) ∈ S ( C ) op × C by X ( c ; d ). Definition D.8.
Let
X, Y ∈ Set S ( C ) × C . For each c ∈ S ( C ), define Y c by the followingcoend formula. Y c ( b ) = Z { a j }∈ Q mj =1 S ( C ) op S ( C ) op ( a + · · · + a m , b ) × m Y j =1 Y ( a j ; c j ) The C -colored circle product of X and Y is given by X ◦ Y ( b ; d ) = Z c ∈ S ( C ) X ( c ; d ) ⊗ Y c ( b )Define the unit object I as follows. I ( c ; d ) = ( c = d ∅ otherwiseThis reduces to the composition monoidal product of species in the case where C ∼ = 1. Proposition D.9. ( Set S ( C ) × C , ◦ , I ) is a monoidal category. Definition D.10.
Let C be a set. Define the category of operads by Opd C = Mon ( Set S ( C ) × C , ◦ , I ) . We refer to an object of
Opd C as a C -colored operad , and a morphism as a color-fixing C -operad functor . Let f : C → D be a function. Let f ( c ) denote ( f ( c ) , . . . , f ( c n )) for c ∈ S ( C ). For a D -operad P , we can pullback along f to get a C -operad given by f ∗ P ( c ; d ) = P ( f ( c ); f ( d )). For a color-fixing D -operad functor φ : P → Q , we get a color-fixing C -operad functor f ∗ φ : f ∗ P → f ∗ Q which sends an operation θ ∈ f ∗ P ( c ; d ) = P ( f ( c ); d ) to φθ ∈ Q ( f ( c ); d ). These assignments give a functor f ∗ : Opd D → Opd D , and we get anindexed category Opd − : Set op → Cat . Define
Opd = R Opd − . We refer to an object of Opd as an operad , and to a morphism as an operad functor .132 .2.2 Operads from symmetric monoidal categories
There is a standard method of constructing an single-colored operad from an object x in a strict symmetric monoidal category C . Namely, we define the set of n -ary operationsto be hom C ( x ⊗ n , x ), and compose these operations using composition in C . This gives theso-called endomorphism operad of x . Here we give the generalization of this idea to themulti-color case, using all the objects of C as the objects of the operad. In what follows welet Ob( C ) be the set of objects of a small category C . Proposition D.11. If C is a small strict symmetric monoidal category then there is an Ob( C ) -colored operad Op ( C ) for which: • the set of operations Op ( C )( c , . . . , c k ; c ) is defined to be hom C ( c ⊗ · · · ⊗ c k , c ) , • given operations f ∈ hom C ( c ⊗ · · · ⊗ c k ; c ) and g i ∈ hom C ( c ij ⊗ · · · ⊗ c ij i , c i ) for ≤ i ≤ k , their composite is defined by f ◦ ( g , . . . , g k ) = f ◦ ( g ⊗ · · · ⊗ g k ) . (D.1) • identity operations are identity morphisms in C , and • the action of S k on k -ary operations is defined using the braiding in C .Proof. The various axioms of a colored operad can be checked for Op ( C ) using the corre-sponding laws in the definition of a strict symmetric monoidal category. The associativityaxiom for Op ( C ) follows from associativity of composition and the functoriality of the ten-sor product in C . The left and right unit axioms for Op ( C ) follow from the unit laws forcomposition and the functoriality of the tensor product in C . The two equivariance axiomsfor Op ( C ) follow from the laws governing the braiding in C . Proposition D.12.
The assignment Op : SymMonCat s → Opd defined on objects as inProposition D.11 and sending any strict symmetric monoidal functor F : C → C ′ to theoperad morphism Op ( F ) : Op ( C ) → Op ( C ′ ) that acts by F on types and also on operations: Op ( F ) = F : hom C ( c ⊗ · · · ⊗ c n , c ) → hom C ′ ( F ( c ) ⊗ · · · ⊗ F ( c n ) , F ( c )) is a functor.Proof. This is a straightforward verification.133 .2.3 Operad Algebras
As a sort of monoid, operads exist to act. The elements of O ( c ; d ) for some operad O are meant to be thought of as “abstract operations” with c as the input types, and d as theoutput type. When O acts on something, it is meant to be thought of as realizing theseabstract operations as real operations on some family of sets indexed by the elements of C . Definition D.13.
Let C be a set. A C -colored set is a functor C → Set , where C isthought of as a discrete category. This is of course the same as a function C → ob Set .For c = ( c , . . . , c n ) ∈ S ( C ), let X c denote the set Q nj =1 X c j . A map of C -colored sets f : X → Y is a natural transformation f : X ⇒ Y . This is the same as a family of functions { f c : X c → Y c } c ∈ C with no further conditions. For c = ( c , . . . , c n ) ∈ S ( C ), let f c : X c → Y c denote the function Q nj =1 f c j : Q nj =1 X c j → Q nj =1 Y c j . Definition D.14.
Let O be a C -colored operad, with operad composition denoted by γ ,and unit operation denoted by I . An O -algebra consists of • a C -colored set X : C → Set , also denoted { X c } c ∈ C • for c ∈ S ( C ) and d ∈ C , a map θ : O ( c ; d ) × X c → X d which makes the following diagrams commute for c, d, c j ∈ C , c, b j ∈ S ( C ), b = P n b j , and σ ∈ S n . • associativity: O ( c, d ) × Q nj =1 O ( b j , c j ) × X b O ( b ; d ) × X b O ( c ; d ) × Q nj =1 [ O ( b j , c j ) × X b j ] O ( c ; d ) × X c X d ∼ =permute γ × θ × Q j θ θ • unity: 1 × X c O ( c ; c ) × X c X cI × ∼ = θ • equivariance: O ( c ; d ) × X c O ( cσ ; d ) × X cσ X dθ σ × σ − θ efinition D.15. A map of O -algebras ( X, θ ) → ( Y, ξ ) consists of a map of C -coloredsets α : X → Y such that the following diagram commutes. O ( c ; d ) × X c O ( c ; d ) × Y c X d Y d × f c θ ξf Let
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