The Category-Theoretical Imperative

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The Category-Theoretical Imperative

DISSERTATION submitted in partial satisfaction of the requirements for the degree of

DOCTOR OF PHILOSOPHY

in Philosophy

Michael Ernst

Dissertation Committee: Distinguished Professor Penelope Maddy, Chair Associate Professor Jeremy Heis Truman P. Handy Professor of Philosophy Colin McLarty Associate Professor James Weatherall Professor Kai Wehmeier

2014 c 2014 Michael Ernst DEDICATION

To my family.

ii TABLE OF CONTENTS

Page

LIST OF FIGURES v

ACKNOWLEDGMENTS vii

CURRICULUM VITAE viii

ABSTRACT OF THE DISSERTATION ix

1 Background 1 1.1 Objections ...... 3 1.1.1 Inadequacy ...... 3 1.1.2 Inappropriateness ...... 6 1.1.3 Coherency Assurance & Conceptual Uniﬁcation ...... 8 1.2 The Allure of Category Theory ...... 9

2 Coherency Assurance 13 2.1 Categories and Graphs ...... 14 2.2 Properties of the Category of R-Graphs ...... 17 2.3 Towards a Contradiction ...... 24 2.3.1 Step 1 ...... 25 2.3.2 Step 2 ...... 26 2.3.3 Step 3 ...... 27 2.3.4 To GRAPH ...... 29 2.3.5 Generalization ...... 29 2.4 Discussion ...... 30

3 Conceptual Uniﬁcation 35 3.1 Introducing Categories ...... 36 3.1.1 Opportunistic Method ...... 38 3.1.2 Constructive Method ...... 40 3.1.3 Axiomatic Method ...... 58 3.1.4 Conclusion ...... 59 3.2 Graph Operations ...... 60 3.2.1 Construction in ZFC ...... 62 3.2.2 Categorial Construction ...... 65

iii 3.2.3 Searching for Guidance ...... 72 3.2.4 Adjunctions ...... 73 3.2.5 Functoriality ...... 75 3.2.6 Adjoint Failure ...... 77 3.2.7 Fixed Inputs ...... 80 3.2.8 G/ ...... 82 3.2.9 [G]...... 83 3.2.10 /G ...... 84 3.2.11 Adjoint Success? ...... 86 3.3 Conclusion ...... 88

Bibliography 89

Appendices 92 A Exponential Completeness Lemma ...... 92 B Tournament Lemma ...... 96 C Consistent Subsystems ...... 103

iv LIST OF FIGURES

Page

2.1 The 1-vertex complete r-graph ...... 18 2.2 The r-graph E ...... 18 2.3 An example product ...... 20 2.4 An example exponential ...... 21 2.5 A subgraph diagram ...... 22 2.6 The subobject classiﬁer for RGRAPH ...... 22 2.7 A characteristic arrow ...... 23 2.8 The 2-vertex complete r-graph ...... 25

3.1 A natural transformation ...... 41 3.2 The composition of natural transformations ...... 41 3.3 An identity natural transformation ...... 42 3.4 A diagram and its reversal ...... 43 3.5 The category Gop ...... 43 3.6 The category Rop ...... 44 3.7 A graph homomorphism ...... 44 3.8 An r-graph homomorphism ...... 45 3.9 A graph with one king ...... 48 3.10 A graph with three kings ...... 49 3.11 The category Uop ...... 50 3.12 An undirected graph homomorphism ...... 51 3.13 The single loop undirected graph ...... 51 3.14 The graph with reversal L ...... 51 3.15 The graph with reversal B ...... 52 3.16 Lexicographic product example: graph G1 ...... 60 3.17 Lexicographic product example: graph G2 ...... 60 3.18 Lexicographic product example: graph G1 with G2 inserted at each vertex . 61 3.19 Lexicographic product example: graph G1[G2]...... 61 3.20 Corona example: graph G1 ...... 61 3.21 Corona example: graph G2 ...... 62 3.22 Corona example: one copy of G1 and two copies of G2 ...... 62 3.23 Corona example: graph G1 /G2 ...... 62 3.24 Representing a graph in Rgraph ...... 66 3.25 An example coproduct ...... 67

v 3.26 An example pushout ...... 67 3.27 A graph represented in Rgraph ...... 68 3.28 Two important comprehensions ...... 68 3.29 Two new subobjects ...... 68 3.30 Three special arrows ...... 69 3.31 Glueing completed ...... 69 3.32 Representations of the lexicographic product and corona in Rgraph ..... 70 3.33 An example counit ...... 74 3.34 An example unit ...... 74

vi ACKNOWLEDGMENTS

There are many people who, one way or another, have helped me to develop the ideas in this dissertation. Parts of the dissertation were written with the ﬁnancial support of UCI’s Department of Logic and Philosophy of Science and from UCI’s School of Social Sciences. I would like to thank the Department of Logic and Philosophy of Science for providing a supportive community in which my ideas could develop and grow. It will always occupy a warm place in my heart.

I have beneﬁted greatly from the feedback and support of my committee in both this work and elsewhere. Colin McLarty has helped me to expand my mathematical skills and enriched my understanding of category theory with his extensive knowledge of its history. Jeremy Heis has helped me to maintain a broader view in my work and to instill in me the importance of teaching. Jim Weatherall has always been available to help me work through the diﬃcult details and broad strokes of my research. Kai Wehmeier has provided a sounding board for my ideas and has provided me an example of the enthusiasm a philosopher can bring to their work.

I would like to thank my advisor, Pen Maddy. I owe much of who I am as a philosopher and as a writer to her guidance. I am immensely grateful to have been able to work with her in my time at UCI.

Finally, I would like to thank my family for their unconditional love and encouragement. Even when they have not understood what I was doing or why they supported me nonethe- less.

vii CURRICULUM VITAE

Michael Ernst

Doctor of Philosophy in Philosophy 2014 University of California, Irvine Irvine, California Master of Arts in Philosophy 2012 University of California, Irvine Irvine, California Bachelor of Science in Philosophy & Joint Mathematics and Computer Science 2008 Harvey Mudd College Claremont, California

viii ABSTRACT OF THE DISSERTATION

The Category-Theoretical Imperative

Michael Ernst

Doctor of Philosophy in Philosophy

University of California, Irvine, 2014

Distinguished Professor Penelope Maddy, Chair

Category theory has been advocated as a replacement for set theory as the foundation for mathematics. It is claimed that as a foundation set theory is both inadequate and inappropriate. Set theory is considered inadequate because it cannot produce all of the mathematical objects of interest. Set theory is considered inappropriate because it provides a poor framework for mathematical research. In this dissertation, I argue that category theory is subject to exactly the same objections by considering the use of category theory for work in graph theory.

ix Chapter 1

Background

Category theory has been put forward as a foundation for mathematics. The more ambi- tious have suggested that it replace set theory as the foundation of mathematics. There are two kinds of arguments behind such a move: those in favor of category theory and those against set theory. I will focus primarily on the objections to set-theoretic foundations. I will not defend set theory but instead I will show that the same objections can be raised against categorial foundations.1 This process will undermine potential advantages of categorial foundations and suggest that no foundation may completely avoid the objections raised.

We will consider objections to set-theoretic foundations of two general ﬂavors: claims that set theory is inadequate as a foundation and claims that set theory is inappropriate as a foundation.2 Set theory is inadequate because it is incapable of producing all of the objects of mathematical interest. There are things that one might want to study in category theory, such as the category of all groups or the category of all sets, that set theory lacks the resources to construct. Set theory is inappropriate because it is too far removed from the

1I follow Kr¨omer(2007) in using ’categorial’ instead of ’categorical’ to avoid confusing diﬀerent mathematical uses of ’categorical’. 2I adopt this division from Feferman (1977).

1 rest of mathematics. This distance is made manifest in two foundational shortcomings. Set theory does not provide guidance for mathematical research. It will not point out which regions of the vast universe of sets are mathematically interesting. Furthermore, the language of set theory inhibits research. Descriptions in set theory can be diﬃcult to follow and can obscure the important features of the objects described. Both of these phenomena reveal the inappropriateness of set-theoretic foundations, a consequence of the distance of those foundations from much of mathematics.

Categorial foundations are supposed to have an advantage over set-theoretic because they are more appropriate. They ”can suggest which research directions are most fruitful, and can test the results of research using relatively explicit measures of elegance and coherence” Goguen (1991). Thus, they have an advantage over set-theoretic foundations with respect to guidance. Furthermore, ”category theory can be seen as providing the language for mathematics” because ”it allows us to talk about mathematical objects in structural terms without having to be about any speciﬁc, constitutive, object” Landry (1998). So, the use of category theory is also superior in the realm of description. This makes categorial foundations more appropriate because they are successful in those two areas which revealed the inappropriateness of set-theoretic foundations.

The rest of this chapter will be devoted to making the claims of the preceding two para- graphs more clear. Section 1 will spell out the two kinds of objections in more detail. There I will point out how the two kinds of objections are asking for very different things from a foundation. So I will introduce two different foundational roles, coherency assurance and conceptual unification, to more accurately capture the desire for adequacy and appropriateness, respectively. Section 2 will look briefly at the history of category theory to illustrate some reasons it might provide a more appropriate foundation, i.e., one that provides better conceptual unification. These two Sections provide the background for the rest of the work, which shows how the two objections of Section 1 can be raised against categorial founda-

2 tions. Chapter 2 will address coherency assurance and in it I will show that inadequacy is not a uniquely set-theoretic shortcoming. Chapter 3 will address conceptual uniﬁcation and there I will explore the adoption of categorial methods in graph theory. I will show how the adoption of those methods creates problems of guidance as well as descriptions that obscure important features of the graphs described.

1.1 Objections

Both the inadequacy and inappropriateness objections can be traced back to Saunders Mac Lane, one of the founders of category theory. He is opposed to the ”rather ﬁxed dogmatic position which reads: Mathematics is what can be done with axiomatic set theory using classical predicate logic” (Mac Lane, 1981, p. 467). Mac Lane dubs that view the ’Grand Set Theoretic Foundation’ and argues that it is damaging to mathematical practice. Our explications of each objection will begin in Mac Lane’s work.

1.1.1 Inadequacy

The inadequacy objection comes from parts of category theory that cannot be captured in a set-theoretic foundation. In category theory ”the Mathematication is working with a variety of objects (categories of all groups, functor categories) which cannot all be described simultaneously as objects of any one [set-theoretic]foundational system” (Mac Lane, 1971a, p.236).3 There are two kinds of categorial objects that set-theoretic foundations have dif- ﬁculty capturing. First, objects like the category of all groups, categories containing all structures of a given type. In the sequel these will be referred to as unlimited categories. Second, functor categories, which are categories containing all the structure preserving maps

3Mac Lane does not explicitly single out set-theoretic foundations, but he suggests that a non-set-theoretic foundation may avoid these shortcomings.

3 from one category to another.

To illustrate the diﬃculties with simultaneously providing these two kinds of categorial objects, we consider some of the more well-known set-theoretic foundations.4 First, there is Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). ZFC has no trouble producing functor categories for any pair of categories it can produce. However, as Mac Lane famously observed it ﬂounders with respect to unlimited categories:

These problems arise in the use of collections such as the category of all sets, of all groups, or of all topological spaces. It is the intent of category theory that the “all” be taken seriously; the usual axiomatizations of set theory do not allow the formation of collections such as the set of all sets, or the set of all groups. (Mac Lane, 1969, p. 192)

The problem is that in ZFC one cannot form collections large enough to serve as unlimited categories. ZFC is exclusively about sets and on pain of contradiction cannot allow the formation of the set of all sets or even the set of all groups. The natural response to this diﬃculty is to introduce another notion of collection above that of sets, one which allows for a collection of all sets or a collection of all groups.

The introduction of classes to the theory does exactly that and so successfully delivers unlimited categories. One can form the class of all sets or the class of all groups. Thus, a theory like Von Neumann-Bernays-G¨odel(NBG) has classes that can serve as unlimited categories and in that way overcomes the limitations of ZFC. In fact the very ﬁrst paper on category theory, Eilenberg and Mac Lane (1945), takes NBG as one of many possible satisfactory foundations for capturing unlimited categories. It is functor categories that cause trouble for NBG and related systems. For any two classes it is not in general possible

4For readers interested in diﬀerent foundations for category theory Shulman (2008) provides a good survey, though it is not directly concerned with unlimited categories.

4 to form the class of all functions between those classes. So, NBG cannot form functor categories. What is needed is a richer theory of these large objects.

Grothendieck universes provide that richer theory. Reminiscent of Zermelo (1996), the idea is to have an increasing hierarchy of universes of sets, each larger than the last. Starting in a universe U we consider C the set of all categories of U. Now, C is too large for U but it is contained in a larger universe U’. Thus, in that larger universe one can form the desired functor categories involving C. However, for every universe U one can only form the category of all categories in U. As Mac Lane observes ”[t]his is still not that will-of-the-wisp, the category of all categories ¨uberhaupt” (Mac Lane, 1971a, p.234). Larger universes have categories not present in U. So, in such a system one can never form the category of all categories, nor can one form any other unlimited category.5 Thus, none of these foundations can simultaneously provide unlimited categories and functor categories.

It has been an open question whether any foundation, set-theoretic or otherwise, can do both. Solomon Feferman has pursued this question at length, starting in Feferman (1977) and most recently in Feferman (2013). He expresses it in the following three requirements:

• (R1) Form the category of all structures of a given kind, e.g. the category Grp of all groups, Top of all topological spaces, and Cat of all categories.

• (R2) Form the category BA of all functors from A to B, where A, B are any two categories.

• (R3) Establish the existence of the natural numbers N, and carry out familiar operations on objects a, b, . . . and collections A, B, . . . , including the formation of [a, b], (a, b),

A A ∪ B,A ∩ B,A − B,A × B,B , ∪A, ∩A, ΠBx[x ∈ A], etc. (Feferman, 2013, p. 9)

(R1) and (R2) make sure Mac Lane’s original requirements are met. The addition of (R3)

5A similar problem arises for Mac Lane’s one universe system. He points out that ”we cannot form the category of all sets or of all groups” for the same reasons I have highlighted here (Mac Lane, 1969, p. 196).

5 is to guarantee that the foundation is also satisfactory for the rest of mathematics. Taken together these three requirements need to be met for a foundation to avoid inadequacy. In Chapter 2, I will show that a consistent foundation cannot meet all three requirements, undercutting the force of the inadequacy objection.

1.1.2 Inappropriateness

The inappropriateness objection has two diﬀerent but related aspects. First there is the quality of guidance, or lack thereof, that a foundation gives for mathematical research. Sec- ond, how much the language provided by the foundation leads to cumbersone or obfuscatory descriptions. In both cases inappropriateness is focused on how the methods provided by a foundation inﬂuence progress in mathematics.

As mentioned, set-theoretic foundations are considered inappropriate because of ”the relatively poor guidance for discovering elegant and coherent theories that they provide” (Goguen, 1991, p.1). The guidance is poor because there is little to none of it. The ’Grand Set Theoretic Foundation’ ”does not adequately describe which are the relevant mathematical structures to be built up from the starting point of set theory” (Mac Lane, 1981, p. 468). All kinds of diﬀerent structures appear within the cumulative hierarchy of set theory. The problem is that only some of those structures are mathematically interesting or important and ”[t]he ’Grand Foundation’ does not provide any way in which to explain the choice of these concepts” (Mac Lane, 1981, p.468). Set theory provides no tools to distinguish the important from the unimportant. It provides no guidance and so is inappropriate.

This inappropriateness of set theory is based on the absence of a positive feature. It is not that set-theoretic foundations do something incorrectly, it is that they fail to do that thing at all. However, one can imagine foundations that do provide guidance that is either incomplete or directed toward the wrong concepts. Foundations are inappropriate when they

6 fail to provide the right kind of guidance. As such, these other kinds of guidance failures would also be grounds for an inappropriateness objection. In Chapter 3, I will pursue this line and argue that categorial foundations are inappropriate.

Set-theoretic foundations are also considered inappropriate because of the ungainliness of set-theoretic language. The difficulties arise when ”[t]he grand doctrine of the new math: ’Everything is a set’ came at the cost of making artificial and clumsy definitions” (Mac Lane, 1991, p.6). Being forced to formulate a field in set-theoretic terms can make work in that field more difficult. Simple concepts do not necessarily have simple or easily accessible set- theoretic realizations.

Constructions can suﬀer from ”the excessive concreteness and representation dependence of set theoretical foundations” (Goguen, 1991, p.1). A set-theoretic construction will introduce properties that are not relevant for the object constructed. These properties often have nothing to do with the object constructed and instead depend solely on the method of construction that is chosen. For example, there is the diﬀerence between the Zermelo and von Neumann ordinals. The choice of construction determines whether or not 2 is an element of 4. That fact is totally irrelevant to mathematical research concerning the natural numbers.

It is important to clarify the exact nature of this inappropriateness objection. The problem lies in the extended use of set-theoretic language, not that set theory is used to construct these objects at all. It would not be a problem if set theory was used to construct the natural numbers and then the language of set theory was left behind. The mistake is the adoption of set-theoretic language for practice in these ﬁelds. Continuing the natural number example, the problem is not constructing 2 as {{∅}} (or {{∅}, ∅}). The problem is subsequently deciding to work with {{∅}} (or {{∅}, ∅}) in place of 2. The set-theoretic construction may play some important role, but it should not supplant the original object in mathematical research and set-theoretic language should not replace the original language of the ﬁeld.6 It

6The extent to which advocates of set-theoretic foundations actually proposed such a widespread adoption

7 is particularly telling that a construction is often immediately left behind once it is veriﬁed that it represents the desired object. This happens because the language of set theory is not appropriate in those cases and so it is jettisoned once it has served its purpose. In Chapter 3, I will extend this objection to category theory by investigating cases where categorial language is not appropriate.

We now have both kinds of objections on the table and can consider how they relate to one another. The two kinds of objections ask very diﬀerent things of a foundation. This suggests that there are in fact two diﬀerent foundational roles in play.

1.1.3 Coherency Assurance & Conceptual Uniﬁcation

For inadequacy, the concern is whether or not a foundation can produce the desired mathematical objects. On this view, the role of a foundation is the role that set theory has traditionally played in mathematics. Speciﬁcally, it is to ”provide a generous, uniﬁed arena to which all local questions of coherence and proof can be referred” (Maddy, 2011, p. 34). We use such a foundation to make sure the concepts and methods of proof that we introduce in mathematics are acceptable. It can protect us against introducing concepts that initially appear reasonable but are in fact not completely well-formed or even inconsistent. I say such a foundation provides ’coherency assurance’ because we use the foundation to assure ourselves that the objects we introduce and study are coherent.7 It does not matter if the objects are constructed in a complicated or cumbersone way. The construction is only needed to provide an assurance of coherency and not to be used after so doing.

In contrast, the inappropriateness objection is fundamentally concerned with how the objects are constructed. A foundation is evaluated on both the context in which it places the of set-theoretic language is beyond our scope here. It is enough for our purposes to understand the objections to such a proposal. 7Of course for real assurance we must have conﬁdence in the foundation used for this process. How that conﬁdence is gained is beyond the scope of this work, interested readers are referred to Maddy (2011).

8 objects and language it provides to build and work with them. On this view, ”[f]oundations should provide general concepts and tools that reveal the structures of the various areas of mathematics and its applications, as well as relationships among them”(Goguen, 1991, p.12-13). We are able to bring all of mathematics under the general concepts of such a foundation. For this reason, I say such a foundation provides (or is) a conceptual uniﬁca- tion for mathematics. Such a foundation ”can suggest which research directions are most fruitful, and can test the results of research using relatively explicit measures of elegance and coherence” (Goguen, 1991, p.13). Thus, a conceptual uniﬁcation avoids both prongs of inappropriateness. It provides guidance for research and a useful language and framework in which to pursue that research.

We have seen how set theory fails to provide coherency assurance for certain categorial objects and how set theory provides a poor conceptual uniﬁcation of mathematics. Now we will take a brief look at the history of category theory to see why it might be able to avoid these diﬃculties.

1.2 The Allure of Category Theory

Category theory was not developed with an eye toward providing conceptual unification (or any other kind of foundation). Instead, ”[c]ategory theory was introduced in 1945 by Eilen- berg and Mac Lane as a language for describing and organizing mathematical constructions, for example the construction of homology groups” (Kao et al., 2010, p.227). Category theory began as an aid for solving certain mathematical problems. It was a new language for expressing them. The usefulness of expressing things in categorial terms was an important part of category theory from the very beginning. As such, it began with the potential to be a conceptual unification. However, as only a language it did not have any potential for providing coherency assurance. This makes it very different from set theory, which had a

9 foundational character from the very beginning.

Category theory did not start becoming a ﬁeld in its own right until the 1950s. One important step in that process was the introduction of adjoints in Kan (1958). Kan ”saw that there were interesting results to be had from studying categories, functors, and relations among them in their own right” (232)(Kao et al., 2010, p.232). While originally introduced as a new way to conceptualize objects of interest, the concepts of category theory became direct objects of study. Through the 1950s and 1960s category theory became more than a tool for use in other ﬁelds, but something to be studied itself. The next step was to consider a categorial foundation for mathematics.

In Lawvere (1966) it was proposed that the category of categories could provide such a foundation. This is referred to as the Category of Categories as a Foundation (CCAF ). Lawvere’s proposal is motivated by certain trends in mathematics at the time:

In the mathematical development of recent decades one sees clearly the rise of the convic- tion that the relevant properties of mathematical objects are those which can be stated in terms of their abstract structure rather than in terms of the elements which the objects were thought to be made of. (Lawvere, 1966, p.1)

It is not the particular constitution of mathematical objects that matters. It is the structural features that matter: how objects relate to other objects. For example, consider the construction of a group in set theory. The particular sets that make up the group elements are not important. What is important is how the group elements relate to one another. An ideal foundation brings out this distinction by describing mathematical objects in terms of their relations to one another.

The category of categories does exactly that. There is no direct description of the internal

10 structure of any of the categories. Instead categories are completely described by the functors between them. The category of categories provides a ’top-down’ description of categories. Rather than build objects up from below, category theory describes them from above in terms of their relation with one another. In this way, CCAF is a foundation that focuses on the relevant properties of mathematical objects.

CCAF is not the only categorial foundation that has been proposed. Two years before CCAF, Lawvere (1964) proposed that the Elementary Theory of the Category of Sets (ETCS) could provide a foundation for all of mathematics. There is also Synthetic Differential Geometry (SDG). SDG is not intended to provide a foundation for all of mathematics, only for differential geometry. It is important to remember that category theory provides a language for description. SDG does not construct the desired objects, it does not provide coherency assurance. SDG describes the objects of differential geometry using the concepts of category theory. The use of category theory as a language suggests it might be a good conceptual unification.

This intuition is reinforced when the concepts of category theory are fruitfully applied in fields not connected to its origins. One might call this the unreasonable applicability of category theory. For example, in first-order logic ”implication and both universal and existential quantification are all adjoints” (Lane, 1996, p.134). The concepts of category theory can be fruitfully applied to the study of first-order logic. The more such examples accrue, the more category theory looks to have captured some truly universal notions in mathematics. This is exactly what one wants from a conceptual unification. These and other successes make some mathematicians and philosophers ”tempted to follow Lawvere and adopt the view that the growth of mathematics should be guided by various categorical slogans or the more widely held view that category theory underlines the general principles common to different areas of mathematics” (Lambek and Scott, 1986, p.126).

Category theory began as a useful language for work in particular ﬁelds. It has turned

11 out to be a useful language in many other ﬁelds of mathematics. Furthermore, category theory appears to focus on the important features of mathematical objects. It deals with abstract structure, while leaving out extraneous internal properties. This naturally raises the question: Might category theory be a useful language for formulating and thinking about all of mathematics? Put another way: Might category theory give us an appropriate and adequate foundation for mathematics? We can call such a position the ’Grand Category- Theoretical Foundation’. I explore the merits of such a proposal in the next couple of chapters. Chapter 2 investigates the adequacy of categorial foundations and Chapter 3 investigates their appropriateness.

12 Chapter 2

Coherency Assurance

A foundation provides coherency assurance when it is able to produce the objects of mathematical interest. Mac Lane considers set-theoretical foundations inadequate because they cannot produce objects important to category theory. Feferman gave us a precise formulation of what a foundation must do in order to avoid that inadequacy:

• (R1) Form the category of all structures of a given kind, e.g. the category Grp of all groups, Top of all topological spaces, and Cat of all categories.

• (R2) Form the category BA of all functors from A to B, where A, B are any two categories.

• (R3) Establish the existence of the natural numbers N, and carry out familiar operations on objects a, b, . . . and collections A, B, . . . , including the formation of [a, b], (a, b),

A A ∪ B,A ∩ B,A − B,A × B,B , ∪A, ∩A, ΠBx[x ∈ A], etc. (Feferman, 2013, p. 9)

In this chapter I derive a contradiction using the unlimited category of graphs, showing that any foundation satisfying all three requirements is inconsistent. Thus, if we take the requirements seriously, every foundation is inadequate.

13 The chapter is divided into three main parts. The ﬁrst part, composed of Sections 1 and 2, is technical background on categories, graphs, and categories of graphs. Section 1 presents the essential features of categories, graphs and the relationship between them. Section 2 highlights the features of the category of graphs most important for our purposes. The second part, Section 3, contains the derivation of the contradiction. The third and ﬁnal part, Sections 4, is an exploration of our situation in light of the results of Section 3.

2.1 Categories and Graphs

A category has objects, A, B, C, . . . and arrows f, g, h, . . . that are related in the following way:

• Each arrow f has a unique domain object A and a unique codomain object B, written f : A → B.

• For arrows f : A → B and g : B → C, there is a speciﬁc composite g ◦ f : A → C. Furthermore, this operation is associative.

• For each object A, there is an identity arrow 1A : A → A, where f ◦ 1A = f and

1A ◦ h = h, for every h : C → A and f : A → B.

These very simple relationships completely describe what it means to be a category. As a result, as soon as you satisfy these axioms you have a category. Such a claim is so obvious as to appear tautologous. If the axioms of category theory are satisﬁed then of course you have a category.

Graphs have a very similar presentation. A (reﬂexive) graph has vertices and edges that are related in the following way:

14 • Each edge e has a unique source vertex u and a unique target vertex v.

• (Reﬂexive) For every vertex v there is a distinguished edge lv such that the source and

target of lv are both v.

We will focus our attention on reﬂexive graphs, also referred to as r-graphs. This is for technical convenience. r-graphs are more amenable to a categorial approach. It will turn out that this focus is not limiting, our result will extend to all graphs reﬂexive and otherwise.

These requirements completely describe what it means to be an r-graph. Thus, in an exact parallel to the case of categories, it is natural to claim that anything satisfying these axioms is an r-graph.

At this point, the close relationship between the category and r-graph axioms should be apparent. The r-graph (and graph) axioms are a proper part of the category axioms, except that ’edge’ is used in the place of ’arrow’ and ’vertex’ is used in the place of ’object’. The use of either pair of words is irrelevant when determining whether or not something satisfies the axioms. So something satisfies the category axioms just in case it satisfies the r-graph axioms and additional axioms for composition. Hence, a category is an r-graph with additional structure. Consequently, it is also a graph with additional structure. Mac Lane makes a similar identification. He uses it to introduce categories in Mac Lane (1971b), his most thorough presentation of category theory and a standard reference in the field.

Categories are related to one another by mappings that preserve their categorial structure. These mappings are called functors and they map objects to objects and arrows to arrows.

We can express precisely how a functor F from category A to category B preserves structure:

• For an A arrow f : A → B, F (f): F (A) → F (B) is an arrow of B.

• For A arrows f : A → B and g : B → C, F (g ◦ f) = F (g) ◦ F (f).

15 • For an A object A, F (1A) = 1F (A).

In this way, F provides a version of A in B. Categorial structure in A is carried over into B.

If we take categories as our objects and functors as our arrows, we have something that satisﬁes the category axioms. Every functor has a unique domain category and a unique codomain category. Functor composition proceeds pointwise, and is associative. Every category clearly has an identity functor. Thus, we have a category of categories. We will not pursue this further here, but this is the idea underlying CCAF (e.g. see Lawvere (1966) and McLarty (1991)). For our purposes, the key is that something similar is true in the case of graphs and r-graphs.

A mapping that preserves graph structure is called a graph homomorphism. An r-graph homomorphism is a graph homomorphism that also preserves the reflexive structure. Graph morphisms map vertices to vertices and edges to edges. So for a (reflexive) graph homomorphism M from a (reflexive) graph G to a (reflexive) graph H:

• For a G edge e with source u and target v, M(e) is an H edge with source M(u) and target M(v).

• (Reﬂexive) For a G vertex v, M(lv) = lM(v).

Much like a functor, M provides a version of G in H.

As with the situation between categories and r-graphs (and graphs), the connection goes even further. The properties of r-graph (and graph) morphisms and functors should make clear that functors preserve r-graph (and graph) structure as part of their preservation of categorial structure. Thus, in exactly the same way that categories are r-graphs (and graphs) with additional structure, functors are a special case of r-graph (and graph) morphisms. A functor is an r-graph homomorphism between categories that preserves their categorial

16 structure in addition to their r-graph structure.

If we take graphs as our objects and graph morphisms as our arrows, we have something that satisﬁes the category axioms. Every graph morphism has a unique domain graph and a unique codomain graph. The composition of graph morphisms is straightforwardly deﬁned and clearly associative. Finally, every graph has an identity graph morphism. Thus, exactly like the above case of categories, we have a category of graphs. We will call this category GRAPH. We have a category RGRAPH of r-graphs from identical reasoning with r- graphs and r-graph homomorphisms.1 Since GRAPH is a category, it follows immediately from earlier that it is itself an r-graph and similarly for RGRAPH. When we consider RGRAPH as an r-graph we will call it U. This allows us to be clear about how we are considering it in a given context (though of course the thing itself does not change). It is from RGRAPH and our ability to consider it as U that we will derive a contradiction (the result will straightforwardly extend to GRAPH). However, before so doing it will be useful to understand some of the categorial properties of RGRAPH.

2.2 Properties of the Category of R-Graphs

RGRAPH has a wealth of categorial properties. This makes it especially amenable to categorial methods because one can leverage all that is already known about such properties when studying it. Here I do not pretend to give an exhaustive or thorough presentation of the categorial properties of RGRAPH. I refer interested readers to Plessas (2011), Bumby and Latch (1986), and Brown et al. (2008) for a more thorough discussion of the diﬀerent properties of categories of graphs. Here I will survery those properties of RGRAPH that feature most prominently in the following derivation of a contradiction.

1RGRAPH can also be introduced as the category whose objects are graphs and whose morphisms preserve adjacency but not type; that is, they may send edges to vertices.

17 RGRAPH has a terminal object 1, shown in ﬁgure 2.1. It has a single vertex and the

Figure 2.1: The 1-vertex complete r-graph

required loop at that vertex. Any r-graph G has exactly one arrow to 1, the morphism that identiﬁes every vertex of G with the single vertex of 1 and every edge of G with the loop at that vertex. It is this uniqueness property that characterizes a terminal object. We denote

the unique arrow from G to 1 as !G.

Arrows from 1 to G serve an important role as well. Each such arrow picks out exactly one vertex of G. In this way, we can characterize the vertices of G as arrows from 1.2 For example !1 : 1 → 1 picks out the unique vertex of 1. This is why we label that vertex !1. In an arbitrary category, arrows from a terminal object are called global elements. Arrows from an arbitrary object O are called generalized elements or O-elements. Global elements are then 1-elements.

In order to characterize the edges of G we introduce the r-graph E, shown in ﬁgure 2.2. Edges in an r-graph G are then represented by arrows from E. For example, the non-loop

1E s t

Figure 2.2: The r-graph E

edge of E is represented by the identity arrow 1E : E → E. Also, the single loop in 1 can be identiﬁed with the arrow !E : E → 1.

This picture of E also gives us a natural way to recover the source and the target of a given edge of G. We introduce the two arrows that pick out the vertices of E, s, t : 1 → E. s

2This is one of the technical conveniences gained by focusing on RGRAPH.

18 picks out the source of an edge and t picks out its target. Thus, for an edge e : E → G, e ◦ s : 1 → G is the source vertex for the edge e and e ◦ t is its target vertex.

There is no structure to an r-graph other than its vertices and edges. Thus, we want a way of saying that an r-graph is fully determined by its arrows from 1 and E. We do this by saying that E and 1 form a family of generators. Let G, H be arbitrary r-graphs and f, h : G → H be morphisms between them. In order for f, h to be distinct they must disagree on a vertex or edge. We express this by saying that f 6= h iﬀ there is either an e : E → G such that f(e) 6= h(e) or a v : 1 → G such that f(v) 6= h(v). In order for f(v) 6= h(v), h and f have to disagree on lv = v◦!E. Thus, we can remove the dependence on 1. By itself, E is a generator for RGRAPH. Thus, f 6= h iﬀ there is an e : E → G such that f(e) 6= h(e).3

Now that we have edges and vertices in RGRAPH we can consider three very special types of r-graphs. A complete r-graph is one where for any pair of vertices there is exactly one edge from one to the other, i.e. for u, v : 1 → G, ∃!e : E → G such that e(s) = u and e(t) = v. Closely related is a tournament, where for any pair of vertices there is exactly one edge between them, i.e. for u, v : 1 → G, ∃!e : e : E → G such that e(s) = u and e(t) = v or e(s) = v and e(t) = u. For example, E is a tournament, but it is not complete because there is no edge from t to s. Discrete graphs are our third important type of graph. A discrete r-graph is an r-graph that contains only degenerate edges, the required loops at every vertex. If D is discrete, then any edge e : E → D of D factors through 1, i.e. there is

0 0 an e : 1 → D such that e = e ◦!E. It is standard practice to adopt a version of the axiom of choice for discrete graphs and so we here adopt it for discrete r-graphs (Plessas, 2011, p. 83). Speciﬁcally, for every arrow f : G → D, if G is nonempty and D is discrete, then there exists a h : D → G such that f ◦ h ◦ f = f.

RGRAPH supports three major categorial constructions that we will use: the product of

3Another technical convenience.

19 r-graphs, the exponential of r-graphs, and the introduction of characteristic arrows.4

For a pair of r-graphs H and G the categorial notion of a product is an r-graph, which we

label G × H, and a pair of projection arrows p1 : G × H → H and p2 : G × H → H. They are such that for any graph T with arrows h : T → G and k : T → H there is a unique arrow hh, ki : T → G × H which makes the diagram in ﬁgure 2.3 commute. To say a diagram

k h hh, ki

> < p ∨ p G < 1 G × H 2 > H

Figure 2.3: An example product

commutes is to say that for any two objects, every path between then is equal. So in this

case p1 ◦ hh, ki = h and p2 ◦ hh, ki = k. We label the unique arrow hh, ki because of its relationship to h and k.

In graph theory there are many diﬀerent kinds of products, this particular kind of product is called the conjunction of G and H (Harary, 1969, p. 25). In this context, we can introduce special arrows between products. Take r-graphs G, G0,H,H0 with arrows f : G → G0 and

0 0 0 0 0 f : H → H . Then we write the unique product arrow hf ◦ p1, f ◦ p2i : G × H → G × H as f × f 0.5

For a pair of r-graphs we want to consider the collection of r-graph morphisms between them. It is natural to ask whether this collection itself has an r-graph structure. In this case of reflexive graphs, Brown et al. (2008) illustrates how that is possible. For any r- graphs G and H we call this new r-graph GH or the exponential of G and H. In a categorial 4I avoid introducing these constructions in terms of adjunctions because, while a useful and powerful concept, adjunctions are not necessary to understand the technical work of this chapter. I do consider them in Chapter 3. 5 Following standard practice we write p1 for the first projection arrow for every product and depend on context to differentiate them.

20 setting, exponentials include not only the object GH but also a special arrow ev that together produce the commutative diagram for arbitrary C and h given in ﬁgure 2.4. For every arrow

ev GH × H > G ∧ >

f f × 1H

C × H

Figure 2.4: An example exponential

f : C × H → G there is a unique f : C → GH such that the above diagram commutes.

H ev is called the evaluation arrow and is connected with G × H in the same way p1 and

p2 are connected to products. f is called the transpose of f, and vice versa. For an arrow h : C → GH we will also use h to also denote its corresponding tranpose from C × H to G. So there is a 1-1 correspondence between arrows from C × H to G and arrows from C to GH . This correspondence is function currying: a function from C × H to G is replaced by a function that takes arguments from C and returns a function from H to G.

The ﬁnal construction we care about is the use of the subobject classiﬁer. Intuitively, if we have R a subgraph of G, then we want a characteristic arrow from G to truth values. Such an arrow should take the parts of G that are in R to true and the other parts to false.

First, we need to make sense of subgraphs in a categorial setting. We want R to be a subgraph of G. A vertex v : 1 → G cannot possibly be a vertex of R because it has the wrong codomain. However, what matters is that R have the same structure as part of G, not that it is identical to some part of G. So we call R a subgraph of G just in case it has a 1-1 embedding into G. In RGRAPH an arrow r : R G is 1-1 just in case it has the following property: r ◦ f = r ◦ g implies that f = g for arbitrary arrows f and g (that are composable with r). In category theory, arrows with that property are called monics, and

21 6 we represent them by arrows with tails, .

If a T -element x : T → G of G can be factored through r : R G, meaning there is an x0 : T → R such that x = r ◦ x0, then that x0 is unique because r is monic. Thus, x is eﬀectively represented in R by x0 and we say that x is a member of R (or r) and write x ∈ R (or x ∈ r). Thus, it is R alongside the arrow r that together make a subgraph of G, where some generalized elements of G are members of R. This relationship is illustrated in ﬁgure 2.5.

x0 T > R ∨ x r

> ∨ G

Figure 2.5: A subgraph diagram

We now want to associate a characteristic arrow to r. In RGRAPH there is a truth value r-graph, standardly called Ω, shown in ﬁgure 2.6. From it we are able to get characteristic

lfalse ltrue

true false

Figure 2.6: The subobject classiﬁer for RGRAPH arrows. Such arrows classify subgraphs, and so Ω is called the subobject classiﬁer. A characteristic arrow for r should go from G to Ω, send vertices of G that are members of

6In some categories, the notion of a monic and a 1-1 embedding come apart. However, they coincide in the case of RGRAPH (and GRAPH).

22 R to true, edges of G that are members of R to ltrue (the reﬂexivity guaranteed loop at the vertex true). The fact that Ω is a subobject classiﬁer guarantees the existence of such

arrow. For r : R G, we are guaranteed a χr : G → Ω that makes the diagram in ﬁgure 2.7 commute for just those x ∈ r (here, as above, x0 is the unique arrow that shows x is a member of r).

!T x 0

> ! > x R R > 1 ∨

r true

> ∨ χ ∨ G r > Ω

Figure 2.7: A characteristic arrow

Notice that x ∈ r just in case true◦!T = χr ◦ x. Hence, this diagram functions exactly as desired. For a vertex of G, v : 1 → G, v ∈ r just in case χr ◦ v = true◦!1 = true. Similarly,

for an edge e : E → G, E ∈ r just in case χr ◦ e = true◦!E = ltrue. Of particularly important

use is that this relationship between r and χr functions in both directions. Given an r we

are guaranteed a χr that characterizes it. Also, given a χ : G → Ω we are guaranteed an

∗ ∗ ∗ ∗ object R and monic arrow r : R G such that χ characterizes r , i.e. χ = χr∗ . This is especially useful because it allows us to produce subgraphs out of characteristic functions.

For example, given an r-graph G we can send every vertex of G to true and so every distin-

guished loop of G is sent to ltrue. We then send every edge of G that is not a distinguished

loop to the edge from true to true that is not ltrue. This gives us a characteristic function for a subgraph of G. In this case, it is the discrete r-graph underlying G, i.e., it contains all the vertices of G but only the distinguished loop edges. In general, for a r-graph G we will

23 write |G| for its underlying discrete r-graph.

Thus, RGRAPH is very rich in categorial constructions, further emphasizing the generality and usefulness of categorial methods. The ﬁnal piece to note is that RGRAPH is a special kind of category called a topos. For readers familiar with category theory that should be clear. For our purposes here, it is only important because it means that certain theorems hold about objects and arrows in RGRAPH. We will make use of some such theorems drawn from McLarty (1995). Now that we have some understanding of RGRAPH we can proceed to the contradiction derivation.

2.3 Towards a Contradiction

The structure of this proof is best understood by drawing parallels to a well known result: the proof that there can be no set of all sets using Cantor’s theorem. That proof proceeds by contradiction and has two distinct parts. Taking a set of all sets, S, one derives a contradiction. The contradiction involves the existence of a surjection f : S → 2S, where 2S is the power-set of S. First, Cantor’s theorem states that there is no surjection between any set and its powerset (Scheinerman, 2012, p. 209). This means that f cannot exist. The second part begins with the realization that every element of 2S is a set. As a result, the set of all sets S must contain 2S as a subset. Hence, there exists a natural surjection f from S onto 2S. The two parts contradict one another and so the existence of S must be rejected.

Our proof proceeds in a similar fashion. We have already claimed that RGRAPH is itself an r-graph, which we called U. Our proof has the following structure:

1. Consider the relationship between U and a relevant exponential, W U .

2. Use a Cantor’s theorem style result to show that there cannot be an onto m : U → W U .

24 3. Use the fact that U is the category of all r-graphs to show that W U is a substructure of U and so U has a mapping onto it.

2.3.1 Step 1

U For this step, we must pick a suitable exponential. We will consider K2 , where K2 is the complete r-graph on two vertices as shown in ﬁgure 2.8.

1 2

Figure 2.8: The 2-vertex complete r-graph

There are two main reasons we work with K2 as the base r-graph for the exponential. First,

K2 has very useful technical properties. The morphism pK2 : K2 → K2 that permutes the

vertices of K2 has no ﬁxed points. This is crucial for step 2. We also have the following lemma:

Lemma (Exponential Completeness). If G is a complete r-graph, then GH is complete for any r-graph H.

U This guarantees that K2 will be complete, which is crucial for step 3. The second reason we

U work with K2 is a bit more subtle. We do it because we cannot easily work with Ω .

Given that our proof is modeled after the universal set argument, it is natural to consider the power-object ΩU as the relevant exponential. The vertices of ΩU correspond to subgraphs of U,ΩU is the ’graph of all subgraphs of U’. Since subgraphs are r-graphs then our step 3 should proceed as quickly and easily as it did in the universal set argument. Simply map subgraphs of U onto vertices of ΩU . However, there is an important sense in which subgraphs are not r-graphs in RGRAPH that blocks exactly this line of argumentation.

25 For an r-graph G, a subgraph is an r-graph H along with a monic i : H G. Thus, two subgraphs of G can share the same H and still be distinct, as long as their corresponding monics are distinct. For example, assume that G has two distinct vertices u : 1 → G and v : 1 → G. Both u and v pick out subgraphs of G and they are distinct by assumption. However, it is clearly the case that 1 = 1. Thus, we can have two distinct subgraphs of G, using only a single r-graph in RGRAPH distinct from G. Hence, we cannot guarantee for every vertex in ΩU that there is a distinct vertex in U that we can map onto it because we cannot guarantee that for every subgraph of U there corresponds a distinct r-graph in RGRAPH.

For this reason we do not work with ΩU . Since we cannot directly follow the speciﬁcs of the universal set argument, we instead choose an exponential with useful properties. This returns us to our ﬁrst reason for picking K2.

2.3.2 Step 2

We want to show that there cannot be an onto m : U → W U . For this result we want to appeal to Corollary 1.2 of Lawvere (1969)7. Here we consider a special case of that result:

Theorem (Lawvere). In any topos C the following holds: For any C-object B, if there exists a C-arrow f : B → B such that f ◦b 6= b for all b : 1 → B, then for no C-object A does there exist a C-arrow A → BA that is onto for global elements.

As mentioned at the end of Section 2, RGRAPH is a topos. Also, recall that pK2 : K2 → K2 permutes the vertices of K2, and so pK2 ◦ b 6= b for all b : 1 → K2. Thus, the combination of

Lawvere’s theorem and the existence of pK2 allows us to establish the following result: 7We only take advantage of the one corollary, but Lawvere (1969) as a whole establishes very interesting connections between results such as Cantor’s theorem and the undeﬁnability of truth. Interested readers should be forewarned, Lawvere reverses the composition notation, using fg when we would write g ◦ f.

26 Theorem (Cantor’s Theorem for R-Graphs). Let G be a reﬂexive graph. Any reﬂexive graph

G homomorphism f : G → K2 is not onto for global elements.

U Speciﬁcally, in RGRAPH there cannot exist an m : U → K2 that is onto for global elements, thereby completing step 2.

2.3.3 Step 3

U In order to complete this step we need to show that there is an m : U → K2 that is onto for global elements. So this section is concerned exclusively with establishing the existence of such an m. This is done in two parts. We begin as before, with a theorem of RGRAPH:

G Lemma (Tournament). For any r-graph G, for each vertex x of K2 there is a tournament

G Qx that is a subgraph of K2 . Also, for distinct vertices x and y, Qx 6= Qy.

We can prove this lemma in RGRAPH using only the properties described in Section 2. Most notably the proof requires the use of choice and the subobject classiﬁer. Full details can be found in the appendix.

U U Consider K2 in light of the above lemma. For every vertex of K2 we have a distinct r-

U graph. There is a one-to-one correspondence between vertices of K2 and some subcollection of the r-graphs in RGRAPH. If we could realize this correspondence as an onto mapping we would have the onto morphism that we need to reach a contradiction. The problem is that the correspondence is between vertices on the one hand and r-graphs on the other.

U What we need is a correspondence between vertices of K2 and vertices of U, not objects of RGRAPH.8

8 U The other possibility is to consider K2 as a category and create a functor realizing the correspondence U between objects of K2 as a category and objects of RGRAPH. As a functor it would also be an r-graph homomorphism and so be the m for which we are looking. I do not take that route here because I want to keep all of the reasoning inside the category and not have to justify rules of functor construction.

27 Up to this point nothing has turned on particular properties of U. Everything established holds for an arbitrary graph. In order to bring in the properties of U we need a statement that expresses the fact that U really is meant to be RGRAPH. I will refer to this as internalization because it will express how RGRAPH is an object in itself.

There are a number of ways we can internalize. The simplest is to assert that since U really is RGRAPH the tournament lemma already expresses the correspondence between U and

U K2 for which we are looking. On the other hand, we can attempt to formalize this kind of reasoning as follows. We extend the language with a new function symbol F taking objects of the category and returning vertices of U. We add an axiom stating that F is bijective. This eﬀectively states that objects of the category are vertices of U, as desired. Finally, since we have extended the language we also extend internal comprehension using the subobject classifer to formulas including F .

Using F , m is now straightforward to deﬁne:

• For a vertex x of U, if x = F (Qy) for some y, send x to y.

U • Pick some vertex of K2 and send all other vertices of U to it.

U • K2 is complete and so a morphism into it is fully determined by the action of that morphism on the vertices of the domain r-graph.

U The construction of m is a straightforward comprehension on the product graph U ×K2 following the outline given above. The desired result then follows directly from the Tournament lemma:

U Theorem (Arrow Existence). In RGRAPH there is an arrow m : U → K2 that is onto for global elements.

This theorem directly contradicts Cantor’s theorem for r-graphs.

28 2.3.4 To GRAPH

In Section 2 I claimed that the focus on reflexive graphs would not limit the breadth of the results, that they would extend to all graphs reflexive and otherwise. Now I am in a position to fulfill that claim. First we need to consider GRAPH, the category of all graphs. It is also important to recall that RGRAPH is a proper part of GRAPH, every r-graph is a graph and every r-graph morphism is a graph morphism. Since GRAPH is a category it is also a reflexive graph. When considered as a reflexive graph we can call it U 0. We work in GRAPH, but exclusively in the proper part of it that is RGRAPH. As before we consider

U 0 9 K2 as our relevant exponential.

0 U 0 Now, any graph morphism between U and K2 is necessarily an r-graph morphism because

U 0 10 0 K2 has exactly one loop at every vertex and so every loop in U gets sent to one of the

U 0 distinguished loops in K2 . Even though we are working in GRAPH and so ignoring the special r-graph structure of the objects, that structure is still preserved by this particular

0 U 0 collection of morphisms. Thus, there cannot be a graph morphism between U and K2 because as we have seen through Theorem 1 there can be no r-graph morphism between them. This means step 2 works for GRAPH. Since we are working in the RGRAPH part of GRAPH all of step 3 goes through for U 0 as it did for U. The only diﬀerence is that the

0 U 0 morphism is now deﬁned between U and K2 ; however, the exact same deﬁnition works. In this way, the result extends to GRAPH.

2.3.5 Generalization

While we can extend the result to GRAPH, there are certain barriers to extending this result to other categories. Here I mention the two most prominent. For step 2 we needed

9 U 0 Here we mean K2 as an exponential in RGRAPH. It will necessarily exist as a graph in GRAPH, but it will not necessarily satisfy the requirements of an exponential there. 10 U 0 This is because K2 is complete.

29 an object B and an arrow f : B → B such that f had no ﬁxed points, i.e. f ◦ b 6= b for all b : 1 → B. In some categories this is impossible. For example, group morphisms and morphisms between monoids preserve the identity element of the group or monoid. Thus, any f : B → B in one of those categories has the identity element of B as a ﬁxed point. So this problem cannot be straightforwardly extended to such categories. For step 3 we needed the category under consideration to be a topos. There are many categories of interest that are not toposes. The most relevant is the category of categories. It is not a topos. Consequently, if the problem is to extend to such categories fundamental changes would have to be made to the proof.

However, a distinct but related contradiction can be derived using the category of rooted well-orderings.11 Rooted well-orderings are well-orderings with a distinguished root element, preserved by the relevant homomorphisms. The category of all such orderings would itself be a rooted well-ordering and so be self-membered. We could then run a Burali-Forti style argument to derive a contradiction. The drawback to this approach is its connection to Feferman’s requirements. One of the advantages to considering graphs is that they are an established ﬁeld of mathematics for which one can make a clear argument that its operations fall under (R3) (as we shall see in the next section). With rooted well-orderings it is more diﬃcult to argue that the operations necessary to derive a contradiction fall under (R3). That said, if someone could make such a case convincingly it would further reinforce the conclusions of this chapter.

2.4 Discussion

We have derived a contradiction. The arrow m must exist according to step 3 and cannot possibly exist according to step 2. At this point, we want to consider how to move forward

11Thank you to Graham Leach-Krouse and Teddy Seidenfeld for both independently making this suggestion

30 in the face of this paradox. The natural thing to do is review the assumptions we made in deriving the contradiction, hoping to locate one we can ﬁnd some grounds to reject.

The bulk of the derivation took place within RGRAPH. Lawvere’s result, the Exponential Completeness lemma and the Tournament lemma are all established by reasoning in the category. For this reason, we begin our search for culprits with the properties of RGRAPH from Section 2. There are three easily separable aspects of RGRAPH which have been used: a handful of specific finite r-graphs, the fact that RGRAPH is a topos, and a particular choice principle. In order to have any claim to being the category of all r-graphs RGRAPH must contain the finite r-graphs that were used, so we focus on the other two properties.

That RGRAPH is a topos plays a crucial role in every proof. The contradiction cannot even be stated without the notions of product and exponential. It is also a property that we must accept if we are to satisfy (R3). Using traditional methods from graph theory one can carry out all of the topos-related constructions. As we saw above, the conjunction of two graphs satisﬁes the properties of a categorial product. Something similar can be done for the exponential and subobject classiﬁer. Consequently, if we are to be able to ‘carry out familiar operations’ in accordance with (R3), RGRAPH must be a topos.

The choice principle we adopted, choice restricted to discrete graphs, is in a similar situation. It follows from normal practice in graph theory. Speciﬁcally, choice for discrete graphs follows from, and is equivalent to, the spanning tree principle (Delhomm´eand Morillon, 2006).12 The spanning tree principle states that every connected r-graph has a spanning tree, which is a minimally connected subgraph containing all of the vertices of the original r-graph. Since the spanning tree principle is a standard part of practice, (R3) forces us to accept the choice principle as well.

While the properties of RGRAPH are required by (R3), they are also necessary if RGRAPH

12(Delhomm´eand Morillon, 2006) establishes the equivalence working in set theory, but the result can be straightforwardly reproduced in the categorial context.

31 is to be considered the category of r-graphs. The category of (not necessarily all) reflexive graphs is typically presented in one of two ways. The most common is as a category of functors from a particular finite category to the category of sets (e.g., Bumby and Latch (1986)). The alternative is to give a direct axiomatization of the category, as in Plessas (2011). Under either presentation the category of reflexive graphs allows for all the constructions appealed to above.13 As a result, this gives us further confirmation that those constructions are well- defined and that we should expect them in RGRAPH. Thus, with respect to unlimited category theory and mathematical practice more generally, RGRAPH has to be a topos with choice for discrete graphs.

Only in the latter half of step 3 was there any reasoning outside of RGRAPH, the introduction of internalization. It is needed to express that U really is RGRAPH as an r-graph. Internalization is not a ‘familiar operation’ and so is not required to satisfy (R3). However, it is required if we are to satisfy (R1). If RGRAPH is to be the category of all r-graphs, then it must contain every r-graph. As a category, RGRAPH is an r-graph and so must contain itself. Internalization is simply a statement of that fact, it expresses that U is the category as contained in itself. Thus, we are not forced to accept internalization as a familiar operation, but we are forced to accept it for RGRAPH to actually be an unlimited category.

These four features are all that is required to derive the contradiction. Furthermore, many different subcollections of them are consistent. If we give up internalization there is a model for the remaining three features. This is also true if RGRAPH is not required to be a topos and, perhaps most interestingly, is also true if we give up on the existence of the handful of finite r-graphs that were used in the proof. The details of these models can be found in the appendix. Their significance is that the problem cannot be located in any one of the four features, it only arises when they are adopted collectively. This would seem to point to the rejection of RGRAPH as a whole, that category simply cannot exist. However, rejection

13Both presentations, as well as a third approach, are further explored in Chapter 3.

32 of the category clearly violates (R1).

We have seen how everything used to derive the contradiction is justified by one of the requirements needed to avoid inadequacy. The existence of RGRAPH and internalization are both necessary to satisfy (R1). Topos, choice, and finite graphs are all necessary to satisfy (R3). Thus, satisfaction of the requirements leads to a contradiction. No foundation can provide for unlimited categories and functor categories and also be sufficient for normal mathematical practice.

There are a number of morals one might draw from this result. One could take the failure of unlimited category theory as an indictment of categorial foundations more generally. This is a tempting conclusion to draw, but it fundamentally misunderstands the role of unlimited categories with respect to foundations. It is true that a mathematically suﬃcient theory of unlimited categories cannot be a foundation for mathematics because it is inconsistent. However, such a theory is not one of the proposals for a categorial foundation of mathematics. Its inconsistency could be used as a case against categorial proposals only if those proposals could provide a foundation for it.

This is certainly false for the most well known proposals. For example, neither CCAF nor ETCS do so. I will not go into the details here but the history of category theory gives us good reason to believe this will be the same for all categorial proposals. The practice of category theory has had only a limited interest in the use of unlimited categories.14 Thus, the result does not show that categorial foundations are inconsistent. However, it does show that they are all inadequate. In fact, any consistent foundation will neccessarily be inadequate.

This illustrates the ultimate untenability of the inadequacy objection. Set theory cannot provide the desired coherency assurance because the concepts involved are jointly incoherent. Furthermore, no consistent foundation, set-theoretic, categorial, or otherwise, can meet all

14For readers interested in a full understanding of the history I recommend Kr¨omer(2007) or Marquis (2009)

33 the requirements. The fact that every foundation is inadequate makes inadequacy an empty objection.

34 Chapter 3

Conceptual Uniﬁcation

Chapter 1 introduced two objections to set-theoretic foundations: inadequacy and inappropriateness. It also introduced two foundational roles corresponding to those objections, coherency assurance and conceptual uniﬁcation respectively. Chapter 2 showed that categorial foundations, and in fact all foundations, are inadequate because they cannot provide coherency assurance for unlimited category theory. This chapter will show that categorial foundations are inappropriate because of situations where they fail to provide conceptual uniﬁcation.

The conceptual unification given by a foundation has two different aspects. First, it provides a framework that guides mathematical research. Second, it provides a language that is useful for doing that work. A foundation can be inappropriate in either of these aspects. It can be inappropriate because it fails to or incorrectly guides mathematics. It can also be inappropriate if the language provided is ungainly. I will argue that the conceptual unification given by category theory is inappropriate in both senses.

It would be natural to consider the two aspects of conceptual uniﬁcation in turn. However, issues of guidance and issues of language can become intertwined and there are often be

35 trade-oﬀs between them. As such, the chapter will not be divided between the two aspects. Instead it will focus on two diﬀerent projects in categorial graph theory and the problems that arise when trying to carry them out.

The ﬁrst project is the realization of categories of graphs. In order to do categorial graph theory one has to introduce such categories. Section 1 will explore three diﬀerent methods for so doing: the opportunistic, constructive, and axiomatic. The primary problem in that section will be one of guidance: the three methods provide either incorrect or no guidance for mathematical research. Some issues of language will arise when considering particular methods, especially the constructive method.

The second project concerns two speciﬁc binary operations used in graph theory: the lexicographic product and corona. Section 2 will investigate the diﬀerent categorial methods available to describe those two operations. The primary problem in that section will be one of language: there is no satisfactory description of either operation using categorial language. Issues of guidance will arise because categorial methods promote particular kinds of description over others.

3.1 Introducing Categories

This section explores ways to introduce categories of graphs. In Chapter 2, we saw how the introduction of unlimited categories leads to contradiction. As mentioned at the end of that chapter, an interest in unlimited categories is highly unusual; ordinary category theory is not concerned with them at all. What we will be considering here is not unlimited categories but the kind of categories that actually appear in mathematical practice.

In practice there is an important technical distinction made between small and large objects. This distinction can be realized in many diﬀerent ways, including class theories and

36 Grothendieck universes. Using Grothendieck universes we cannot form the category of all graphs but we can form the category of all graphs in some universe U, the category of all U-graphs. Working with only one universe we can call the sets in U small and the other sets large.1 This allows us to form the category of all small graphs and similar related categories with coherency assurance provided by the theory of Grothendieck universes.2 Thus, the small/large distinction gives us categories of graphs with which we can work without the danger of inconsistency.

The small/large distinction actually does more for category theory than just help mathematicians avoid contradiction. Some of the central results in category theory depend upon it. For example,

Freyd’s Special Adjoint Functor Theorem . . . is always one of the ﬁrst results I[Shulman] quote when people ask me ”are there any real theorems in category theory?” So it is all the more striking that it involves in an essential way notions . . . which refer explicitly to the diﬀerence between sets and proper classes (or between small sets and large sets) Shulman (2008)

Without a small/large distinction important parts of category theory are lost. With the distinction come the aforementioned central results and our ability to consider a wide variety of categories of graphs. This returns us to the investigations of this section.

We know that we can legitimately introduce categories of small graphs, the issue is what methods we employ for so doing. In this section we will consider three approaches to the introduction of categories of graphs: opportunistic, constructive, and axiomatic. Our primary focus is on the constructive method; the ﬁrst and third are included for contrast.

1 Readers familiar with set theory can think of U as Vκ for κ inaccessible. 2There are many other systems that can provide this coherency assurance. Shulman (2008) provides a good survey of the many diﬀerent alternatives. I use Grothendieck universes because of their introduction in Chapter 1.

37 3.1.1 Opportunistic Method

The opportunistic method can be characterized in two steps. First, begin with a kind of object and morphisms between objects of that kind. Second, introduce a category of them as the collection of all of these objects and morphisms.

The more careful mathematicians will invoke some foundational system in order to justify the second step. For example, we could appeal to the theory of Grothendieck universes and a comprehension principle. However, many mathematicians transition directly from the objects and morphisms to the category itself. Those mathematicians depend on the implicit understanding that the objects are appropriately small and so no problems will arise.

This method is particularly well-exempliﬁed by Hell (1979).3 Hell (1979) begins with graphs that are built out of sets and homomorphisms as structure preserving functions between those sets. When it comes to the introduction of the category, the process is both brief and straightforward: ”Now we form the category of graphs, Graph, by taking for its objects all graphs, and for its morphisms all homomorphisms” (Hell, 1979, p. 123). Hell does not appeal to any particular principles to introduce his Graph. All he has is his deﬁnition of categories as classes and so the introduction is implicitly licensed by a class theory operating in the background.

The key feature of the opportunistic method is that the language of category theory is not used throughout the entire practice. It does not appear in that ﬁrst step where the objects and morphisms are introduced. This allows work to be done with the language used for those introductions instead of with the language of category theory. The use of categorial language becomes restricted to particular applications, those applications where it appears most promising. For example, Hell (1979) describes graphs and graph homomorphisms using

3Knauer (2011) is another good example of the opportunistic method, where it is used to introduce a large number of categories.

38 set-theoretical language. The categorial language introduced in that paper is an addition that is intended to help with particular problems, not to be used for all work in graph theory.

Thus the opportunistic method represents a situation where category theory is being employed but does not serve as a conceptual uniﬁcation. Instead categorial language is one tool among many that can be applied when appropriate to the problem at hand. This gives us a starting point for considering the level of conceptual uniﬁcation exhibited by the other methods.

Issues of conceptual unification aside, the opportunistic method has at least two serious drawbacks. Most prominently, when one introduces a category opportunistically it is not guaranteed to have any particular properties. Each property of the category must be individually verified.4 For example, if we consider the Graph of Hell (1979) we do not know if it has products or not. In order to determine if it has products we would have to return to the original description of graphs. Using that description we would have to produce a construction that can fulfill the role of product within that category. This is exactly the kind of work we had to with RGRAPH in Chapter 2, the individual verification of particular properties of the category. A lot of work using the opportunistic method is devoted toward this type of verification, verification that has to be done before those categorial properties can be leveraged for mathematical research.

The opportunistic method also provides very little information about how different categories may relate to one another. For example, imagine we were to construct a category of reflexive graphs to parallel the construction of Graph in Hell (1979), call it Rgraph. We would know next to nothing about how the two categories are related to one another. The only thing we could easily conclude is that Rgraph is a subcategory of Graph because reflexive graphs are a special case of graphs. However, there are many more interesting relationships

4To be especially careful one would need to verify that it is even a category but this step is usually omitted due to its triviality.

39 aside from being a subcategory and those relationships are not made apparent using the opportunistic method. Instead those relationships must also be individually veriﬁed. As with the above, the opportunistic method leaves a lot of work to be done after the categories are introduced but before they can be fruitfully employed.

3.1.2 Constructive Method

For the constructive method, we build our categories of graphs out of other categories. As a result, we are able to immediately deduce many of the properties of the categories from the properties of the categories used to construct them. Similar advantages are gained with respect to the relationship between constructed categories. Furthermore, the language and methods of category theory are used throughout instead of being added on top of an independent preexisting practice.5 This makes the constructive method a real case of conceptual uniﬁcation as the diﬀerent categories can be given a similar treatment using the methods of category theory.

We construct categories of graphs as speciﬁc functor categories. To understand what that means, we will need additional technical background. That background will allow us to construct diﬀerent categories of graphs and examine the advantages of those constructions.

After considering these successes we will investigate potential shortcomings of the constructive method. That investigation will begin with what is only an apparent problem of language to illustrate some of the extra resources available to the method. A genuine problem of language will be revealed by the method’s diﬃculty in characterizing undirected graphs. Finally, we will consider the many interesting and important categories that are missed by the constructive method, which constitutes an important failure of guidance.

5For categorial language to be completely pervasive the categories used for the construction must be given in some categorial framework such as CCAF.

40 Functor & Opposite Categories

Recall from Chapter 2 that functors are maps between categories that preserve their categorial structure. If we are given two categories A, B then the collection of functors from A to B itself has the structure of a category. This can be seen as exponentiation in the category of categories and we write BA for the category of functors from A to B.

The objects of BA are clearly the functors, but we also need to describe the arrows. The arrows of BA are what are called natural transformations between functors. Let F,G be functors from A to B and f : A → A0 be an arbitrary A-arrow. A natural transformation ϕ : F → G between F and G is a collection of arrows of B indexed by objects of A such that for every f the diagram in ﬁgure 3.1 commutes.

ϕ F (A) A > G(A)

F (f) G(f) ∨ ∨ ϕ 0 F (A0) A > G(A0)

Figure 3.1: A natural transformation

Let H : A → B be a functor and φ : G → H a natural transformation. Consider the diagram in ﬁgure 3.2. Since the two smaller rectangles commute, the larger rectangle commutes.

ϕ φ F (A) A > G(A) A > H(A)

F (f) G(f) H(f) ∨ ∨ ∨ ϕ 0 φ 0 F (A0) A > G(A0) A > H(A0)

Figure 3.2: The composition of natural transformations

41 This gives us φ ◦ ϕ as a natural transformation from F to H (associativity of composition for natural transformations follows immediately from associativity of composition for arrows in B). Finally, for the identity natural transformation ψ : F → F , simply use 1A = ψA for every object of A. Note that we will typically write 1F to stand for the identify natural transformation on the functor F . It gives us the diagram in ﬁgure 3.3, which obviously commutes. Thus, we have both identity and composition and so with functors as objects

1 F (A) A > F (A)

F (f) F (f) ∨ ∨ 1 0 F (A0) A > F (A0)

Figure 3.3: An identity natural transformation and natural transformations as arrows BA forms a category.6

The constructive method uses functor categories whose domain is an opposite category. For a category C we deﬁne its opposite category, Cop, as follows:

• The objects of Cop are the same as the objects of C.

• The arrows of Cop are the arrows of C with their domain and codomain reversed.

For example consider ﬁgure 3.4, if the diagram on the left is a diagram from C, then we get the diagram on the right as a diagram in Cop. Notice that (g ◦ f)op = f op ◦ gop. It is straightforward to verify that Cop is in fact a category. This gives us all the resources we need to carry out the constructive method.

Categories will be built as functor categories of the form BA, where B = SET the category of 6It is also possible to introduce functor categories explicitly as exponential objects in the category of all small categories, but I consider the approach used here to be more accessible.

42 g gop B > C B < C ∧ > op f g ◦ op ◦ f g f op f

∨ < A A

Figure 3.4: A diagram and its reversal sets and A = Cop for some small category C. This will guarantee that the resulting category is a topos.

Constructing Categories of Directed Graphs

Here we will construct both Graph and Rgraph, the category of all small graphs and the category of all small reﬂexive graphs, respectively. Our constructions will closely follow the constructions of these categories that appear in Bumby and Latch (1986) and Lawvere (1989), though both those papers are concerned with issues beyond our current scope.7 Each construction is characterized by the choice of C. We will ﬁrst look at those choices and then the interesting consequences that follow.

Let Gop be the category pictured in ﬁgure 3.5 and Rop be the category pictured in ﬁgure 3.6. Identity arrows have been suppressed, but all objects and all other arrows are included. We

s V E t

Figure 3.5: The category Gop

op op can now consider the categories SETG and SETR .

7Brown et al. (2008) provides alternative constructions of some of the categories considered here. I highly recommended it to readers interested in a diﬀerent way of thinking about the relevant graphs.

43 s l◦s l V - EY t l◦t

Figure 3.6: The category Rop

op Let G and H be objects of SETG and so functors from Gop to SET. G and H both pick out two sets and a pair of functions from one set to the other. This ﬁts exactly with

our description of a graph once we have restricted ourselves to small graphs. Gop has been suggestively labeled to illustrate the connection. G(V ) is the set of vertices of G, G(E) is its set of edges, and G(s) and G(t) are the functions that pick out the sources and targets of edges, respectively.

op Since SETG is a functor category the arrows will be natural transformations. A natural transformation ψ : G → H will be composed of two arrows ψV : G(V ) → H(V ) and

ψE : G(E) → H(E) such that the diagrams in ﬁgure 3.7 commute. ψV and ψE are functions

ψ ψ G(E) E > H(E) G(E) E > H(E)

G(s) H(s) G(t) H(t)

∨ ψ ∨ ∨ ψ ∨ G(V ) V > H(V ) G(V ) V > H()

Figure 3.7: A graph homomorphism

between the vertex sets and edge sets, respectively. Let e be an edge of G and so an element of G(E) with source G(s)(e) and target G(t)(e). The fact that the two diagrams commute means that the image of e under ψ, ψE(e), has a source and target that are the images under

ψ of the source and target of e, H(s)(ψE(e)) = ψV (G(s)(e)) and H(t)(ψE(e)) = ψV (G(t)(e)). This is exactly the requirement we gave in our deﬁnition of graph homomorphism in Chapter 2. The notion of graph homomorphism naturally falls out of the functor construction.

44 The situation is very similar for the constructive treatment of the category of small reﬂexive

op op graphs, SETR . Let G and H now be objects of SETR . Everything said above about the

op op situation in SETG holds in SETR . In addition we also have G(l) and H(l) which map vertices to their distinguished loops and require an additional commutative diagram, given in ﬁgure 3.8, to have a natural transformation ψ : G → H.8 That that diagram commutes

ψ G(V ) V > H(V )

G(l) H(l)

∨ ψ ∨ G(E) E > H(E)

Figure 3.8: An r-graph homomorphism

op guarantees that natural transformations between functors in SETR preserve loops: for a vertex v ∈ G(v) the image of the loop of v is the loop of the image of v, ψE(l(v)) = l(ψV (v)). Thus, just as above, our original notion of reﬂexive graph homomorphism falls naturally out of the functor construction.

op Because they are functor categories from an opposite category to SET both SETG and

op SETR are toposes. We gave a description of only some of the features of a topos in Chapter 2. Drawing on those descriptions we immediately know that both categories have products, exponentials, and a subobject classifier. As toposes they also have many other interesting and useful constructions that we have not yet considered. Furthermore, there is a uniform method for determining the structure of those constructions in the categories. We do not have to do any individual verification of these properties of the categories, a clear advantage over the opportunistic method. The properties are direct consequences of our uniform method of construction. Here we have a solid success for category theory as conceptual unification. The same simple construction technique was used in both cases and produced the desired

8We ignore the diagrams for l ◦ s and l ◦ t because they contribute no additional information.

45 categories along with immediately guaranteeing many of the properties of interest.

There is a further advantage we get from this use of categorial methods. There is an obvious

inclusion of Gop into Rop, simply map V to V , E to E, s to s and t to t. Let us call this inclusion I : Gop → Rop, which is trivially a functor between the two categories. If we take a reﬂexive graph R : Rop → SET, then R ◦ I : Gop → SET is a normal graph where we have simply forgotten that the reﬂexive loops are meant to be distinguished from other loops. I

op op gives us a new functor SETI : SETR → SETG that does this forgetting for every reﬂexive graph. In addition, because of how SETI was introduced using I we get two more functors

op op from SETG to SETR (Bumby and Latch, 1986, p.4).9 The ﬁrst takes graphs and adds a distinguished loop at every vertex turning them into reﬂexive graphs. The second takes a graph G and maps it to its loop graph. The loop graph of G has as vertices the loops of G and as edges those edges of G that are between vertices of G that have loops. I will not go into further details here and instead refer the interested reader to Bumby and Latch (1986).

For our purposes, the importance of these results is to highlight the advantages gained from categorial methods. Using the categorial constructions many results about both the categories and the relationships between them are easily and immediately established. For these categories, the categorial approach was immediately fruitful and relatively straightforward to pursue.

Unfortunately, categorial graph theory is concerned with a wider array of categories than simply these two. Using the constructive method to create categories of undirected graphs creates problems of language. The constructive method also has trouble with those categories of graphs that are not toposes, both those that appear as subcategories of constructed categories and others that do not. Those misses represent an important failure of guidance. We will consider each of these problems in turn. However, before we consider those problems we will brieﬂy consider what is only an apparent problem for the constructive method. This

9These turn out to be right and left adjoints to SETI , a notion we introduce later in this chapter.

46 will illustrate how it avoids a particular problem of language.

An Apparent Problem of Language

An established limitation of categorial language is that ”a property of objects, morphisms,

etc., in a category C can be characterized category-theoretically if and only if it is invariant under the automorphisms (self-equivalences) of C” (Clark and Bergman, 1973, p. 80). Any properties not invariant under automorphism represent a serious problem for categorial language, the inability to express certain properties is a serious limitation. As many categories do not possess non-trivial automorphisms this is often not a problem.10 Here we will con-

struct a non-trivial automorphisms of Gop and exhibit important properties of graphs that they do not preserve.

However, this will represent only an apparent and not an actual disadvantage of the constructive method. The constructive method provides more resources for description than simply the categories it constructs. We will be able to express more by taking advantage of the method of construction.11

First, let us introduce the property of interest: that of possessing a single king. The notion of king is generally only deﬁned for tournaments. Recall from Chapter 2 that tournaments are graphs that possess exactly one edge between every pair of vertices. A particular vertex v is the king of a tournament T just in case for every other vertex u of T at least one of the following obtains:

• the edge between v and u has v as its source and u as its target.

• there is another vertex w, the edge from v to w has v as its source and the edge from

10Clark and Bergman (1973) shows that it is a problem for the category of rings by illustrating non-trivial automorphisms of that category. 11While I will not pursue it further here, this certainly suggests that methods other than the constructive may lack the resources to avoid this problem.

47 w to u has w as its source.

A king of a tournament ’defeats’ every other vertex in either one or two steps. That a tournament has exactly one king turns out not to be preserved by all automorphisms of

Gop.12

From a simple relationship between Gop and Rop we were able to make interesting connections

op op between SETG and SETR . We can do something similar if we focus on Gop (or Rop) on its own. There is an obvious non-trivial automorphism of Gop, call it A : Gop → Gop, which simply swaps s and t. Then for any graph G : Gop → SET we can consider G ◦ A, which is a graph similar to G but with all of the edges reversed. As with I : Gop → Rop, this gives us a

op op functor SETA : SETG → SETG that takes every graph to its edge-reversed counterpart.

op Furthermore, this functor is an automorphism of SETG .

As an example, if G is the graph in figure 3.9, then SETA(G) is the graph pictured in figure 3.10. Both G and SETA(G) are tournaments, but they have importantly different

Figure 3.9: A graph with one king

structure. G has one king, while SETA(G) has three kings. Thus, SETA does not preserve the possession of a sole monarch. Following Clark and Bergman (1973) this means it cannot be characterized category-theoretically. Thus, we have an important graph-theoretic concept, having a single monarch, that would seem to be lost when we adopt the language of category theory. Here we are well-served by further investigation.

12Every tournament has at least one king (West, 2001, p.62).

48 Figure 3.10: A graph with three kings

Let us focus on the two vertex graph with a single edge between those vertices, E. If we

op are living exclusively in SETG we only have access to the objects and arrows, not to any structure the objects may have that is not expressed by the arrows in the category. So we know that E has two p-elements where p is the single vertex graph.13 We know that it cannot be mapped onto the two vertex discrete graph and so that there must be an edge between its vertices. We also know that there is no homomorphism permuting its vertices, so any edges must be oriented in the same direction. In fact, we know that E has only one endomorphism and so it has only one edge. The problem is that we do not know which direction that edge points. That is why the notion of king cannot be deﬁned using simply the arrows of the category. Coronation depends crucially on the orientation of the edges of a graph and we do not have access to that orientation working in the category.

However, we know more about the objects of the category than simply how they are related by the arrows. We know that they are functors from Gop to SET. Using that fact we can get a complete description of any object in the category. For a given edge e with endpoints u and v of a graph G, if we want to know the orientation of e we need simply ask if G(s)(e) = u or G(s)(e) = v. While the category on its own does not have enough resources to uncover edge direction, the constructive method guarantees that the objects of the category have their own internal structure. Furthermore, we can use that internal structure to determine edge direction and so can ultimately distinguish between the possession of 1 king and the

op 13In SETG the object that picks out vertices is not the terminal object in the category and so p 6= 1.

49 possession of 3. Thus, this apparent problem of language turns out not to be a problem after all.

Categories of Undirected Graphs: A Problem of Language

So far we have only constructed categories of directed graphs and uncovered no instances of genuine inappropriateness. In what follows we will investigate a number of situations where the construcive method is inappropriate, beginning with categories of undirected graphs.

op We have to work with a diﬀerent domain category (the Cop in SETC ) to construct a category of undirected graphs. Problems arise because ”directed graphs are easier to describe than undirected graphs” (Bumby and Latch, 1986, p. 2). These new domain categories will produce undesired artifacts in the constructed category. This is a problem of language because the construction introduces additional properties that make it more diﬃcult to work with and describe undirected graphs.

To construct a category of undirected graphs the normal move is to consider them as directed graphs with edges in both directions (e.g. the Bumby and Latch (1986) construction). That is, every edge has a counterpart edge going in the opposite direction. To build our functor category we start with Uop, shown in ﬁgure 3.11. The role of r is to pair every edge with an

s r V E t

Figure 3.11: The category Uop edge going in the opposite direction, so not pictured in the diagram is that s ◦ r = t, t ◦ r = s and r ◦ r = 1E.

For objects G, H : Uop → SET and a natural transformation ψ : G → H we get a new

50 commutative diagram, shown in ﬁgure 3.12. This guarantees that the pairing between two

ψ G(E) E > H(E)

G(r) H(r)

∨ ψ ∨ G(E) E > H(E)

Figure 3.12: An undirected graph homomorphism

edges given by r is preserved by the morphisms of the category. Without this we would get more morphisms than the category should have, speciﬁcally we would include those morphisms that split undirected edges and recombine them in diﬀerent ways.

op The problem with SETU is how to represent the single loop undirected graph, shown in ﬁgure 3.13. If we are going to represent it as a directed graph with edge-reversal, we have

Figure 3.13: The single loop undirected graph two choices. In ﬁgure 3.14 we have L and in ﬁgure 3.15 we have B. The edges are labeled

e = r(e)

Figure 3.14: The graph with reversal L

to show exactly how r is working in both cases. We might be tempted to identify the single loop undirected graph with L and identify B with the undirected graph with a single vertex

op and two loops. However, in SETU there is no arrow from L to B. You cannot map the edge of L onto either edge of B without violating the commutative diagram given above for r. However, the single loop undirected graph should clearly have two maps to the double loop undirected graph.

51 e r(e)

Figure 3.15: The graph with reversal B

op The diﬀerence between L and B illustrates the fundamental problem with SETU . Undi- rected graphs normally have vertices, edges, and loops as a special kind of edge with only

op one endpoint. SETU introduces another special kind of edge called a band (Brown et al., 2008, p.1). Bands have very speciﬁc relationships to other edges. If we call every edge that

op is neither a loop or a band an arc then the homomorphisms in SETU obey the following rules:

• An arc can be mapped to an arc, a loop, or a band.

• A band can be mapped to a band or to a loop.

• A loop can only be mapped to a loop.

We ran into these constraints in our treatment of B and L. L has exactly one loop and B has no loops but does have one band. Bands are an undesired byproduct of the constructive method. They add extra complexity to any investigation of undirected graphs because they force us to be attentive to properties introduced by the construction in which we were not previously interested. As such, the constructive method description of undirected graphs is inappropriate due to an ungainliness of language.

op A defender of the constructive method may object that while SETU has these unneces-

op sary features we could consider a subcategory of SETU without them. Speciﬁcally, we

op could consider the subcategory of SETU containing only those undirected graphs without bands.14 However, this is to trade a problem of language for a problem of guidance. To claim that the subcategory is really the category of interest is to admit that the constructive method has directed us toward the wrong category.

14This category cannot be constructed because it is not a topos.

52 Even if that move is not made to defend the constructive method against the shortcomings

op of SETU , this problem of subcategories arises in other contexts. There are many categories of graphs that are not constructed (in our technical sense), but appear as subcategories of constructed categories. We now consider a particular instance of that general phenomena.

An Unconstructed Subcategory: A Problem of Guidance

Here we will investigate another genuine case of inappropriateness for the constructive method. It is inappropriate for the treatment of simple undirected loopless graphs.

First we will illustrate the important place simple undirected loopless graphs have in categorial graph theory and graph theory more generally. What are called coloring problems in graph theory are concerned speciﬁcally with properties of simple undirected loopless graphs. However Sgraph, the category of all small simple undirected loopless graphs, cannot be constructed because it is not a topos.15 Furthermore, we will see how the categories favored by the constructive method are not suitable for the investigation of coloring problems. This represents a failure of guidance for the constructive method, it guides away from some of the central problems of graph theory.

Simple undirected loopless graphs are undirected graphs with no parallel edges (simple) and no vertices that are self-adjacent (loopless). Sgraph is consequently the subcategory

op of SETU containing just those undirected graphs with no loops, bands, or parallel edges. This is particularly interesting because it reverses how diﬀerent types of graphs are normally introduced. Typically, simple undirected loopless graphs are referred to as ’graphs’ with parallel edges and loops introduced afterwards as increases in complexity (Harary, 1969, p. 9). The constructive method reverses this process because of ”a diﬀerence between the combinatorial and categorical viewpoints in graph theory” (Bumby and Latch, 1986, p.8):

15It does not possess a terminal object.

53 The categorical approach searches for general concepts which may be simply described and proceeds from there to particular or special examples while the combinatorial approach assumes that such things as multiple edges or loops will be complicated. (Bumby and Latch, 1986, p.8).

Categorial graph theory begins with the combinatorially more complex and works towards the combinatorially more simple. While this is an interesting change from the norm, it in itself is not problematic. In order to understand where the problem arises we need to understand coloring problems.

A coloring of a graph is an assignment of colors to vertices such that no two adjacent vertices have the same color. It is obviously impossible to color graphs with loops or bands because such graphs have vertices which are self-adjacent. Since additional parallel edges do not aﬀect the adjacency relation, simple graphs are the objects of interest with respect to coloring. For these reasons, graph coloring is concerned with simple undirected loopless graphs.

Every simple undirected loopless graph can be colored by assigning a distinct color to each vertex. The more interesting and historically important question is ﬁnding the minimum number of colors required. One of the most famous results in graph theory is the Four Color Theorem, which states that any graph that can be embedded in the plane without crossings can be colored using at most four colors (proved in Appel et al. (1977) and Appel et al. (1977)). Now, graph coloring has a natural description in terms of homomorphisms in Sgraph. A simple undirected loopless graph is n-colorable just in case there is a homomorphism from it to (the simple undirected version of) Kn. The close connections between colorings and homomorphisms are one of the reasons homomorphisms are introduced into graph theory in the ﬁrst place.

Homomorphisms play an important rule in attempts to make progress on Hedetniemi’s conjecture, an important coloring problem. Hedetniemi’s conjecture is that if G is n-colorable

54 and H is m-colorable, then G × H can be colored with min{n, m} colors Hedetniemi (1966). The introduction of categorial methods and focus on Sgraph have helped to make progress on the problem (for example see Plessas (2011)). While this represents a success of adopting certain categorial methods, it actually counts against the constructive method.

Use of the constructive method focuses attention on graphs that can have loops. As a result, papers which use the constructive method do not address coloring problems in graph theory. For example, Brown et al. (2008) admits that ”[t]he results of this article are not immediately relevant to colouring problems”. This is not a limitation of these papers, it is not a requirement that all research in graph theory relate to coloring problems. However, it is a problem of guidance for the constructive method. The constructive method directs research away from coloring problems because the categories where coloring problems are interesting and relevant are not constructed categories. Thus, while the constructive method is certainly useful for some research it is inappropriate because of this limitation with respect to guidance.

This problem is a direct result of the fact that Sgraph is not constructed. While the constructive method does not guide toward Sgraph, it at least has access to it as a subcategory of a constructed category.16 In this next section we will consider a category where even that access is in doubt.

An Unconstructed Category: Another Problem of Guidance

In this section we consider a third example of genuine inappropriateness for the constructive method: Cgraph, the category of all small simple graphs under continuous graph homomorphisms. First of all this category does not form a topos and so cannot be constructed(in

16This requires some principle for extracting subcategories but I see no reason to disallow such a principle in this context. Granting it certainly provides no help avoiding the problems of guidance.

55 our technical sense).17 Second, we can show that it is not even a subcategory of any of the natural constructed categories. Even so we will see the important role it plays in categorial graph theory. Thus, Cgraph is an example of an important category of graphs overlooked by the constructive method and so represents another failure of guidance.

One natural way to build categories of graphs is to fix a notion of graph and vary the notion of homomorphism.18 There are at least six distinct notions of morphisms between graphs (Knauer, 2011, p. 8). The homomorphisms in Cgraph are continuous graph morphisms, or comorphisms. Graph comorphisms are continuous in the sense that they require structure in their image to be present in the pre-image. Knauer (2011) defines them only for simple graphs. We will follow that practice to avoid the technical complications of extending the definition to all graphs. Specifically, let G, H be simple graphs. A comorphism f between G and H maps vertices to vertices and satisfies the following requirement for edges:

• if there is an e ∈ H(E) and x, y ∈ G(V ) such that s(e) = f(x) and t(e) = f(y), then there must be an e0 ∈ G(E) such that s(e0) = x and t(e0) = y.

This is the exact sense in which structure in the image must be present in the pre-image. Notice that if H is a non-empty discrete simple graph then there is at least one map from every simple graph G to H: simply collapse all the vertices of G to some vertex of H and since H is discrete the continuity requirement is trivially satisﬁed. This hints at some of the diﬃculties with recovering Cgraph as a subcategory.

The continuity requirement is what causes the diﬃculty with recovering Cgraph as a subcategory of a constructed category. Continuity is what gives discrete graphs so many incoming

arrows, more than discrete graphs have in Gop, Rop, or Uop. This means it cannot be a

17 To see this consider D2 the 2 vertex discrete graph. There is no way to distinguish between D2’s two 2 vertex subgraphs using an arrow out of D2. Thus, Cgraph does not have a subobject classifier. 18One of the first problems confronting those looking to apply categorial methods to graph theory was finding the ’right’ notion of homomorphism for graphs (Hell, 1979, p.123).

56 subcategory of any of those categories. Furthermore, the continuity requirement appears backwards for a natural transformation. If f : G → H is a comorphism, the continuity requirement is looking for a map from the edges of H in the image of f back to the edges of G. A natural transformation will always take elements of the domain and map them to the codomain, not vice versa.19

This reveals a certain inflexibility of the constructive method. The problem is that ”[i]f graphs are functors, then the appropriate definition of ’graph morphism’ should be a natural transformation between these functors” (Bumby and Latch, 1986, p.2). We do not have the freedom to fix the objects and vary the morphisms. Once functors are chosen as the objects the constructive method gives us the morphisms. At best we are able to look at subcategories with smaller collections of morphisms. However, that misses categories like Cgraph that are not subcategories. In addition, we have already seen in the previous section how a turn to subcategories represents a failure of guidance for the constructive method.

Finally, Cgraph is important from the categorial perspective because it allows for a natural categorial formulation of a number of graph theoretic operations. Cgraph has products, as well as what are called coproducts and tensor products. These operations in Cgraph correspond to the traditional graph-theoretic operations of disjunction, join, and complete product (Knauer, 2011, p. 80). Thus, the use of comorphisms allows those constructions to be easily described using the language of category theory in such a way that tells us a lot about them.

Here again we have a category of important mathematical interest that does not appear to be captured by the constructive method. It is even worse than the previous section because Cgraph may not even appear as a subcategory of one of the constructed categories. It would be a mistake to look to the constructive method for guidance concerning which categories

19Neither of these limitations guarantee that Cgraph cannot be extracted as a subcategory of some constructed category. Clever choice of domain category or further elaborations on the constructive method may be able to get around these diﬃculties.

57 are of mathematical interest because there are mathematically important categories (like Cgraph) that the method seems to miss entirely.20

Constructive Method: A Review

We are now in a position to take stock of the constructive method. There are certain areas of graph theory where the constructive method is fruitfully employed such as the investigation of directed graphs under the standard graphs morphisms. However, there are a number of areas where the constructive method is inappropriate. It is inappropriate for undirected graphs because construction introduces artifacts that make the language ungainly. It is inappropriate for coloring problems because it guides away from the relevant categories. It is inappropriate for treatment of continuous graph morphisms (and other variant morphisms) because it misses the relevant category entirely. Thus, the constructive method is ultimately incapable of providing conceptual uniﬁcation.

3.1.3 Axiomatic Method

The axiomatic method is to introduce categories of graphs by giving an axiomatization of them. This approach is by far the least common in the literature, Plessas (2011) being its most notable example. It has the advantage of using categorial language throughout the entire method. It has the disadvantages of providing eﬀectively no guidance as to which are the categories of interest and requiring the individual veriﬁcation of properties.

The axiomatic method at least allows for conceptual uniﬁcation along the aspect of language. The language of category theory is used for the axiomatization and so any advantages of the use of that language are available. This does not guarantee appropriateness as that question

20This is not a failure of categorial methods to provide coherency assurance because categorial methods have more resources than the constructive method.

58 requires a further investigation of the use of categorial language for work in graph theory (something pursued in the following section).

The axiomatic method is inappropriate with respect to guidance and shares important limitations with the opportunistic method. The only requirement it places on categories is that they can be axiomatized using categorial language, which is a very weak requirement. As a result, the axiomatic method itself does not provide any real guidance as to which categories are the categories of interest.21 Furthermore, one has to decide if an axiomatization is the correct axiomatization of a given category of graphs. In order to do so one has to individually verify the properties the category purportedly possesses in much the same way as is required under the opportunistic method. Thus, the axiomatic method allows for the use of categorial language but also has serious limitations.

3.1.4 Conclusion

We have considered the three methods used for the introduction of categories of graphs: opportunistic, constructive, and axiomatic. The opportunistic method does not provide any conceptual uniﬁcation because the language and methods of category theory are only employed in a patchwork way. The constructive method is useful in some parts of graph theory but inappropriate in others. Its primary failing was providing incorrect guidance for category selection. The axiomatic method leaves the door open for the use of categorial language but does not provide guidance for category selection. Thus, none of the methods for the introduction of categories are able to provide good guidance among the vast universe of categories as to which are the interesting categories to pursue and which are not. If categorial methods are going to provide any guidance it must come from elsewhere.

21The use of categorial language itself may provide some guidance, but that is a question of the appropriateness of categorial language and not a question of the appropriateness of the axiomatic method.

59 3.2 Graph Operations

This chapter explores situations where categorial methods are inappropriate. Section 1 showed that all of the categorial methods used for the introduction of categories are inappropriate. This section will show that categorial methods are also inappropriate for two graph operations of interest.

We are concerned here with capturing graph operations in a categorial setting. We have already seen some examples of this, such as the product or exponential. The focus of this section will be on two operations: the lexicographic product and corona. According to Knauer (2011) these operations ”do not have a categorical description” (Knauer, 2011, p.81). If Knauer is correct, then the lexicographic product and corona will show the inappropriateness of category theory due to a limitation of categorial language. The rest of this section is devoted to determining whether or not Knauer is correct.

Before we can investigate them in a categorial context we need to know what the two operations are. They are both binary operations that take two input graphs and return a

new graph built out of the inputs. For graphs G1 and G2, given in ﬁgures 3.16 and 3.17

respectively: ”Take the ﬁrst graph G1, pump up its vertices and insert the second graph

Figure 3.16: Lexicographic product example: graph G1

Figure 3.17: Lexicographic product example: graph G2

G2 in each vertex” (as in ﬁgure 3.18)(Knauer, 2011, p. 82). ”An edge between two vertices of G1 then means that each vertex of G2 inside the one pumped-up vertex of G1 is adjacent to every vertex inside the other pumped-up vertex of G1”(as in ﬁgure 3.19)(Knauer, 2011,

60 Figure 3.18: Lexicographic product example: graph G1 with G2 inserted at each vertex

p. 82). That is how to construct the lexicographic product of G1 and G2, written G1[G2].

Figure 3.19: Lexicographic product example: graph G1[G2]

Multiple edges between vertices of G1 lead to multiple edges between the vertices inside the pumped up vertices. The lexicographic product can be seen as a generalization of the lexicographic ordering from orderings to arbitrary graphs.

You construct the corona of graphs G1 and G2, given in ﬁgures 3.20 and 3.21 respectively:

Figure 3.20: Corona example: graph G1

”by taking one copy of G1 (which has p1 points[vertices]) and p1 copies of G2,” (as in ﬁgure

3.22) ”and then joining the ith point of G1 to every point in the ith copy of G2” (as in ﬁgure

3.23) (Harary, 1969, p. 167-168). That is how to construct the corona of G1 and G2, which

22 we write G1 /G2. One can think of it as putting a copy of G2 above each vertex of G1.

Now that we have descriptions of the lexicographic product and corona we can investigate how they might be described or constructed in diﬀerent contexts. To provide some contrast we will consider how to build them in ZFC. After that we will consider a categorial

22There is some freedom in the deﬁnition. Since the edges are directed we could have instead put a copy of G2 ’below’ each vertex of G1. None of the results will turn on this choice.

61 Figure 3.21: Corona example: graph G2

Figure 3.22: Corona example: one copy of G1 and two copies of G2

Figure 3.23: Corona example: graph G1 /G2 construction.

3.2.1 Construction in ZFC

In ZFC a graph can be represented as an ordered quadruple hVG,EG, sG, tGi, where sG and

tG are functions from EG to VG. If we have two graphs G and H then their lexicographic product will be the following quadruple:

• VG[H] = VG × VH .

• EG[H] = (VG × EH ) t (EG × VH × VH )

• sG[H](x) =

– hx1, sH (x2)i for x ∈ (VG × EH )

– hsG(x1), x2i for x ∈ (EG × VH × VH ).

• tG[H](x) =

– hx1, tH (x2)i for x ∈ (VG × EH )

62 – htG(x1), x3i for x ∈ (EG × VH × VH ).

That is the goal, we want to see how to actually build that working in ZFC. For this discussion I will assume the reader is familiar with the basic axioms of ZFC, a complete description can be found in Enderton (1977).

First, we need to make sense of ordered tuples. We will use the Kuratowski definition so ha, bi = {{a}, {a, b}}. An ordered quadruple will then be an ordered pair of ordered pairs, and an ordered tuple is an ordered pair of a set and an ordered pair. In order to create an ordered pair from its two components requires 3 applications of the pairing axiom, {a} from a paired with a, {a, b} from a paired with b, and {{a}, {a, b}} from {a} paired with {a, b}. To recover the first element of an ordered pair requires 2 uses of Union and 1 of Separation. Union gives {a, b}, then do Separation on that to get {x ∈ {a, b}|{x} ∈ {{a}, {a, b}}} = {a}. Finally, Union on {a} gives a. These steps generalize to ordered quadruples and ordered triples based on their respective definitions.

We need to be able to take unions of sets, Cartesian products of sets, and disjoint unions of sets. For two sets a, b we get a ∪ b from 1 use of Pairing and 1 use Union, get {a, b} from a paired with b and then a ∪ b from Union applied to {a, b}. The Cartesian product is more involved and requires 2 uses of Powerset and 1 use of Separation, but starts with a ∪ b. Starting with a and b, produce a ∪ b as above, and then apply Powerset to it twice to get PP (a ∪ b). Then apply separation to get {x ∈ PP (a ∪ b)|∃x1∃x2(x1 ∈ a ∧ x2 ∈ b ∧ x =

{{x1}, {x1, x2}})} = {hx1, x2i|x1 ∈ a ∧ x2 ∈ b} = a × b. Disjoint unions are very similar to normal unions, but require 2 uses of Replacement in addition to what unions require. For two sets a, b, do replacement on both sets to get a0 = {hx, bi|x ∈ a} and b0 = {hx, ai|x ∈ b}, then a t b = a0 ∪ b0.

Functions are subsets of Cartesian products. So a function f from a to b is a subset of a × b. Thus, in general a function is product from a single application of Separation on a Cartesian

63 product.

Putting this all together to make G[H] proceeds as follows. Starting with ordered quadruples

hVG,EG, sG, tGi and hVH ,EH , sH , tH i, use instances of Union and Separation to get each of

the eight components individually. Then build VG[H] = VG × VH following the process

given above for Cartesian products. Build EG[H] = (VG × EH ) t (EG × VH × VH ) using the processes given for Cartesian products and for disjoint unions of sets. This allows us to build VG[H] ×EG[H], to which we can apply Separation to build sG[H](x) and tG[H](x). Finally,

use multiple instances of Pairing to construct hVG[H],EG[H], sG[H], tG[H]i out of the individual components. That is how to build the lexicographic product working in ZFC.

The corona is constructed in a similar fashion. We will not go into the speciﬁcs, as they are not fundamentally diﬀerent. Instead, we will simply note that for graphs G and H the corona in ZFC will be the following quadruple:

• VG/H = VG t (VG × VH ).

• EG/H = EG t (VG × VH ) t (VG × EH )

• sG/H =

– sG(x) for x ∈ EG

– x1 for x ∈ (VG × VH )

– hx1, sH (x2)i for x ∈ (VG × EH ).

• tG/H (x) =

– tG(x) for x ∈ EG

– x for x ∈ (VG × VH )

– hx1, tH (x2)i for x ∈ (VG × EH ).

64 Now we have a relatively clear idea of how to produce both the lexicographic product and the corona working in ZFC. The process is not particularly complicated, but does involve many steps. From the perspective of inappropriateness, they would certainly count as clumsy definitions and potentially artificial as well. It is certainly much more convoluted than the description in the preceding section. This simply reinforces what we already knew, set theory is not appropriate as a conceptual unification for mathematics. We shall now look at how the same process goes using categorial methods.

3.2.2 Categorial Construction

For the categorial construction we encounter a slight dilemma. On the one hand, we would like to build the lexicographic product and corona for arbitrary pairs of graphs. So we should

op work in Graph = SETG . On the other hand, working in Graph is inconvenient because the product and exponential of Graph represent some of the less useful graph-theoretic

op constructions. We would prefer to work in Rgraph = SETR . We get around this dilemma using the following pair of functors. We begin in Graph and use the functor that turns graphs into r-graphs by adding a distinguished loop at every vertex.23 Then we do the constructions in Rgraph. Once the constructions are complete we map what we have built back into Graph using the functor that deletes the distinguished loops from every vertex of an r-graph to turn it into a graph.24 Thus, we can safely work in Rgraph and still produce results that extend to Graph.

Both constructions will follow the same general structure. Working in Rgraph we will ﬁrst represent the graph we are attempting to construct, call it G, as a pair of r-graphs, where

23This functor was introduced in Section 1.2.2. 24Strictly speaking this second functor is actually from the subcategory of Rgraph where homomorphisms do not map anything onto distinguished loops except for distinguished loops. However, that subcategory does contain all the objects of Rgraph and since we are only concerned with the objects and not the morphisms this detour causes no problems.

65 25 EG is discrete, with two homomorphisms between them, pictured in ﬁgure 3.24. Here the

αG - E V G ωG - G

Figure 3.24: Representing a graph in Rgraph

vertices of VG represent the vertices of G and the vertices of EG represent the edges of G, while α and ω give us the sources and targets for those edges, respectively. If we already had a graph H we could construct this representation using |HE| for its edges and H for its

26 s t E vertices. For the arrows, αH = H ◦ i and ωH = H ◦ i where i is the inclusion of |H | into HE.

Once we have this representation we consider EG ×E, which is the collection of edges for our desired G except that they are entirely disconnected from one another. The ﬁnal step will be to glue together those edges by associating their endpoints with one other in the appropriate way. In order to do all of this we need to introduce a few new pieces of categorial machinery.

Coproduct & Pushout

We need two new binary operations, the coproduct of two r-graphs and the pushout of two r-graph morphisms. The coproduct is the dual of the product. This means the deﬁnition of the coproduct comes from the deﬁnition of the product by reversing all of the arrows. So, for a pair of r-graphs H and G the coproduct is an r-graph, which we label G + H, and a

pair of injection arrows i1 : G → G + H and i2 : H → G + H. They are such that for any h graph T with arrows h : G → T and k : H → T there is a unique arrow k : G + H → T which makes the diagram in ﬁgure 3.25 commute. In the case of r-graphs the coproduct is

25 It is not necessary for VG to be discrete and we take advantage of that freedom in the subsequent example to set VH = H. 26Recall from Chapter 2 that |G| is the discrete graph underlying G.

66 T< >∧ k h h k i i G 1 > G + H < 2 H

Figure 3.25: An example coproduct

just the disjoint union of the two graphs. As with products we can introduce special arrows between coproducts. Take r-graphs G, G0,H,H0 with arrows f : G → H and f 0 : G0 → H0.

i1◦f 0 0 0 We write the unique coproduct arrow 0 : G + H → G + H as f + f . i2◦f

The pushout requires two arrows with common domain as input. Given r-graph morphisms h : A → H and k : A → K the pushout of h and k is an r-graph P and a pair of arrows

i1 : H → P , i2 : K → P such that i1 ◦ h = i2 ◦ k. In addition, for any pair m : H → T , n : K → T such that m ◦ h = n ◦ k there is a unique u : P → T such that the diagram in ﬁgure 3.26 commutes. In the case of r-graphs is what is called the amalgam of the two

k A > K

h i2

∨ i ∨ H 1 > P n

m u

> >> T

Figure 3.26: An example pushout

graphs H and K and involves glueing them together by associating the image of h in H with the image of k in K. This gives us all the machinery we need to do the constructions.

67 General Glueing

We begin be taking an arbitrary diagram of the form given in ﬁgure 3.27 and show how to construct an r-graph from it. As mentioned above, EG × E gives us the edges that we need

αG - E V G ωG - G

Figure 3.27: A graph represented in Rgraph to glue together. We will use a pushout to glue the edges together. However, in order to do so we must ﬁrst create the correct arrows to use in the pushout. First, we need to do the pair of comprehensions given in ﬁgure 3.28. There i is the inclusion of |VG| into VG and

i × αG χ∆ |VG| × EG > VG × VG > Ω

i × ωG χ∆ |VG| × EG > VG × VG > Ω

Figure 3.28: Two important comprehensions

χ∆ is the classifying arrow for ∆ : VG × VG → Ω. This comprehension gives us the two new subobjects of |VG| × EG shown in ﬁgure 3.29. S is a collection of pairs whose ﬁrst element

sG S > > |VG| × EG

tG T > > |VG| × EG

Figure 3.29: Two new subobjects is a vertex and whose second element is an ’edge’ with that vertex as a ’source’ and T is a collection of pairs whose ﬁrst element is a vertex and whose second element is an ’edge’

68 with that vertex as a ’target’. The quotes around the terms are to denote that these are not really edges or sources or targets, but represent them in our ﬁrst diagram.

With these two subobjects we can deﬁne the following three arrows shown in ﬁgure 3.30.

There p1, p2 are the projection arrows from the product andu ˆ : EG → EG × 1 is the unitary

sG tG p1 S + T > |VG| × EG > |VG|

sG p2 uˆ 1EG × s S > > |VG| × EG > EG > EG × 1 > EG × E

tG p2 uˆ 1EG × t T > > |VG| × EG > EG > EG × 1 > EG × E

Figure 3.30: Three special arrows

isomorphism. Call the second and third arrows h : S → EG × E and k : T → EG × E. Using h those two arrows and the coproduct we can get k : S + T → EG × E.

We now have two arrows with the same domain and can deﬁne a pushout using them. This pushout is given in ﬁgure 3.31. G is the r-graph we were trying to construct. It has a

h k S + T > EG × E

sG p1 ◦ i2 tG ∨ ∨ i1 |VG| > G

Figure 3.31: Glueing completed

vertex for every vertex in VG and it has a non-distinguished loop edge for every vertex in

EG. Furthermore, the sources and targets of those edges agree with αG and ωG. As we did in the ZFC case we will gloss over the veriﬁcation process and simply claim that we have produced the correct thing.

69 Representations of the Lexicographic Product & Corona

In order to construct G[H] and G/H we have to get representations of the form given in ﬁgure 3.32.

α α G[H-] G/H- E ω V EG/H ω VG/H G[H] G[H-] G[H] G/H-

Figure 3.32: Representations of the lexicographic product and corona in Rgraph

E s Recall that for an arbitrary r-graph G we can get VG,EG, αG and ωG as G, |G |,G ◦ i and Gt ◦i respectively. We use these conventions to deﬁne the representations of G[H] and G/H. For G[H] the following objects and arrows give the desired representation:

• VG[H] = VG × VH

• EG[H] = (VG × EH ) + (EG × (VH × VH ))

• αG[H] : EG[H] → VG[H]

1V ×αH G :(VG × EH ) + (EG × (VH × VH )) → VG × VH αG×p1

• ωG[H] : EG[H] → VG[H]

1V ×ωH G :(VG × EH ) + (EG × (VH × VH )) → VG × VH ωG×p2

where p1, p2 are the appropriate project arrows from (VH × VH ). For G/H the following objects and arrows give the desired representation:

• VG/H = VG + (VG × VH )

• EG/H = EG + ((VG × VH ) + (VG × EH ))

70 • αG/H : EG/H → VG/H

αG ( + p1 × αH ) ◦ u˜ where p1

αG + p1 × αH :(EG + (VG × VH )) + (VG × EH ) → VG × VH and p1

u˜ : EG + ((VG × VH ) + (VG × EH )) → (EG + (VG × VH )) + (VG × EH ) is the associativity arrow.

• ωG/H : EG/H → VG/H

1V ×V ωG + G H : EG + ((VG × VH ) + (VG × EH )) → VG + (VG × VH ) p1×ωH

where p1 stands for both the projection arrow from (VG × VH ) or the projection arrow from

(VG × EH ), depending on context.

These two representations, along with the general glueing construction, allow us to construct the lexicographic product and corona in Rgraph. Then we simply map back into Graph to get both in their most general form. Thus, it would appear that Knauer is incorrect because there is a description of both operations in a categorial context. However, that description is relatively complicated and involves many steps.27 In terms of inappropriateness, it certainly counts as a clumsy definition, since it would not be at all easy to work with. Is it more clumsy or artificial than the construction in ZFC? That is difficult to say, though also not essential to our purpose. The characterizations we get here are certainly much clumsier than the intuitive ideas we used to introduce the lexicographic product and corona. Thus it appears that both ZFC and category theory (through Graph and Rgraph) are inappropriate because of their treatment of these two operations.

However, we may have been too quick to count this example against category theory. We did not take any guidance from category theory as to the method of these constructions. Certainly we are correct that if we build the lexicographic product and corona in the above

27In fact the above work does not actually complete the process. In order to raise the velvet curtain and reveal the completed lexicographic product and corona we would need to verify that the construction has produced the correct graphs. That is omitted in the interest of space.

71 ways we get a description that is inappropriate. However, there may be other categorial methods that we can use to describe them that is cleaner and easier to use. That is, category theory may suggest an alternative method for describing exactly the same operations but that does so in a more appropriate way.

3.2.3 Searching for Guidance

When looking to characterize operations on objects in a category, the most prominent recom- mendation is to search for adjunctions. Adjunctions (and the adjoint functors which make them up) represent a central part of categorial reasoning. It is diﬃcult to overstate the amount of importance many authors place on adjunctions. According to R.J. Wood: ”To some, including this writer, adjunction is the most important concept in category theory” (Wood, 2004, p.6). They are sometimes put on the same footing as categories themselves (Mac Lane, 1968, 290). Furthermore, ”[i]t is now widely recognized that adjoint functors characterize the structures that have importance and universality in mathematics” (Eller- man, 2006, p.128). Without evaluating the truth of his claim, it is clear that Ellerman thinks adjunctions are the pinnacle of categorial reasoning and an essential part of the applications of category theory.

The importance of adjuncts is perhaps nowhere better represented than in the introduction to the chapter on adjunctions in Awodey (2009):

This chapter represents the high point of this book, the goal toward which we have been working steadily. The notion of adjoint functor, ﬁrst discovered by D. Kan in the 1950s, applies to everything that we have learned up to now to unify and subsume all of the diﬀerent universal mapping properties that we have encountered, from free groups to limits to exponentials. But more importantly, it also captures an important mathematical phenomena that is invisible without

72 the lens of category theory. Indeed, I will make the admittedly provocative claim that adjointness is a concept of fundamental logical and mathematical importance that is not captured elsewhere in mathematics.

Many of the most striking applications of category theory involve adjoints, and many important and fundamental mathematical notions are instances of adjoint functors. As such, they share the common behavior and formal properties of all adjoints, and in many cases this fact alone accounts for all of their essential features. (Awodey, 2009, p.207)

Adjunctions are a crucial component that uniﬁes large swaths of categorial reasoning. In addition, they allow category theory to unify large swaths of mathematical reasoning more generally.

Adjunctions are considered both very important and very useful. Thus, if we are to adopt the view that the growth of mathematics should be guided by various categorial slogans, ’investigate and look for adjunctions’ appears to clearly be one of those slogans. For this reason, we will now attempt to capture the lexicographic product and corona with adjunctions. It may have been our failure to think of them using adjunctions that is responsible for the inappropriateness of the previous section. Perhaps adjunctions can provide a more appropriate description of the two operations. Of course in order to investigate this question we must ﬁrst understand adjunctions.

3.2.4 Adjunctions

An adjunction is a pair of mutually composable functors satisfying certain properties. The

functors in an adjunction are called adjoint functors. Consider a pair of functors F : A → B, G : B → A, and let A be an object of category A and B an object of category B. An

73 adjunction provides a bijective correspondence between arrows f : F (A) → B of B and arrows g : A → G(B) of A. That is, for any arrow of the form f : F (A) → B, we can associate it with exactly one arrow of the form g : A → G(B).

This correspondence is established by a natural transformation called the counit or a natural transformation called the unit. The counit, : F ◦ G → 1B, has an arrow B : F (G(B)) → B for every object in B. Furthermore, B has the property that for any B-arrow of the form f : F (A) → B for some object A of A, there is a unique A-arrow g : A → G(B) that makes the diagram in ﬁgure 3.33 commute.

G(B) F (G(B)) B > B ∧ ∧ > g F (g) f

A F (A)

Figure 3.33: An example counit

The unit functions in a similar fashion. It is written η and has an arrow ηA : A → G(F (A)) for every object in A. Furthermore, ηA has the property that for any A-arrow of the form g : G(B) → A for some object B of B, there is a unique B-arrow f : F (A) → B that makes the diagram in ﬁgure 3.34 commute.

η A A> G(F (A)) F (A)

g G(f) f > ∨ ∨ G(B) B

Figure 3.34: An example unit

74 Both the unit and the counit are individually suﬃcient to establish the correspondence. The existence of one uniquely guarantees the existence of the other. That is, given two functors and a counit there is exactly one corresponding unit and vice versa. We write F a G to represent an adjunction. In such a situation, we call F the left adjoint, G the right adjoint and both of them adjoint functors.

Many of the operations we have already considered are adjoint functors. For example, the product of two graphs and the exponential are both adjoints. However, that merely scratches the surface. There are a whole host of other operations that feature in adjunctions. Here what we want to know is if the lexicographic product or corona are adjoint functors.

3.2.5 Functoriality

For either operation to be an adjoint functor and so parts of an adjunction, they must ﬁrst at least be functors. We begin by making sure this is the case. For this work we will be considering all graphs and so, as with the earlier constructions, our work will center around

op Graph = SETG .

Functoriality of the Lexicographic Product

The lexicographic product is a binary operation on graphs. As such, as a functor it would have domain Graph × Graph and codomain Graph. We need to determine its action on objects and arrows of Graph × Graph and make sure that it preserves identity and composition. We will write the functor [ ] and deﬁne it as follows:

• For an object hG, Hi of Graph × Graph, [ ](hG, Hi) = G[H].

• For an arrow hf, gi : hG, Hi → hI,Ji, let [ ](hf, gi) = f[g]. Here the natural transfor-

75 mation f[g] is deﬁned by its vertex and edge components:

– (f[g])V :

Takes hx, yi of V (G) × V (H) to hfV (x), gV (y)i of V (I) × V (J).

– (f[g])E :

0 0 0 For he, hy, y ii ∈ E(G)×(V (H)×V (H)), (f[g])E(he, hy, y ii) = hfE(e), hgV (y), gV (y )ii

0 0 0 and for hx, e i ∈ V (G) × E(H), (f[g])E(hx, e i) = hfV (x), gE(e )i.

That f[g] is a natural transformation and so a graph homomorphism is straightforward to verify. Furthermore, since the action of f[g] proceeds pointwise it is also straightforward to recognize that [ ] preserves identity and composition. Hence, [ ] is in fact a functor.

Functoriality of the Corona

Just like the lexicographic product, the corona is a binary operation on graphs. So it too will have domain Graph × Graph and codomain Graph. Once we deﬁne its action on the objects and arrows of Graph×Graph, it will clearly preserve identity and composition just like [ ]. For the corona, we will write / to denote the functor, and deﬁne it as follows:

• For an object hG, Hi of Graph × Graph, / (hG, Hi) = G/H.

• For an arrow hf, gi : hG, Hi → hI,Ji, let / (hf, gi) = f / g. Here the natural transformation f / g is deﬁned by its vertex and edge components:

– (f / g)V :

For x ∈ V (G), (f / g)V (x) = fV (x)

and for hx, yi ∈ V (G) × V (H), (f / g)V (hx, yi) = hfV (x), gV (y)i.

– (f / g)E :

For e ∈ E(G), (f / g)E(e) = fE(e),

76 for hx, yi ∈ V (G) × V (H), (f / g)E(hx, yi) = hfV (x), gV (y)i

0 0 0 and for hx, e i ∈ V (G) × E(H), (f / g)E(hx, e i) = hfV (x), gE(e )i.

That f / g is a graph homomorphism is straightforward to verify. Also, the action of f / g proceeds pointwise, just like f[g], and so it is similarly easy to recognize that / preserves identity and composition. Hence, / is in fact a functor.

Thus, we were able to easily show that both the lexicographic product and corona are at the very least functors. However, we will now see that neither of them is an adjoint functor.

3.2.6 Adjoint Failure

We will consider both the lexicographic product and the corona and show that neither of them is a left adjoint nor a right adjoint and so neither of them is an adjoint functor.

In order to show that [ ] is not a right adjoint, it is enough to show that it does not preserve products (by Exercise 10.11 from McLarty (1995)). Thus, we want graphs A, B, C, D such

that [ ](hA, Ci × hB,Di) ∼= A × B[C × D] 6∼= A[C] × B[D]. Consider the following

A = D = and B = C =

In this situation,

A × B[C × D] =

and

A[C] × B[D] =

These two graphs are clearly not isomorphic and so [ ] does not preserve products and is consequently not a right adjoint.

77 In order to show that [ ] is not a left adjoint, it is enough to show that it does not preserve coproducts (again by Exercise 10.11 from McLarty (1995)). Thus, we want graphs A, B, C, D such that [ ](hA, Ci + hB,Di) ∼= A + B[C + D] 6∼= A[C] + B[D]. Consider the following

A = B = C = D =

In this situation,

A + B[C + D] =

and

A[C] + B[D] =

These two graphs are clearly not isomorphic and so [ ] does not preserve coproducts and is consequently not a left adjoint. Thus, we have shown that the lexicographic product is not a right nor left adjoint and so is not an adjoint functor. Now we do something similar for the corona.

In order to show that / is not a right adjoint, it is enough to show that it does not preserve

products. Thus, we want graphs A, B, C, D such that / (hA, Ci×hB,Di) ∼= A×B/C ×D 6∼= A/C × B/D. Consider the following28

A = B = D = and C =

In this situation,

A × B/C × D =

and

A/C × B/D =

28Here C is the empty graph.

78 These two graphs are clearly not isomorphic and so / does not preserve products and is consequently not a right adjoint.

In order to show that / is not a left adjoint, it is enough to show that it does not

preserve coproducts. Thus, we want graphs A, B, C, D such that / (hA, Ci + hB,Di) ∼= A + B/C + D 6∼= A/C + B/D. Consider the following

A = D = and B = C =

In this situation,

A + B/C + D =

and

A/C + B/D = These two graphs are clearly not isomorphic and so / does not preserve coproducts and is consequently not a left adjoint.

Thus, we have shown that / is not a left nor a right adjoint and so is not an adjoint functor.

Even though we have shown that neither operation is an adjoint functor there is more to

be done. Consider some category C with products and exponentials. In that situation × A a A for a fixed object A of C. Normally, is not an adjoint functor, but if we fix one of its arguments we get an adjoint functor. In a similar fashion, we can investigate the functors [G], G[ ], /G, and G/ for a fixed G to see if they are adjoint functors. We know they are at least functors because they are generated by fixing an argument in a functor. Also, because neither operation we are considering is a commutative operation [G] and G[ ] are going to be distinct functors, as are /G and G/ .29

29In the product case we do not investigate A × independently from × A because the two functors are naturally isomorphic.

79 3.2.7 Fixed Inputs

We begin by ﬁxing the left input to each functor.

G[ ]

Whether or not G[ ] has a left or right adjoint turns out to depend upon the properties of the graph G. Consequently, we have to break up the discussion into distinct cases.

The two simplest cases are if G = ∅ or G = . In the former case, ∅[ ] takes every graph to the empty graph and so behaves identically to the functor × ∅. That behavior is well known, but unhelpful for the study of the lexicographic product. In the latter case, G[ ] behaves exactly like the identity functor and so is similarly uninteresting.

In order to discuss the further cases, some additional terminology needs to be introduced. We are going to want to refer to the number of vertices and the number of edges in a given

graph. For the graph G, we will use vG to refer to the number of vertices and eG to the number of edges.

After the empty and singleton graphs, the next level of complexity is discrete graphs. Assume

that G is discrete. Then for any graph H, G[H] is vg copies of H with no edges between any

vG of the copies. This makes it the same as the vG-ary coproduct of H with itself. Let 4 :

vG vG GRAPH → GRAPH be the vG-ary diagonal functor, +vG : GRAPH → GRAPH

vG be the vG-ary coproduct functor and ×vG : GRAPH → GRAPH be the vG-ary product

30 vG vG vG functor. Then G[ ] is the same as +vG ◦ 4 . In addition, since +vG a 4 and 4 a ×vG

vG vG 31 we know that +vG ◦ 4 a ×vG ◦ 4 . Thus, if G is discrete it has a right adjoint.

30 This is on the assumption that we can take vG indexed products of categories. 31 It most likely does not have a left adjoint because +vG does not, but this has not been conﬁrmed.

80 The final case to consider is when G is not discrete.32 This would complete our treatment of all graphs. However, here we will also require that G be finite, as the problem gets significantly more complex in the infinite case. We begin with some simple observations. For two finite graphs, G and H we can easily compute the number of vertices and edges in their product, coproduct, and lexicographic product:

• vG×H = vG × vH and eG×H = eG × eH .

• vG+H = vG + vH and eG+H = eG + eH .

2 • vG[H] = vG × vH and eG[H] = vG × eH + eG × (vH ) .

Now, let I,J be two non-discrete ﬁnite graphs. Then: 2 2 eG[I]×G[J] = (vGeI + eGvI )(vGeJ + eGvJ )

2 2 2 2 2 2 = vGeI eJ + eGvI vJ + vGeg(eI vJ + eJ vI )

2 2 > vGeI eJ + eGvI vJ

= eG[I×J] The strict inequality is due to the fact that all of the individual values are greater than 0.

Thus, we know that G[I × J] 6∼= G[I] × G[J] and so G[ ] is not a right adjoint because it does not preserve products.

Along a similar vein: 2 eG[I+J] = vG(eI + eJ ) + eG(vI + vJ )

2 2 = vGeI + eGvI + vGeJ + eGvJ + 2eGvI vJ

= eG[I]+G[J] + 2eGvI vJ ∼ Thus, since eGvI vJ 6= 0 we know that G[I + J] 6= G[I] + G[J] and so G[ ] is not a left adjoint because it does not preserve coproducts. Hence, in the most general case (assuming ﬁnitude), G[ ] is not an adjoint functor. This means we are really no better oﬀ in this case than we were in the general case of [ ].

32Notice that this means G has at least 1 vertex.

81 3.2.8 G/

The situation for G/ is much simpler than it was for G[ ]. Here we have only two possibilities to consider, G = ∅ and G 6= ∅. In the former case, we have yet another functor that takes every graph to the empty graph. As a result, we can lump ∅ / together with ∅[ ] and × ∅ as being eﬀectively the same.

In the nonempty case, we restrict ourselves again to the ﬁnite and work with vertex and edge formulas. For the corona of two ﬁnite graphs G and H:

• vG/H = vG + vGvH and eG/H = eG + vGvH + vGeH .

Now, let I,J be two non-discrete ﬁnite graphs. Then:

vG/I×G/J = (vG + vGvI )(vG + vGvJ )

2 = vG(vG + vGvI vJ ) + vG(vI + vJ )

2 = vGvG/I×J + vG(vI + vJ ) ∼ Since all individual values are non-zero, vG/I×G/J 6= vG/I×J . Thus, we know that G/I × J 6= G/I × G[J] and so G/ is not a right adjoint because it does not preserve products.

Along a similar vein:

vG/I+G/J = vG + vGvI + vG + vGvJ

= vG/I+J + vG ∼ Thus, since vG 6= 0 we know that G/I + J 6= G/I + G/J. So, G/ is not a left adjoint because it does not preserve coproducts. Hence, in the most general case (assuming ﬁnitude), G/ is not an adjoint functor. As in the case of the lexicographic product, ﬁxing the right argument has not meaningfully improved our position with respect to adjointness.

Fixing the left input did not lead to any meaningful adjoint functors for the lexicographic product or the corona. However, ﬁxing the right input does.

82 3.2.9 [G]

Fixing the right input of the lexicographic product will split into two cases. On the one hand, we will see that for nearly all Gs it is not a right adjoint. On the other hand, it is a left adjoint to what we will call the lexicographic exponential, written ( )[G].

To show that it is not a right adjoint, we pursue a line of reasoning similar to that used for G[ ]. We restrict our attention to ﬁnite graphs and consider the relevant vertex and edge formulas. If G is the empty graph or the single vertex discrete graph, then it just like the × ∅ or identity functors, respectively. We have already addressed both cases. Otherwise,

[G] does not preserve products because for non-discrete graphs I and J, I ×J[G] 6∼= (I[G])× (J[G]). To see this, consider the vertex formulas:

v(I×J[G]) = vI vJ vG

2 v(I[G])×(J[G]) = vI vJ vG Hence, if G has at least two vertices, the two graphs cannot be isomorphic. If G has one vertex and at least one edge, then it is the edge formula that shows us it does not preserve products:

2 2 e(I[G])×(J[G]) = (vI eG + eI vG)(vJ eG + eJ vG)

2 2 2 2 = vI vJ eG + eI eJ vG + vI eJ eGvG + eI vJ eGvG

2 > vI vJ eG + eI eJ vG

= e(I×J[G]) The inequality requires G to have at least one edge. Thus, in all non-trivial cases [G] is not a right adjoint.

In order to illustrate that [G] is a left adjoint, we have to introduce the lexicographic exponential functor ( )[G] : GRAPH → GRAPH. In order to understand it completely, s t we need four special constants: the graphs E = and K1 = , and the two graph

homomorphisms between them. s : K1 → E maps the lone vertex of K1 to the vertex s of

83 E and t : K1 → E does likewise for the vertex t.

With these we can deﬁne ( )[G] as follows:

• For an object H of GRAPH,( )[G](H) = (H)[G], which is deﬁned as follows:

[G] – V ((H) ) = {All Homomorphisms, l : K1[G] → H}.

– E((H)[G]) = {All Homomorphisms, m : E[G] → H}.

– s(H)[G] : For m : E[G] → H, s(H)[G] (m) = m ◦ (s[1G]) : K1[G] → H.

– t(H)[G] : For m : E[G] → H, t(H)[G] (m) = m ◦ (t[1G]) : K1[G] → H.

• For an arrow f : I → J,( )[G](f) = (f)[G], which is deﬁned as follows:

[G] [G] – (f)V takes the homomorphism l : K1[G] → I ∈ V ((I) ) to the homomorphism

[G] f ◦ l : K1[G] → J ∈ V ((J) ).

[G] [G] – (f)E takes the homomorphism m : E[G] → I ∈ E((I) ) to the homomorphism f ◦ m : E[G] → J ∈ E((J)[G]).

That all objects are well-deﬁned and that ( )[G] is a functor are both straightforward and so again the details are omitted.

Now, with only a little work one can show that [G] a [G]. Thus, we have exactly one non-trivial instance where a functor based upon the lexicographic product is an adjoint functor.

3.2.10 /G

Consideration of /G splits into two cases just as the investigation of [G]. It also fails to be a right adjoint for non-trivial G, but is a left adjoint for any G. We call its right adjoint in such situations the corona power, written ( )/G.

84 Now, if G is the empty graph, then /G behaves just like the identity functor and so we can dismiss it as we have similar cases in the above. If we restrict our attention to ﬁnite G, then /G fails to preserve products for any non-empty G. As usual, let I and J be non-discrete graphs. Using vertex formulas we will show that I × J/G 6∼= (I/G) × (J/G):

v(I/G)×(J/G) = (vI + vI vG)(vJ + vJ vG)

2 = vI vJ + 2vI vJ vG + vI vJ vG

2 = v(I×J/G) + vI vJ vG + vI vJ vG Hence, for non-empty G, I × J/G 6∼= (I/G) × (J/G). Thus, in all non-trivial cases /G is not a right adjoint.

In order to illustrate that /G is a left adjoint, we have to introduce the corona power functor ( )/G : GRAPH → GRAPH. In order to understand it completely, we again need

/G the K1, E, s and t. Then ( ) is deﬁned as follows:

• For an object H of GRAPH,( )/G(H) = (H)/G, which is deﬁned as follows:

/G – V ((H) ) = {All Homomorphisms, l : K1 /G → H}.

– E((H)/G) = {All Homomorphisms, m : E/G → H}.

– s(H)/G : For m : E/G → H, s(H)/G (m) = m ◦ (s / 1G): K1 /G → H.

– t(H)/G : For m : E/G → H, t(H)/G (m) = m ◦ (t / 1G): K1 /G → H.

• For an arrow f : I → J,( )/G(f) = (f)/G, which is deﬁned as follows:

/G /G – (f)V takes the homomorphism l : K1 /G → I ∈ V ((I) ) to the homomorphism

/G f ◦ l : K1 /G → J ∈ V ((J) ).

/G /G – (f)E takes the homomorphism m : E/G → I ∈ E((I) ) to the homomorphism f ◦ m : E/G → J ∈ E((J)/G).

We again omit the details verifying that everything is well-deﬁned and that ( )/G is in fact a functor.

85 Now, with only a little work one can show that /G a /G. Thus, we have exactly one non-trivial instance where a functor based upon the corona is an adjoint functor.

3.2.11 Adjoint Success?

We have shown that both [G] and /G appear in adjunctions as left adjoints. Our ultimate goal in this section on graph operations is to determine if categorial methods are appropriate in this situation. So, we now want to know if we have captured both the lexicographic product and corona in an accessible and understandable way using category theory.

In order to really address this question we have to return to Knauer’s remark that the lexicographic product and corona do not have categorial descriptions. The constructions of both in Rgraph gave a clear sense in which Knauer was wrong. Working in Rgraph we were able to construct, and so to describe, both the lexicographic product and corona. However, Knauer had something further in mind. His comment about the lexicographic product and corona comes after a thorough investigation of many diﬀerent graph operations wherein Knauer illustrates how each of those operations can be captured in an adjunction. What he was claiming is that something similar cannot be done for the lexicographic product or corona (U. Knauer, personal communication, November 22, 2013).

On that count Knauer looks correct. We showed that neither operation is an adjoint functor and so they cannot be captured in an adjunction. However, we do have the partial success of [G] and /G. Do they provide us with a categorial description of the lexicographic product or corona?

There are two reasons why [G] and /G cannot be considered successful categorial descriptions of the lexicographic product and corona. The ﬁrst problem is that they are both left adjoints. In order to get at the graph-theoretic structure of an object in Graph we look

86 at the homomorphisms from E and from the single vertex graph, call it V . An adjunction

F a G where F : A → B and G : B → A provides a correspondence between arrows of the form F (A) → B and arrows of the form A → G(B). The problem is that we are interested in arrows of the form E → [G](A), V → [G](A), E → /G(A), and E → /G(A). Since [G] and /G are both left adjoints, the adjunction does not help us get at these arrows, the arrows we would need to get at the graph-theoretic structure of the lexicographic product or corona.

The second problem is that any description of the lexicographic product and corona in terms of these adjunctions will always include /G and [G]. If we cannot get independent descriptions of one side of the adjunction or the other then we will never have a description of what the individual operations do on their own.33 For these reasons, Knauer is entirely correct if ’categorical description’ is (as he intended) understood as ’described as part of certain adjunctions’.

There are two important lessons to take from this section. First, category theory can in fact provide a description of the lexicographic product and corona. However, that description is ultimately inappropriate due to the ungainliness of the construction. Second, though adjunctions are important and useful in some parts of mathematical practice they do not provide appropriate guidance. They cannot capture, and so guide away from, the lexicographic product and corona. Consequently, one of central concepts of category theory is inappropriate for mathematical practice in graph theory. Ultimately, category theory seems to lack the resources for an appropriate description of the lexicographic product and corona.

33This is possible in the case of × because we get a characterization of the diagonal functor that is independent from its role in the adjunction.

87 3.3 Conclusion

We have considered the use of category theory for conceptual unification along two different avenues. First, we considered unification that could be achieved through the methods we use to introduce categories. This approach failed because none of the three methods were able to provide appropriate guidance for the selection of categories.

We moved on to consider conceptual uniﬁcation at the level of individual operations. Follow- ing the suggestion of Knauer (2011), we considered the lexicographic product and corona as particular problem cases. We showed that the only descriptions category theory has provided are inappropriate and that one of the central concepts of category theory, adjunctions, is incapable of providing descriptions at all. Thus, category theory appears unable to provide an appropriate description of the lexicographic product and corona.

We have shown a number of situations where category theory is inappropriate for work in graph theory suggesting that category theory is ultimately inappropriate as a conceptual uniﬁcation of mathematics.

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91 Appendices

A Exponential Completeness Lemma

Lemma (Exponential Completeness). If G is a complete r-graph, then GH is complete for any r-graph H.

We could verify this directly by considering two arbitrary vertices of GH and showing that there is exactly one edge between them. However, a detour through the following lemma signiﬁcantly shortens the work:

s Lemma. G is complete iﬀ G(t) : GE → G1+1 is an isomorphism.

s t is the pushout arrow from 1 + 1 to E based on the arrows s and t. We consider a proof of Exponential Completeness ﬁrst, and then a proof of the above lemma upon which it depends.

Proof(Exponential Completeness):

s Let G be a complete graph. By our lemma, G(t) : GE → G1+1 is an isomorphism. Now, H

s is a functor and so preserves isomorphisms. Consequently, (G(t))H :(GE)H → (G1+1)H is an isomorphism.

92 Now, consider the diagram in ﬁgure A.1. There are familiar natural isomorphisms between

(GH )E > GH×E > GE×H > (GE)H

s (1 × s ) ( s ×1 ) s (GH )(t) G H (t) G (t) H (G(t))H ∨ ∨ ∨ ∨ (GH )(1+1) > GH×(1+1) > G(1+1)×H > (G(1+1))H

Figure A.1: A collection of natural isomorphisms each of the following functors:

(GH ) ∼= GH× ∼= G ×H ∼= (G )H

Since these diﬀerent functors are naturally isomorphic, each of the squares in the above diagram commutes and each horizontal arrow is an isomorphism. Hence, if the far right arrow

s is an isomorphism then the arrow on the far left is as well. Thus, (GH )(t) :(GH )E → (GH )1+1 is an isomorphism. Then by our lemma, GH is complete 4

Proof(of lemma):

⇐:

s Assume that G(t) : GE → G1+1 is an isomorphism. Let x, y be vertices of G. We want to show that there is exactly one e : E → G such that e ◦ s = x and e ◦ t = y.

x x 1+1 Consider y : 1 + 1 → G. Let p y q : 1 → G be the name of that arrow. Then s (t) −1 x E (G ) ◦ p y q is an arrow from 1 to G and so names an arrow from E to G. Let s s (t) −1 x (t) x e : E → G be the arrow that it names, so peq = (G ) ◦ p y q. Then G ◦ peq = p y q.

s (t) x s x Now, if we take transposes G ◦ peq = p y q gives us that e ◦ t ◦ p2 = y ◦ p2 where

93 s x p2 : 1 × (1 + 1) → 1 + 1 is the unitary isomorphism. Thus, e ◦ t = y and so e ◦ s = x and e ◦ t = y, meaning e is the desired edge.

0 0 0 0 s x Let e : E → G be such that e ◦ s = x and e ◦ t = y. Then e ◦ t ◦ p2 = y ◦ p2 and so s s (t) 0 x (t) 0 0 G ◦ pe q = p y q. Since G is an isomorphism this means that pe q = peq and so e = e, establishing the uniqueness of e. Thus, there is exactly one e : E → G such that e ◦ s = x and e ◦ t = y. Therefore, G is complete.

⇒:

s Assume that G is complete. We want to show that G(t) : GE → G1+1 is an isomorphism.

s Since RGRAPH is a topos, it is enough to show that G(t) is monic and epic.

Monic:

s s Let h, k : H → GE be such that G(t) ◦ h = G(t) ◦ k. We need to show that h = k. Suppose not. Then since E is a generator, there must be some e : E → H such that h ◦ e 6= k ◦ e.

s s Let h0 = h ◦ e and k0 = k ◦ e. So we have G(t) ◦ h0 = G(t) ◦ k0 and h0 6= k0. Taking transposes

0 0 0 s 0 s h 6= k but h ◦ (E × t ) = k ◦ (E × t ).

In what follows we will use pictures to represent the numerous equalities we can extract from these two equations, rather than using a system of equations. In order to do so, ﬁrst

s consider ﬁgure A.2 with E × E on the left and the image of E × t : E × (1 + 1) → E × E on the right.

Now, we represent our two equations as images in G of what the arrows do to E × E. The equality means that the two images in ﬁgure A.3 are pointwise equal.

The inequality means that the two images in ﬁgure A.4 are somewhere distinct.

However, in order to be distinct they must disagree on edges, since the equality guarantees

94 hs◦!E, 1Ei hs◦!E, s◦!Ei hs, si hs, ti hs◦!E, t◦!Ei (hs◦!E, s◦!Ei) (hs, si) (hs, ti) (hs◦!E, t◦!Ei)

h1E, 1Ei (h1E, s◦!Ei) (h1E, t◦!Ei) h1E, s◦!Ei h1E, t◦!Ei

ht◦!E, 1Ei (ht◦!E, s◦!Ei) (ht, si) (ht, ti) (ht◦!E, t◦!Ei) ht◦!E, s◦!Ei ht, si ht, ti ht◦!E, t◦!Ei

s Figure A.2: E × E and the image of E × t

0 0 0 0 0 0 0 0 h (hs◦!E, s◦!Ei) h (hs, si) h (hs, ti) h (hs◦!E, t◦!Ei) k (hs◦!E, s◦!Ei) k (hs, si) k (hs, ti) k (hs◦!E, t◦!Ei)

0 0 0 0 h (h1E, s◦!Ei) h (h1E, t◦!Ei) k (h1E, s◦!Ei) k (h1E, t◦!Ei)

0 0 0 0 0 0 0 0 h (ht◦!E, s◦!Ei) h (ht, si) h (ht, ti) h (ht◦!E, t◦!Ei) k (ht◦!E, s◦!Ei) k (ht, si) k (ht, ti) k (ht◦!E, t◦!Ei)

Figure A.3: Equality as appearing in G

0 0 h (hs◦!E, 1Ei) k (hs◦!E, 1Ei) 0 0 0 0 0 0 0 0 h (hs◦!E, s◦!Ei) h (hs, si) h (hs, ti) h (hs◦!E, t◦!Ei) k (hs◦!E, s◦!Ei) k (hs, si) k (hs, ti) k (hs◦!E, t◦!Ei)

0 0 h (h1E, 1Ei) k (h1E, 1Ei) 0 0 0 0 h (h1E, s◦!Ei) h (h1E, t◦!Ei) k (h1E, s◦!Ei) k (h1E, t◦!Ei)

0 0 h (ht◦!E, 1Ei) k (ht◦!E, 1Ei) 0 0 0 0 0 0 0 0 h (ht◦!E, s◦!Ei) h (ht, si) h (ht, ti) h (ht◦!E, t◦!Ei) k (ht◦!E, s◦!Ei) k (ht, si) k (ht, ti) k (ht◦!E, t◦!Ei)

Figure A.4: Inequality as appearing in G they agree on vertices. However, since G is complete and the relevant edges have the same sources and targets they are necessarily equal. Thus, h0 = k0 which is a contradiction. So

s our supposition was false, meaning that h = k, and so G(t) is monic.

Epic:

s s Let h, k : G1+1 → H be such that h ◦ G(t) = k ◦ G(t). We need to show that h = k. Suppose not. Then since E is a generator, there must be some e : E → G1+1 such that h ◦ e 6= k ◦ e. Consider e : E × (1 + 1) → G. Since G is complete, we know that the conﬁguration given in ﬁgure A.5 obtains in G, where the unlabeled parts are present but not in the image of e (we

95 maintain the labels for the vertices and edges of E × (1 + 1) that it inherits from E × E).

e(hs◦!E, s◦!Ei) e(hs, si) e(hs, ti) e(hs◦!E, t◦!Ei)

e(h1E, s◦!Ei) e(h1E, t◦!Ei)

e(ht◦!E, s◦!Ei) e(ht, si) e(ht, ti) e(ht◦!E, t◦!Ei)

Figure A.5: Conﬁguration appearing in G

That conﬁguration represents an arrow from E × E :→ G, call it e0. Recall from above how

s 0 s E × t embeds E × (1 + 1) into E × E. The image then also shows that e ◦ E × t = e and s s s so G(t) ◦ e0 = e. Then h ◦ e = h ◦ G(t) ◦ e0 = k ◦ G(t) ◦ e0 = k ◦ e, this contradicts h ◦ e 6= k ◦ e.

s So our supposition was false and in fact h = k. Thus, G(t) is epic.

s We have shown that G(t) is both monic and epic. Since we are working in a topos, this is

s suﬃcient to prove that G(t) is an isomorphism.

This completes the second direction of the proof.

B Tournament Lemma

G Lemma (Tournament). For any r-graph G, for each vertex x of K2 there is a tournament

G Qx that is a subgraph of K2 . Also, for distinct vertices x and y, Qx 6= Qy.

Proof:

Let G be an r-graph. Recall that K2 is complete. From the exponential completeness lemma

G we may conclude that K2 is complete. Since it is complete, it has exactly one arrow between

96 each pair of vertices and a loop at every vertex. We use this fact to introduce the following notation: ≺ u, v is the edge whose source is u and whose target is v.

G Our goal is to produce a distinct tournament for each vertex x in K2 . We do this in two

G steps. First, we produce a tournament Q over K2 that hits every vertex and has a transitive adjacency relation. Then we associate a vertex x with the subgraph of Q induced by those vertices that have edges to x, call it Qx. Finally, we show that for two distinct vertices x and y, Qx 6= Qy.

Construction of Q

There is a close relationship between the desired tournament and a well-ordering of the

G G vertices of the graph. Thus, consider |K2 |, the discrete subgraph underlying K2 . This

G G comes along with an monic i : |K2 | K2 . We have a version of the axiom of choice G for discrete graphs. As a result, we can produce a (non-strict) well-ordering of |K2 |, call

0 G G 0 G G it w : W |K2 | × |K2 |. We look at (i × i) ◦ w : W K2 × K2 , call this arrow w. We know it is monic because it is composed of monics. Hence, it has a classifying arrow

G G χw : K2 × K2 → Ω. This is all represented in the pullback given in ﬁgure B.6.

W > 1 ∨

w0 ∨ G G |K2 | × |K2 | true ∨ i × i ∨ ∨ G G χw K2 × K2 > Ω

Figure B.6: The pullback describing χw

97 Now, consider the following arrows:

G s G t G E h(K2 ) ,(K2 ) i G 1 G 1 u×u G G χw (K2 ) −−−−−−−−→ (K2 ) × (K2 ) −−→ K2 × K2 −→ Ω

G 1 G G E where u is the standard isomorphism from (K2 ) to K2 . This gives us an r : R (K2 ) , and the pullback in ﬁgure B.7.

R > 1 ∨

r true

∨ G s G t ∨ G E h(K2 ) , (K2 ) i G 1 G 1 u × u G G χw (K2 ) > (K2 ) × (K2 ) > K2 × K2 > Ω Figure B.7: A second important pullback

Finally, the properties of the two above pullbacks guarantee the existence of an arrow o : R → W that allows us to combine the two pullbacks as in ﬁgure B.8.

o R > W > 1 ∨ ∨

w0 ∨ G G r |K2 | × |K2 | true ∨ i × i

∨ G s G t ∨ ∨ G E h(K2 ) , (K2 ) i G 1 G 1 u × u G G χw (K2 ) > (K2 ) × (K2 ) > K2 × K2 > Ω Figure B.8: Pullbacks combined

Now, we want to know what membership in r looks like. Consider some graph P and an

G E G s G t arrow k : P → (K2 ) such that k ∈ r. Then we know that hu ◦ (K2 ) ◦ k, u ◦ (K2 ) ◦ ki ∈ w.

Further computation shows that hev ◦ hk, s◦!P i, ev ◦ hk, t◦!P ii ∈ w. So if P = 1, k would

G name an edge of K2 whose source comes before its target in the well-ordering w.

0 G s G t 0 Now, consider an edge of R, h : E → R. First, u × u ◦ h(K2 ) , (K2 ) i ◦ r ◦ h = hev ◦ hr ◦

0 0 h , s◦!Ei, ev ◦ hr ◦ h , t◦!Eii. Since the lefthand rectangle commutes we know that u × u ◦

G s G t 0 h(K2 ) , (K2 ) i◦r◦h = i×i◦w ◦o◦h. This is most important because it means the resulting

98 G G G G arrow from E → K2 × K2 factors through a discrete graph, |K2 | × |K2 |. Therefore, it

0 0 must be a loop. In fact, ev ◦ hr ◦ h , s◦!Ei and ev ◦ hr ◦ h , t◦!Ei must both be loops because

G they factor through |K2 |.

The most important feature of a loop l : E → P , where P is some arbitrary graph, is

0 0 that it factors through 1. This means there is some l : 1 → P , where l = l ◦!E. Hence,

0 0 0 0 l ◦ s◦!E = l ◦!E ◦ s◦!E = l ◦!E = l and l ◦ t◦!E = l ◦!E ◦ t◦!E = l ◦!E = l. Call this the loop property.

G E G G From r : R (K2 ) we get r : R × E → K2 . This is not yet a subgraph of K2 because r is not monic. However, we are working in a topos and so image factorization, a la Theorem 16.4 of McLarty[1992], allows us to introduce the following arrows r0 and q, which are epic and monic respectively:

0 r G q G R × E −→ [b.K2 |(∃c.R × E)b = rc] −→ K2

0 G G Such that q ◦ r = r. Let Q = [b.K2 |(∃c)b = rc], then Q is clearly a subgraph of K2 under

G 0 the arrow q. Now, an edge e : E → K2 is in q just in case there is an edge e : E → R × E such that r ◦ e0 = e.

Properties of Q

G We want to argue that an edge e : E → K2 is in q just in case its endpoints, e ◦ s◦!E and

e◦t◦!E, respect W . That is, he◦s◦!E, e◦t◦!Ei ∈ w. We begin with the following equivalences

G for e : E → K2 :

e ∈ q

99 iﬀ

∃hh1, h2i : E → R × E : r ◦ hh1, h2i = e

iﬀ

ev ◦ hr ◦ h1, h2i = e

Now, we want to determine the value of e◦s◦!E and e◦t◦!E. We begin with a few observations. h2 : E → E and so must be s◦!E, t◦!E or 1E. Also h1 : E → R, so we know that ev ◦ hr ◦ h1, s◦!Ei and ev ◦ hr ◦ h1, t◦!Ei must be loops. Furthermore, r ◦ h1 is obviously in r and so we know that hev ◦ hr ◦ h1, s◦!Ei, ev ◦ hr ◦ h1, t◦!Eii ∈ w. We now argue by case on the possible values of h2.

1. h2 = s◦!t

Then e = ev ◦ hr ◦ h1, h2i = ev ◦ hr ◦ h1, s◦!Ei is a loop and so e ◦ s◦!E = e ◦ t◦!E. Since

w is non-strict, he ◦ s◦!E, e ◦ t◦!Ei ∈ w.

2. h2 = t◦!E

This is analogous to case 1 since we know that ev ◦ hr ◦ h1, t◦!Ei must also be a loop.

3. h2 = 1E

Now, e◦s◦!E = ev ◦hr◦h1, 1ei◦s◦!E = ev ◦hr◦h1 ◦s◦!E, s◦!Ei = ev ◦hr◦h1, s◦!Ei◦s◦!E.

Also, ev ◦ hr ◦ h1, s◦!Ei is a loop and so hr ◦ h1, s◦!Ei ◦ s◦!E = ev ◦ hr ◦ h1, s◦!Ei.

Similar reasoning show that e ◦ t◦!E = ev ◦ hr ◦ h1, t◦!Ei. Thus, he ◦ s◦!E, e ◦ t◦!Ei =

hev ◦ hr ◦ h1, s◦!Ei, ev ◦ hr ◦ h1, t◦!Eii and so he ◦ s◦!E, e ◦ t◦!Ei ∈ w.

G Thus, we have established that for an edge e : E → K2 , e ∈ q iﬀ he ◦ s◦!E, e ◦ t◦!Ei ∈ w.

100 G G Now, we construct the Qx’s. Let x : 1 → K2 be a vertex of K2 . Let wx ⊆ w be the restriction

of the well-ordering to elements occuring before x and let Wx stand for the domain of wx.

G G Thus, for hu, vi : 1 → K2 × K2 , hu, vi ∈ wx iﬀ hu, vi ∈ w&hu, xi ∈ w&hv, xi ∈ w.

G G From Wx and wx we can construct qx : Qx → K2 in exactly the same way q : Q → K2 was

G constructed from W and w. Identical reasoning also means that for an edge e : E → K2 ,

e ∈ qx iﬀ he ◦ s◦!E, e ◦ t◦!Ei ∈ wx. Combining this with the above fact, we see that e ∈ qx iﬀ

he ◦ s◦!E, e ◦ t◦!Ei ∈ w&he ◦ s, xi ∈ w&he ◦ t, xi ∈ w (This also depends on the fact that for

G u, v : 1 → K2 , hu, vi ∈ wx iﬀ hu◦!E, v◦!Ei ∈ wx, which follows straightforwardly from the

fact that 1 is terminal). We have produced the Qx’s and must now establish distinctness.

Distinctness

G G Now, let x, y : 1 → K2 be distinct vertices of K2 . We want to argue that since x 6= y,

Qx 6= Qy. We do this by showing that Qx and Qy are not isomorphic. This is because they both encode well-orderings, one of which is a proper initial segment of the other. The rest of the proof is concerned with establishing that result in this context, so readers familiar with well-orderings may wish to skip it.

Suppose there is an isomorphism γ : Qx → Qy. Without loss of generality, we assume that

−1 hx, yi ∈ w. Now, if qx = qy ◦ γ then qx ◦ γ = qy and so qy ⊆ qx. Now, hy, yi ∈ wy and so

y◦!E ∈ qy. Then, since qy ⊆ qx, y◦!E ∈ qx. Thus, hy◦!E ◦ s, xi ∈ wx, which means hy, xi ∈ wx

and so hy, xi ∈ w. This contradicts hx, yi ∈ w and x 6= y. Thus, qx 6= qy ◦ γ.

00 Since qx 6= qy ◦ γ and E is a generator, there must be some h : E → Qx such that

00 00 G qx ◦ h 6= qy ◦ γ ◦ h . Recall that K2 is complete and so for those two edges to be distinct,

they must disagree on an endpoint. Thus, if qx 6= qy ◦γ there must be some vx : 1 → Qx such

that qx◦vx 6= qy ◦γ◦vx. Let vx be the one where qx◦vx is least in w. Since qx◦vx ∈ Qx we know

that hqx ◦ vx, xi ∈ w and so hqx ◦ vx, yi ∈ w by transitivity. Consequently, hqx ◦ vx, yi ∈ wy

101 and so ≺ qx ◦ vx, y ∈ qy. Thus, there is a vy : 1 → Qy such that qy ◦ vy = qx ◦ vx.

We establish two useful facts about vy:

−1 1. vx 6= vy ◦ γ

−1 If vx = vy ◦ γ , then qx ◦ vx = qy ◦ vy = qy ◦ γ ◦ vx. That is a contradiction.

−1 2. hqx ◦ vx, qx ◦ γ ◦ vyi

−1 −1 If not, then by our minimality assumption qx ◦ (γ ◦ vy) = qy ◦ γ ◦ (γ vy) = qy ◦ vy =

−1 qx ◦ vx. Then since qx is monic we would have vx = vy ◦ γ , which we have ruled out.

G Now, as subgraphs of K2 , Qx and Qy have at most one edge between any two of their vertices. Thus we use the ≺, notation for their edges as well.

Since, qx ◦ vx 6= qy ◦ γ ◦ vx there are two possibilities:

1. hqx ◦ vx, qy ◦ γ ◦ vxi ∈ w

Thus, ≺ qx ◦vx, qy ◦γ◦vx ∈ qy and speciﬁcally, qy◦ ≺ vy, γ◦vx =≺ qx ◦vx, qy ◦γ◦vx .

−1 −1 Since, ≺ vy, γ ◦ vx is an edge of Qy, γ ◦ ≺ vy, γ ◦ vx =≺ γ ◦ vy, vx is an edge

−1 −1 of Qx. Then ≺ qx ◦ γ ◦ vy, qx ◦ vx ∈ qx and so hqx ◦ γ ◦ vy, qx ◦ vxi ∈ w. This

−1 −1 contradicts the fact that vx 6= vy ◦ γ and hqx ◦ vx, qx ◦ γ ◦ vyi ∈ w.

2. hqy ◦ γ ◦ vx, qx ◦ vxi ∈ w

Then since hqx ◦ vx, xi ∈ w we know that hqy ◦ γ ◦ vx, xi ∈ w and so there must be a

0 0 0 0 v : 1 → Qx such that qx ◦ v = qy ◦ γ ◦ vx. Clearly, v 6= vx. Since hqx ◦ v , qx ◦ vxi ∈ w

0 0 0 0 0 we know that qx ◦ v = qy ◦ γ ◦ v . Then qy ◦ γ ◦ v = qx ◦ v = qy ◦ γ ◦ vx. Then v = vx, a contradiction.

In both cases we derived a contradiction. Thus, there can be no γ : Qx → Qy that is an

G isomorphism. Thus, we have shown for distinct vertices x, y of K2 , that Qx and Qy are are not isomorphic. Therefore, for distinct vertices x and y, Qx 6= Qy, completing the proof.

102 4

C Consistent Subsystems

Here we consider the following three properties of a category, C:

1. Topos: C is a topos.

2. Internalization: C has an object U and there is an isomorphism between the objects of C and the arrows from 1 to U.

3. E: C has an object E with two global elements s and t and the only endomorphisms

of E are 1E, s◦!E, and t◦!E.

We will produce models for each pair of these properties. These models will also satisfy the spanning tree principle: if an object C is connected then it has a spanning tree. In the categorial context, an object is connected just in case there is no epic from it to 1 + 1.

Topos & Internalization

Let C be the singleton category, containing only one object and that object’s identity arrow. It is trivially a topos. Furthermore, it has one arrow and one object and so satisﬁes ∼ internalization. Finally, in C, 1 + 1 = 1. Thus, the only object is not connected and so C satisﬁes the spanning tree principle.

103 Topos & E

For these two properties, we look to Bumby and Latch (1986) where the category is deﬁned as a category of functors from a given ﬁnite category to the category of sets. Readers who are interested can refer to that paper for further details. For those concerned with the category of sets, it can be adequately modeled by Vκ for κ inaccessible.

Internalization & E

Let C consist of the following four graphs and all homomorphisms between them: 1, 1+1, E, and E+E. It is clear that this category has E with all of its desired properties. Furthermore, the only connected graphs, E and 1, are already their own spanning trees. Finally, E + E has four vertices and so there are four arrows from 1 to E + E and there are four objects in the category, allowing for internalization.

104