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From Finite Sets to Feynman Diagrams

John C. Baez and James Dolan

Department of Mathematics, University of California

Riverside, California 92521

USA

email: baez@math.ucr.edu, jdolan@math.ucr.edu

April 14, 2000

to appear in Mathematics Unlimited - 2001 and Beyond,

eds. Bjorn Engquist and Wilfried Schmid

Abstract

‘Categorification’ is the process of replacing equations by isomor-

phisms. We describe some of the ways a thoroughgoing emphasis on

categorification can simplify and unify mathematics. We begin with

elementary arithmetic, where the category of finite sets serves as a

categorified version of the set of natural numbers, with disjoint union

and Cartesian product playing the role of addition and multiplication.

We sketch how categorifying the integers leads naturally to the infi-

nite loop space Ω∞S∞, and how categorifying the positive rationals

leads naturally to a notion of the ‘homotopy cardinality’ of a tame

space. Then we show how categorifying formal power series leads to

Joyal’s especes des structures, or ‘structure types’. We also describe

a useful generalization of structure types called ‘stuff types’. There is

an inner product of stuff types that makes the category of stuff types

into a categorified version of the Hilbert space of the quantized har-

monic oscillator. We conclude by sketching how this idea gives a nice

explanation of the combinatorics of Feynman diagrams.

1 Introduction

Prediction is hard, especially when it comes to the future, but barring someunforeseen catastrophe, we can expect the amount of mathematics producedin the 21st century to dwarf that of all the centuries that came before. Bythe very nature of its subject matter, mathematics is capable of limitless

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expansion. Thanks to rapid improvements in technology, our computationalpower is in a phase of exponential growth. Even if this growth slows, wehave barely begun to exploit our new abilities. Thus the interesting questionis not whether the 21st century will see an unprecedented explosion of newmathematics. It is whether anyone will ever understand more than the tiniestfraction of this new mathematics — or even the mathematics we already have.

Will mathematics merely become more sophisticated and specialized, orwill we find ways to drastically simplify and unify the subject? Will we onlybuild on existing foundations, or will we also reexamine basic concepts andseek new starting-points? Surely there is no shortage of complicated andinteresting things to do in mathematics. But it makes sense to spend at leasta little time going back and thinking about simple things. This can be a bitembarrassing, because we feel we are supposed to understand these thingscompletely already. But in fact we do not. And if we never face up to thisfact, we will miss out on opportunities to make mathematics as beautiful andeasy to understand as it deserves to be.

For this reason, we will focus here on some very simple things, startingwith the notion of equality. We will try to show that by taking pieces ofelementary mathematics and categorifying them — replacing equations byisomorphisms — we can greatly enhance their power. The reason is thatmany familiar mathematical structures arise from a process of decategorifica-

tion: turning categories into sets by pretending isomorphisms are equations.In what follows, we start with a discussion of equality versus isomorphism,

and a brief foray into n-categories and homotopy theory. Then we describein detail how the natural numbers and the familiar operations of arithmeticarise from decategorifying the category of finite sets. This suggests that adeeper approach to arithmetic would work not with natural numbers butdirectly with finite sets. This philosophy has already been pursued in manydirections, especially within combinatorics. As an example, we describe howformal power series rise from decategorifying Andre Joyal’s ‘structure types’,or ‘especes de structures’ [15, 16]. A structure type is any sort of structureon finite sets that transforms naturally under permutations; counting thestructures of a given type that can be put on a set with n elements is one ofthe basic problems of combinatorics. We also describe a useful generalizationof structure types, which we call ‘stuff types’.

To hint at some future possibilities, we conclude by showing how to un-derstand some combinatorial aspects of quantum theory using stuff types.

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First we define an inner product of stuff types and note that, with this innerproduct, the category of stuff types becomes a categorified version of theHilbert space of a quantized harmonic oscillator. Then we show that thetheory of Feynman diagrams arises naturally from the study of ‘stuff opera-tors’, which are gadgets that turn one stuff type into another in a linear sortof way.

2 Equality and Isomorphism

One of the most fundamental notions in mathematics is that of equality.For the most part we take it for granted: we say an equation expresses theidentity of a thing and itself. But the curious thing is that an equation isonly interesting or useful to the extent that the two sides are different! Theone equation that truly expresses identity,

x = x,

is precisely the one that is completely boring and useless. Interesting equa-tions do not merely express the identity of a thing and itself; instead, theyhint at the existence of a reversible transformation that takes us from thequantity on one side to the quantity on the other.

To make this more precise, we need the concept of a category. A set haselements, and two elements are either equal or not — it’s a yes-or-no business.A category is a subtler structure: it has objects but also morphisms betweenobjects, which we can compose in an associative way. Every object x has anidentity morphism

1x: x → x

which serves as a left and right unit for composition; this morphism de-scribes the process of going from x to x by doing nothing at all. But moreinterestingly, we say a morphism

f : x → y

is an isomorphism if there is a morphism g: y → x with fg = 1y, gf =1x. An isomorphism describes a reversible transformation going from x toy. Isomorphism is not a yes-or-no business, because two objects can beisomorphic in more than one way. Indeed, the notion of ‘symmetry’ arises

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precisely from the fact that an object can be isomorphic to itself in manyways: for any object x, the set of isomorphisms f : x → x forms a group calledits symmetry group or automorphism group, aut(x).

Now, given a category C, we may ‘decategorify’ it by forgetting aboutthe morphisms and pretending that isomorphic objects are equal. We areleft with a set (or class) whose elements are isomorphism classes of objectsof C. This process is dangerous, because it destroys useful information. Itamounts to forgetting which road we took from x to y, and just rememberingthat we got there. Sometimes this is actually useful, but most of the timepeople do it unconsciously, out of mathematical naivete. We write equations,when we really should specify isomorphisms. ‘Categorification’ is the attemptto undo this mistake. Like any attempt to restore lost information, it nota completely systematic process. Its importance is that it brings to lightpreviously hidden mathematical structures, and clarifies things that wouldotherwise remain mysterious. It seems strange and complicated at first, butultimately the goal is to make things simpler.

We like to illustrate this with the parable of the shepherd. Long ago,when shepherds wanted to see if two herds of sheep were isomorphic, theywould look for a specific isomorphism. In other words, they would line upboth herds and try to match each sheep in one herd with a sheep in theother. But one day, a shepherd invented decategorification. She realized onecould take each herd and ‘count’ it, setting up an isomorphism between itand a set of ‘numbers’, which were nonsense words like ‘one, two, three, . . . ’specially designed for this purpose. By comparing the resulting numbers, shecould show that two herds were isomorphic without explicitly establishingan isomorphism! In short, the set N of natural numbers was created bydecategorifying FinSet, the category whose objects are finite sets and whosemorphisms are functions between these.

Starting with the natural numbers, the shepherds then invented the basicoperations of arithmetic by decategorifying important operations on finitesets: disjoint union, Cartesian product, and so on. We describe this in detailin the next section. Later, their descendants found it useful to extend N

to larger number systems with better formal properties: the integers, therationals, the real and complex numbers, and so on. These make it easier toprove a vast range of theorems, even theorems that are just about naturalnumbers. But in the process, the original connection to the category of finitesets was obscured.

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Now we are in the position of having an enormous body of mathematics,large parts of which are secretly the decategorified residues of deeper truths,without knowing exactly which parts these are. For example, any equationinvolving natural numbers may be the decategorification of an isomorphismbetween finite sets. In combinatorics, when people find an isomorphism ex-plaining such an equation, they say they have found a ‘bijective proof’ of it.But decategorification lurks in many other places as well, and wherever wefind it, we have the opportunity to understand things more deeply by go-ing back and categorifying: working with objects directly, rather than theirisomorphism classes.

For example, the ‘representation ring’ of a group is a decategorified versionof its category of representations, with characters serving as stand-ins for therepresentations themselves. The ‘Burnside ring’ of a group is a decategorifiedversion of its category of actions on finite sets. The ‘K-theory’ of a topologicalspace is a decategorified version of the category of vector bundles over thisspace. And the ‘Picard group’ of a variety is obtained by decategorifying thecategory of line bundles over this variety. In all mathematics involving theseideas and their many generalizations, there is room for categorification.

Actually, in the first three examples above, decategorification has been fol-lowed by the process of adjoining formal additive inverses, just as we obtainedthe integers by first decategorifying FinSet and then adjoining the negativenumbers. Apart from ignorance of categories, the desire for additive inversesmay be the main reason mathematicians indulge in decategorification. Thismakes it very important to understand additive inverses in the categorifiedcontext. We touch upon this issue in the next section.

When one digs more deeply into categorification, one quickly hits uponhomotopy theory. The reason is as follows. Starting with a topological spaceX, we can form a category Π1(X) whose objects are the points of X andwhose morphisms f : x → y are homotopy classes of paths from x to y. Thiscategory is a groupoid, meaning that all its morphisms are isomorphisms[12, 29]. We call it the fundamental groupoid of X.

If we decategorify Π1(X), we obtain the set Π0(X) of path components ofX. Note that Π1(X) is far more informative than Π0(X), because it recordsnot just whether one can get from here to there, but also a bit about thepath taken. In fact, one can make quite precise the sense in which Π1 is astronger invariant than Π0. We say X is a homotopy n-type if it is locallywell-behaved (e.g. a CW complex) and any continuous map f : Sk → X is

5

contractible to a point when k > n. If X is a homotopy 0-type, we canclassify it up to homotopy equivalence using Π0(X). This is not true if X isa homotopy 1-type. However, if X is a homotopy 1-type, we can classify itup to homotopy equivalence using Π1(X).

There is a suggestive pattern here. To see how it continues, we needto categorify the notion of category itself! This gives the concept of a ‘2-category’ or ‘bicategory’ [10, 20]. We will not give the full definitions here, butthe basic idea is that a 2-category has objects, morphisms between objects,and also 2-morphisms between morphisms, like this:

Just as a category allows us to distinguish between equality and isomorphismfor objects, a 2-category allows us to make this distinction for morphisms.

From any space X we can form a 2-category Π2(X) whose objects arepoints of X, whose 1-morphisms are paths in X, and whose 2-morphisms arehomotopy classes of ‘paths of paths’. In fact this 2-category is a 2-groupoid,meaning that every 2-morphism is invertible and every morphism is invertibleup to a 2-morphism. We call Π2(X) the fundamental 2-groupoid of X. If Xis a homotopy 2-type, we can classify it up to homotopy equivalence usingΠ2(X). Moreover, if we decategorify Π2(X) by treating homotopic paths asequal, we get back Π1(X).

Going still further, Grothendieck [14] proposed that if we take the notionof ‘set’ and categorify it n times, we should obtain a notion of n-category:a structure with objects, morphisms between objects, 2-morphisms betweenmorphisms and so on up to n-morphisms. Any space X should give us ann-category Πn(X) whose objects are the points of X, whose morphisms arepaths in X, whose 2-morphisms are paths of paths, and so on... with then-morphisms being homotopy classes of n-fold paths. In this n-category allthe j-morphisms are equivalences, meaning that for j = n that they areinvertible, while for j < n they are invertible up to equivalence. We callan n-category with this property an n-groupoid. Given any decent definitionof n-categories, n-groupoids should be essentially the same as homotopy n-types.

6

In short, by iterated categorification, the whole of homotopy theory shouldspring forth naturally from pure algebra! This dream is well on its way tobeing realized, but it still holds many challenges for mathematics. Variousdefinitions of n-category have been proposed [5, 9, 23, 28], and for someof these the notion of fundamental n-groupoid has already been explored.However, nobody has yet shown that these different definitions are equiva-lent. Finding a clear treatment of n-categories is a major task for the centuryto come.

3 Natural Numbers and Finite Sets

Starting from the realization that the set of natural numbers is obtained bydecategorifying FinSet, let us see how the basic operations of arithmetic havetheir origins in the theory of finite sets. First of all, addition in N arises fromdisjoint union of finite sets, since

|S + T | = |S| + |T |

where S + T denotes the disjoint union of the finite sets S and T . However,it is worth noting that there is really no such thing as ‘the’ disjoint unionof two sets S and T ! Before forming their union, we must use some trickto ensure that they are disjoint, such as replacing the elements s ∈ S byordered pairs (s, 0) and replacing the elements t ∈ T by ordered pairs (t, 1).But there are many tricks we could use here, and it would be unfair, eventyrannical, to demand that everybody use the same one. If we allow freedomof choice in this matter, we should really speak of ‘a’ disjoint union of S andT . But this raises the question of exactly what counts as a disjoint union!

This question has an elegant and, at first encounter, rather surprisinganswer. A disjoint union of sets S and T is not just a set; it is a set S + Tequipped with maps i: S → S + T and j: S → S + T saying how S and Tare included in it. Moreover, these maps must satisfy the following universalproperty: for any maps f : S → X and g: T → X, there must be a unique

7

map h: S + T → X making this diagram commute:

S

S + T X

T

@@

@@R

i

HHHHHHHHHHj

f

-h

j

*

g

In other words, a map from a disjoint union of S and T to some set containsthe same information as a pair of maps from S and T to that set.

The wonderful thing about this definition is that now, if you have yourdisjoint union and I have mine, we automatically get maps going both waysbetween the two, and a little fiddling shows that these maps are inverses. (Ifyou have never seen this done, you should do it yourself right now.) Thus,while there is not a unique disjoint union of sets, any two disjoint unionsare canonically isomorphic. Experience has shown that knowing a set up tocanonical isomorphism is just as good as knowing it exactly. For this reason,we may actually speak of ‘the’ disjoint union of sets, so long as we bear inmind that we are using the word ‘the’ in a generalized sense here.

Defining disjoint unions by a universal property this way has a number ofadvantages. First, it allows freedom of choice without risking a descent intochaos, because it automatically ensures compatibility between different tricksfor constructing disjoint unions. With the rise of computer technology, theimportance of this sort of ‘implementation-independence’ has become veryclear: for easy interfacing, we need to be able to ignore the gritty details ofhow things are done, but we can only do this if we have a clear specificationof what is being done.

Second, since the above definition of disjoint union relies only upon theapparatus of commutative diagrams, it instantly generalizes. Given two ob-jects A and B in any category, we define their coproduct to be an objectA + B equipped with morphisms i: A → A + B, j: B → A + B such thatfor any morphisms f : A → X, g: B → X there exists a unique morphism

8

h: A + B → X making this diagram commute:

A

A + B X

B

@@

@@@R

i

HHHHHHHHHHj

f

-h

j

*

g

The coproduct may or may not exist, but when it does, it is unique up tocanonical isomorphism. For example, the coproduct of topological spaces istheir disjoint union with its usual topology, while the coproduct of groups istheir free product. This realization, and others like it, help us save time bysimultaneously proving theorems for all categories instead of one category ata time.

As our discussion has shown, addition of natural numbers is just a decat-egorified version of the coproduct of finite sets. What about multiplication?This is a decategorified version of the Cartesian product:

|S × T | = |S||T |.

Like the disjoint union, the Cartesian product is really defined only up tocanonical isomorphism: we say S × T is the set of ordered pairs (s, t) withs ∈ S and t ∈ T , but if you carefully examine the set-theoretic fine print,you will see that there is some choice involved in defining ordered pairs! VonNeumann had the clever idea of defining (s, t) to be the set s, s, t.This trick gets the job done, but there is nothing sacrosanct about it, so inlong run it turns out to be better to define Cartesian products via a universalproperty.

In this approach, we say a Cartesian product of S and T is any set S ×Tequipped with maps p: S × T → S, q: S × T → T such that for any mapsf : Y → S and g: Y → T , there is a unique map h: Y → S × T making this

9

diagram commute:

S

S × T X

T

@@

@@I

p

q

hHHHHHHHHHHY

f

g

Again, thanks to the magic of universal properties, this definition determinesthe Cartesian product up to canonical isomorphism. And again, we canstraightforwardly generalize this definition to any category, obtaining thenotion of a product of objects. For example, the product of topological spacesis their Cartesian product with the product topology, while the product ofgroups is their Cartesian product with operations defined componentwise.

Now let us see how the basic laws satisfied by addition and multiplicationin N come from properties of FinSet. The natural numbers form a rig underaddition and multiplication: that is, a ‘ring without negatives’. We thusexpect FinSet to be some sort of categorified version of a rig. In fact, thedefinition of coproduct automatically implies that it is commutative andassociative up to canonical isomorphism. (Again, anyone who has not seenthe proof should prove this right now!) Since the definition of product isobtained simply by turning all the arrows around, the same arguments showthe commutativity and associativity of the product. On the other hand, thedistributivity of products over coproducts is not an automatic consequenceof general abstract nonsense; this is really a special feature of FinSet and aclass of related categories.

What about 0 and 1? In any category, we say an object X is initial ifthere is a unique morphism from X to any object in the category. Dually,we say X is terminal if there is a unique morphism from any object in thecategory to X. Since these definitions are based on universal properties,initial and terminal objects are unique up to canonical isomorphism whenthey exist. In FinSet, the empty set is initial and the 1-element set — i.e.,any 1-element set — is terminal. We shall call these sets 0 and 1, because

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they play the same role in FinSet that the numbers 0 and 1 do in N. Inparticular, it is easy to see that in any category with an initial object 0, wehave a canonical isomorphism

X + 0 ∼= X.

Turning the arrows around, the same argument shows that in any categorywith a terminal object 1, we have a canonical isomorphism

X × 1 ∼= X.

In short, FinSet is a categorified version of a commutative rig, with allthe usual laws holding up to canonical isomorphism. Moreover, these isomor-phisms satisfy various ‘coherence laws’ which allow us to manipulate themwith the ease of equations. We thus call FinSet a symmetric rig category; fordetails, see the work of Kelly [19] and Laplaza [21]. Many of the categoriesthat mathematicians like are symmetric rig categories. Often the additionand multiplication are given by coproduct and product, but not always. Forexample, the category of vector spaces is a symmetric rig category with di-rect sum as addition and tensor product as multiplication. The direct sum ofvector spaces is the coproduct, but their tensor product is not their productin the category-theoretic sense.

The most primitive example of a symmmetric rig category is actuallynot FinSet but FinSet0, the subcategory whose objects are all the finite setsbut whose morphisms are just the bijections. Disjoint union and Cartesianproduct still serve as ‘addition’ and ‘multiplication’ in this smaller category,even though they are not the coproduct and product anymore. The decat-egorification of FinSet0 is still N, and FinSet0 plays a role in the theory ofsymmetric rig categories very much like that N plays in the theory of com-mutative rigs, or Z plays in the more familiar theory of commutative rings.In the next section, we shall see more of the importance of FinSet0.

Having categorified the natural numbers, it is tempting to categorify othernumber systems: the integers, rational numbers, the real numbers and so on.For quantum mechanics it might be nice to categorify the complex numbers!However, there are already severe difficulties with categorifying the integers.Doing this would be like inventing ‘sets with negative cardinality’. However,there are strong limitations on our ability to cook up a category where objectshave ‘additive inverses’ with respect to coproduct. For suppose the object

11

X has an object −X satisfying −X + X ∼= 0 where 0 is the initial object.Then for any object Y there is only one morphism from −X + X to Y . Buta morphism of this sort is the same as a morphism from −X to Y togetherwith a morphism from X to Y . This means that both X and −X are initial!In short,

−X + X ∼= 0 ⇐⇒ X ∼= 0 & − X ∼= 0.

This does not mean that categorifying the integers is doomed to fail; itjust means that we must take a more generous attitude towards what countsas success. Schanuel [26] has proposed one approach; Joyal [17] and theauthors [6] have advocated another, based on ideas from homotopy theory.Briefly, it goes like this. Given a symmetric rig category R, we can form itsunderlying groupoid R0: that is, the subcategory with all the same objectsbut only the isomorphisms as morphisms. This groupoid inherits the struc-ture of a symmetric rig category from R. Since groupoids are essentially thesame as homotopy 1-types, we can form a homotopy 1-type B(R0), calledthe classifying space of R0, whose fundamental groupoid is equivalent to R0.With suitable tinkering, we can use the addition in R0 to make B(R0) into atopological monoid [27]. Then, by a well-known process called ‘group com-pletion’ [8], we can adjoin additive inverses and obtain a topological groupG(B(R0)). Thanks to the multiplication in R0, G(B(R0)) has a structuremuch like that of a commutative ring, but with the usual laws holding onlyup to homotopy. Homotopy theorists call this sort of thing an ‘E∞ ring space’[24].

If we perform this series of constructions starting with FinSet, we obtainthe E∞ ring space

G(B(FinSet0)) = Ω∞S∞

where the space on the right is a kind of limit of the n-fold loop space ΩnSn asn → ∞. This space is fundamental to a branch of mathematics called stablehomotopy theory [1, 13]. It has nonvanishing homotopy groups in arbitrarilyhigh dimensions, so we should really think of it as an ‘ω-groupoid’. Whatthis means is that to properly categorify subtraction, we need to categorifynot just once but infinitely many times!

Seeing the deep waters we have gotten ourselves into here, we might be abit afraid to go on and categorify division. However, historically the negativenumbers were invented quite a bit after the nonnegative rational numbers —in stark contrast to the usual textbook presentation in which Z comes before

12

Q. Apparently half an apple is easier to understand than a negative apple!This suggests that perhaps ‘sets with fractional cardinality’ are simpler than‘sets with negative cardinality’. And in fact, they are.

The key is to think carefully about the actual meaning of division. Theusual way to get half an apple is to chop one into ‘two equal parts’. Ofcourse, the parts are actually not equal — if they were, there would be onlyone part! They are merely isomorphic. So what we really have is a Z/2symmetry group acting on the apple which interchanges the two isomorphicparts. Similarly, if a group G acts on a set S, we can ‘divide’ the set bythe group by taking the quotient S/G, whose points are the orbits of theaction. If S and G are finite and G acts freely on S, this construction reallycorresponds to division, since |S/G| = |S|/|G|. However, it is crucial thatthe action be free.

For example, why is 6/2 = 3? We can take a 6-element set S with a freeaction of the group G = Z/2 and identify all the elements in each orbit toobtain a 3-element set S/G:

Pictorially, this amounts to folding the set S in half, so it is not surprisingthat |S/G| = |S|/|G| in this case. Unfortunately, if we try a similar trickstarting with a 5-element set, it fails miserably:

We don’t obtain a set with 212

elements, because the group action is not free:the point in the middle gets mapped to itself. To define ‘sets with fractionalcardinality’, we need a way to count the point in the middle as ‘half a point’.

To do this, we should first find a better way to define the quotient of S byG when the action fails to be free. Following the policy of replacing equationsby isomorphisms, let us define the weak quotient S//G to be the category with

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elements of S as its objects, with a morphism g: s → s′ whenever g(s) = s′,and with composition of morphisms defined in the obvious way. The figuresabove are really just pictures of such categories — but to reduce clutter, wedid not draw the identity morphisms.

Next, let us figure out a good way to define the ‘cardinality’ of a category.Pondering the examples above leads us to the following recipe: for eachisomorphism class of objects we pick a representative x and compute thereciprocal of the number of automorphisms of this object; then we sum overisomorphism classes. In short:

|C| =∑

isomorphism classes of objects [x]

1

|aut(x)| .

It is easy to see that with this definition, the point in the middle of theprevious picture gets counted as ‘half a point’, so we get a category withcardinality 21

2. In general,

|S//G| = |S|/|G|

whenever G is a finite group acting on a finite set S. In fact, this formulacan be viewed as a simplified version of ‘Burnside’s lemma’, which givesthe cardinality of the ordinary quotient instead of the better-behaved weakquotient [11].

The cardinality of a category C is a well-defined nonnegative rationalnumber whenever C has finitely many isomorphism classes of objects andeach object has finitely many automorphisms. More generally, we can makesense of |C| whenever the sum defining it converges; we call categories forwhich this is true tame.

Here are some easily checked and reassuring facts about these notions.Any category equivalent to a tame category is tame, and equivalent tamecategories have the same cardinality. Moreover, one can define the ‘coprod-uct’ and ‘product’ of categories in the usual way; these operations preservetameness, and if C and D are tame categories we have

|C + D| = |C| + |D|, |C × D| = |C||D|.

Finally, there is a standard trick for thinking of sets as special categories:given a set S we form the discrete category whose objects are the elements

14

of S and whose morphisms are all identity morphisms. As one would hope,applying this trick to a finite set gives a tame category with the same cardi-nality.

In the next section we describe a way to compute the cardinality of a largeclass of tame categories — the tame categories whose objects are finite setsequipped with extra structure. The simplest example is the category of finitesets itself! A simple computation gives the following marvelous formula:

|FinSet| =

∞∑

n=0

1

n!= e.

It would be nice to find a natural category whose cardinality involves π, butso far our examples along these lines are rather artificial.

The cardinality of a category C equals that of its underlying groupoid C0.This suggests that this notion really deserves the name groupoid cardinality.It also suggests that we should generalize this notion to n-groupoids, or evenω-groupoids. Luckily, we don’t need to understand ω-groupoids very well totry our hand at this! Whatever ω-groupoids are, they are supposed to be analgebraic way of thinking about topological spaces up to homotopy. Thus wejust need to invent a concept of the ‘cardinality’ of a topological space whichhas nice formal properties and which agrees with the groupoid cardinality inthe case of homotopy 1-types. In fact, this is not hard to do. The key is touse the homotopy groups πk(X) of the space X.

The homotopy cardinality of a topological space X is defined as the al-ternating product

|X| = |π1(X)|−1 |π2(X)| |π3(X)|−1 · · ·

when X is connected and the product converges; if X is not connected, wedefine its homotopy cardinality to be the sum of the homotopy cardinalitiesof its connected components, when the sum converges. We call spaces withwell-defined homotopy cardinality tame. The product or coproduct of twotame spaces is again tame, and we have

|X + Y | = |X| + |Y |, |X × Y | = |X||Y |,

just as one would hope.

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Even better, homotopy cardinality gets along well with fibrations, whichwe can think of as ‘twisted products’ of spaces. Namely, if

F → X → B

is a fibration and the base space B is connected, we have

|X| = |F ||B|whenever two of the three spaces in question are tame (which implies thetameness of the third). This fact is an easy consequence of the long exacthomotopy sequence

· · · → πn+1(B) → πn(F ) → πn(X) → πn(B) → πn−1(F ) → · · ·As an application of this fact, recall that any topological group G has a

classifying space BG, meaning a space with a principal G-bundle over it

G → EG → BG

whose total space EG is contractible. Since EG is contractible it is tame,and |EG| = 1. Thus G is tame if and only if BG is, and

|BG| =1

|G| .

In other words, we can think of BG as a kind of ‘reciprocal’ of G.This curious idea is already lurking in the usual approach to equivariant

cohomology. Suppose X is a space on which the topological group G acts.When the action of G on X is free, it is fun to calculate cohomology groups(and other invariants) of the quotient space X/G. When the action is notfree, this quotient can be very pathological, so people usually replace it bythe homotopy quotient

X//G = EG ×G X.

A careful examination of this construction shows that instead of identifyingpoints of X that lie in the same G-orbit, we are sewing in paths betweenthem — thus following the dictum of replacing equations by isomorphisms,or more generally, equivalences. There is a fibration

X → X//G → BG,

so when X and G are tame we have

|X//G| = |X| |BG| = |X|/|G|just as one would hope.

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4 Power Series and Structure Types

In combinatorics, a standard problem is to count the number of structuresof a given type that can be put on a finite set. For example, the number ofways to linearly order an n-element set is n!, while the number of ways topartition it into disjoint nonempty subsets is some more complicated functionof n. Such counting problems can often be solved with almost magical easeusing formal power series [11, 25, 30].

The idea is simple. Let F be any type of structure that we can put onfinite sets, and let Fn be the set of all structures of this type that can beput on your favorite n-element set (which we will simply call ‘n’). Define thegenerating function of F to be the formal power series

|F |(x) =

∞∑

n=0

|Fn|n!

xn.

Using this definition, operations on structure types correspond to operationson their generating functions.

For example, suppose we have two structure types F and G. Then wecan make up a new structure type F + G by saying that a structure of typeF + G on a finite set is the same as a structure of type F or a structure oftype G on this set. We count the structures of these two types as disjoint,so number of ways to put a structure of type F + G on the n-element set is

|(F + G)n| = |Fn| + |Gn|.

We thus have the following relation between generating functions:

|F + G| = |F | + |G|.

Similarly, we can make up a structure type FG as follows: to put astructure of type FG on a finite set, we first choose a way to chop this setinto two disjoint parts, and then put a structure of type F on the first partand a structure of type G on the second part. The number of ways to dothis for the n-element set is

|(FG)n| =n

∑

m=0

(

n

m

)

|Fm| |Gn−m|,

17

so a little calculation shows that

|FG| = |F ||G|.

Simple structure types have simple power series as their generating func-tions. For example, consider the impossible structure: by definition, thereare no ways to put this structure on any set. If we use use 0 to stand for theimpossible structure, we have

|0| = 0.

For that matter, consider the structure of being the empty set. There is oneway to put this struture on a set with 0 elements, and no ways to put it on aset with n elements when n > 0, so the generating function of this stucturetype is just the power series 1. If we call this structure type ‘1’, we thus have

|1| = 1.

More generally, the structure of being an n-element set has the generatingfunction xn/n!. It follows that the vacuous structure — the structure of‘being a finite set’ — has the generating function

ex = 1 + x +x2

2!+

x3

3!+ · · ·

Let us use these ideas to count something. Let B denote the structuretype of binary rooted trees [11]. For example, here is a way to put a structureof type B on the set 1, . . . , 7:

2

1

4

3

5

7

6

We can recursively define B as follows. First, there is exactly one way to puta structure of type B on the empty set. Second, to put a structure of type

18

B on nonempty finite set S, we pick an element r ∈ S called the root, chopthe set S −r into two parts L and R called the left and right subtrees, andput a structure of type B on L and on R. Translating this definition into anequation using the method described above, we have

B = 1 + xB2.

In short: a binary rooted tree is either the empty set or a one-element settogether with two binary rooted trees! Thanks to the way things are set up,this equation gives rise to an identical-looking equation for the generatingfunction |B|:

|B| = 1 + x|B|2.Now comes the miracle. We can solve this equation for |B|:

|B|(x) =1 −

√1 − 4x

2x,

and doing a Taylor expansion we get:

|B|(x) = 1 + x + 2x2 + 5x3 + 14x4 + 42x5 + · · ·

The coefficients of this series are called the Catalan numbers, cn:

|B|(x) =

∞∑

n=0

cnxn

A little messing around with the binomial theorem shows that

cn =1

n + 1

(

2n

n

)

,

so the number of ways to put a binary rooted tree structure on the n-elementset is

|Bn| = n!cn =(2n)!

(n + 1)!.

What is really going on here? Joyal clarified the subject immensely bygiving a precise definition of the ‘structure types’ we have already been usinghere in an informal way. A structure type is a functor F : FinSet0 → FinSet.For the reader who is not comfortable with functors, we can spell out exactly

19

what this means. First, a structure type assigns to any finite set S a set FS,which we think of as the set of all structures of type F on S. Second, itassigns to any bijection f : S → T a map Ff : FS → FT . These maps describehow structures of type F transform under bijections. Third, we require thatFfg = FfFg whenever f and g are composable.

Structure types form a category where the morphisms are ‘natural trans-formations’ between functors — any decent book on category theory willexplain what these are [22], but it doesn’t much matter here. The categoryof structure types, which we call FinSetx, is a kind of categorified versionof the rig of formal power series

Nx = ∞

∑

n=0

an

n!xn : an ∈ N.

We have already seen how any structure type F determines a formal powerseries |F | ∈ Nx. In fact, the category FinSetx is a symmetric rig categorywith the addition and multiplication operations already mentioned. As aresult, its decategorification is a commutative ring. This commutative ringis not Nx, since the isomorphism class of a structure type F contains moreinformation than the numbers |Fn|. However, there is a homomorphism fromthe decategorification of FinSetx onto Nx.

In fact, one can make the analogy between power series and structuretypes very precise [5]. But instead of doing this here, let us give an illustrationof just how far the analogy goes. Suppose F is a structure type and X is afinite set. Define the groupoid F (X) as follows:

F (X) =∞

∑

n=0

(Fn × Xn)//n! .

Here n! stands for the group of permutations of the set n. This group actson Fn thanks to the definition of structure type, and it also acts on Xn inan obvious way, so we can form the weak quotient (Fn ×Xn)//n!, which is agroupoid. Taking the coproduct of all these groupoids, we get F (X). This isthe categorified version of evaluating a power series at a number. But unlikethe case of ordinary power series, there is no issue of convergence!

The issue of convergence arises only when we decategorify. Since groupoidcardinality gets along nicely with coproducts and weak quotients, we have

20

the formula|F (X)| = |F |(|X|)

whenever both sides make sense. The left-hand side makes sense when thegroupoid F (X) is tame. The right-hand side makes sense when the powerseries |F |(x) converges at x = |X|. These two conditions are actually equiv-alent.

This formula is good for calculating groupoid cardinalities. The groupoidF (1) is particularly interesting because it is equivalent to the groupoid ofF -structured finite sets. The objects of this groupoid are finite sets equippedwith a structure of type F , and the morphisms are structure-preserving bi-jections. When this groupoid is tame, its cardinality is |F |(1). For example,if F is the vacuous structure, its generating function is |F |(x) = ex, and F (1)is the groupoid of finite sets. Thus the cardinality of this groupoid is e, aswe have already seen. For a more interesting example, take a structure oftype P on a finite set to be a partition into disjoint nonempty subsets. Then

|P |(x) = eex−1,

so the cardinality of the groupoid of finite sets equipped with a partition isee−1. On the other hand, the groupoid of binary rooted trees is not tame,because the power series |B|(x) diverges at x = 1. By analytic continuationusing the formula

|B|(x) =1 −

√1 − 4x

2x

one might assign this groupoid a cardinality of 12−

√3

2i, ignoring the issue

of branch cuts. These ‘analytically continued cardinalities’ have interestingproperties, but it is far from clear what they mean!

Finally, let us mention some interesting generalizations of structure types.As we have seen, any structure type F determines a groupoid F (1), thegroupoid of F -structured finite sets. There is a functor U : F (1) → FinSet0

sending each F -structured finite set to its underlying set. It turns out thatthis functor contains all the information in the structure type. Given suitablefiniteness conditions, everything we have said about structure types actuallyworks for any groupoid G equipped with a functor U : G → FinSet0. Sincethe objects of G can be thought of as finite sets together with extra ‘stuff’,we call a groupoid equipped with a functor to FinSet a stuff type. For a good

21

example of a stuff type that is not a structure type, take G to be the groupoidwhose objects are ordered n-tuples of finite sets and whose morphisms areordered n-tuples of bijections, and let U be the projection onto one entry. Aswe shall see in the next section, stuff types are important for understandingthe combinatorics of Feynman diagrams.

In fact, we can categorify the concept of ‘stuff type’ infinitely many times!If we do, we ultimately reach the notion of a space equipped with a contin-uous function to the classifying space B(FinSet0). This notion allows fora fascinating interplay between Feynman diagrams and homotopy theory.Unfortunately, for the details the reader will have to turn elsewhere [7].

5 Feynman Diagrams and Stuff Operators

From here we could go in various directions. But since we are dreamingabout the future of mathematics, let us choose a rather speculative one, anddiscuss some applications of categorification to quantum theory. By now itis clear that categorification is necessary for understanding the connectionsbetween quantum field theory and topology. It has even played a role insome attempts to find a quantum theory of gravity. But having reviewedthese subjects elsewhere [4, 2, 3], we restrict ourselves here to some of thesimplest aspects of quantum physics.

One of the first steps in developing quantum theory was Planck’s newtreatment of electromagnetic radiation. Classically, electromagnetic radia-tion in a box can be described as a collection of harmonic oscillators, onefor each vibrational mode of the field in the box. Planck ‘quantized’ theelectromagnetic field by assuming that energy of each oscillator could onlytake the discrete values (n + 1

2)~ω, where n is a natural number, ω is the

frequency of the oscillator in question, and ~ is a constant now known asPlanck’s constant. Planck did not really know what to make of the num-ber n, but Einstein and others subsequently interpreted it as the numberof ‘photons’ occupying the vibrational mode in question. However, far frombeing particles in the traditional sense of tiny billiard balls, these photonsare curiously abstract entities — for example, all the photons occupying agiven mode are indistinguishable from each other.

The treatment of this subject was originally quite ad hoc, but with furtherwork the underlying mathematics gradually became clearer. It turns out

22

that states of a quantized harmonic oscillator can be described as vectorsin a Hilbert space called ‘Fock space’. This Hilbert space consists of formalpower series in a single variable x that have finite norm with respect to acertain inner product. For a full treatment of the electromagnetic field wewould need power series in many variables, one for each vibrational mode,but to keep things simple let us consider just a single mode. Then the powerseries 1 corresponds to the state in which no photons occupy this mode.More generally, the power series xn/n! corresponds to a state with n photonspresent.

Now, the relation between power series and structure types suggests anodd notion: perhaps FinSetx is a kind of categorified version of Fock space!This notion becomes a bit more plausible when one recalls that the powerseries xn/n! correspond to the structure type of being an n-element set. Infact, if we think about it carefully, what Einstein really did was to categorifyPlanck’s mysterious number ‘n’ by taking it to stand for an n-element set:a set of photons. So perhaps categorification can help us understand thequantized harmonic oscillator more deeply.

To test this notion more carefully, the first thing to check is whether wecan categorify the inner product on Fock space. In fact we can! Given powerseries

f(x) =

∞∑

n=0

fn

n!xn , g(x) =

∞∑

n=0

gn

n!xn ,

their Fock space inner product is given by

〈f, g〉 =

∞∑

n=0

fngn/n! .

Categorifying, we define the inner product of structure types F and G asfollows:

〈F, G〉 =∞

∑

n=0

(Fn × Gn)//n! .

Note that the result is a groupoid. If we take the cardinality of this groupoid,we get back the inner product of the generating functions of F and G:

|〈F, G〉| = 〈|F |, |G|〉.

23

Even better, this groupoid has a nice interpretation: its objects are finitesets equipped with both a structure of type F and a structure of type G, andits morphisms are bijections preserving both structures.

To go further, we should categorify linear operators on Fock space, since inquantum theory observables are described by self-adjoint operators. For this,we really need the ‘stuff types’ mentioned at the end of the previous section,and we need to use a little more category theory. The inner product onstructure types extends naturally to an inner product on stuff types, the innerproduct of U : F → FinSet0 and V : G → FinSet0 being a groupoid 〈F, G〉called their ‘weak pullback’. We define a stuff operator to be a groupoid Oequipped with a functor U : O → FinSet0 × FinSet0. Stuff operators reallywork very much like operators on Fock space. We can act on a stuff type Fby a stuff operator O and get a stuff type OF . There is a nice way to addstuff operators, there is a nice way to compose stuff operators using weakpullbacks, and any stuff operator O has an adjoint O∗ with

〈F, OG〉 ∼= 〈O∗F, G〉

for all stuff types F and G. Just as any stuff type F satisfying a certainfiniteness condition has a generating function |F | which is a vector in Fockspace, any stuff operator O satisfying a certain finiteness condition gives anoperator on Fock space, which we call |O|. This operator is characterized bythe fact that

|OF | = |O|(|F |)for any stuff type F for which |F | lies in Fock space. And as one would hope,we have

|O + O′| = |O| + |O′|, |OO′| = |O| |O′|, |O∗| = |O|∗.

for any stuff operators O and O′.In physics, the most important operators on Fock space are the annihi-

lation and creation operators, a and a∗. The annihilation operator acts asdifferentiation on power series:

(af)(x) = f ′(x),

while the creation operator acts as multiplication by x:

(a∗f)(x) = xf(x).

24

As their names suggest, these operators decrease or increase the number ofphotons by 1. These operators are adjoint to one another, but not inverses.Instead, they satisfy the famous relation

aa∗ − a∗a = 1.

While this relation plays a crucial role throughout quantum theory, it remainsslightly mysterious: why should first creating a photon and then destroyingone give a different result than first destroying one and then creating one?If categorification is going to help us understand the quantized harmonicoscillator, it must give an explanation of this fact.

To give such an explanation, we must first describe the stuff operator Athat serves as the categorified version of the annihilation operator. Since Aactually maps structure types to structure types, instead of giving its fulldefinition we shall take the more intuitive course of saying how it acts onstructure types. Given a structure type F , a structure of type AF on thefinite set S is defined to be a structure of type F on S + 1, where 1 is the1-element set. Note that

|AF |(x) =∞

∑

n=0

|(AF )n|n!

xn =∞

∑

n=0

|Fn+1|n!

xn =d

dx|F |(x),

or in short:|A| = a.

Next let us describe a categorified version of the creation operator. Wecould simply define A∗ to be the adjoint of A, but A∗ actually maps structuretypes to structure types, so let us say how it does so. For any structure typeF , a structure of type A∗F on S consists of a choice of element s ∈ S togetherwith a structure of type F on S − s. This is the same thing as choppingS into two disjoint parts and putting the structure of ‘being a 1-element set’on one part and a structure of type F on the other, so

A∗F = xF

and thus|A∗| = a∗.

Next we we should check that our annihilation and creation operatorssatisfy a categorified version of the commutation relations aa∗ − a∗a = 1. Of

25

course we wish to avoid minus signs, and having categorified we should speakof natural isomorphism rather than equality, so we write this as

AA∗ ∼= A∗A + 1.

We will not give a rigorous proof of this here; instead, we will we just sketchthe proof that AA∗F ∼= A∗AF + F when F is is a structure type. To put astructure of type AA∗F on a finite set S is to put a structure of type A∗Fon the set S + 1. This, in turn, is the same as choosing an element s ∈ S + 1and putting a structure of type F on S +1−s. Now either s ∈ S or s /∈ S.In the first case we are really just choosing an element s ∈ S and putting astructure of type AF on S − s. This is the same as putting a structure oftype A∗AF on S. In the second case we are really just putting a structureof type F on S. We thus have

AA∗F ∼= A∗AF + F

as desired.The interesting thing about this calculation is that it is purely combina-

torial. It reduces a fact about quantum theory to a fact about finite sets. Ifwe examine it carefully, it boils down to this: if we have a box with someballs in it, there is one more way to put an extra ball in and then take a ballout than there is to take a ball out and then put one in!

Starting from here one can march ahead, categorifying huge tracts ofquantum physics, all the way to the theory of Feynman diagrams. For nowwe content ourselves here with the briefest sketch of how this would go,leaving the details for later [7]. First, we define the field operator to be thestuff operator

Φ = A + A∗.

Our normalization here differs from the usual one in physics because we wishto avoid dividing by

√2, but all the usual physics formulas can be adapted

to this new normalization. More generally, we define the Wick powers ofΦ, denoted :Φp:, to be the stuff operators obtained by taking Φp, expandingit in terms of the annihilation and creation operators, and moving all theannihilation operators to the right of all the creation operators ‘by hand’,ignoring the fact that they do not commute. For example:

:Φ0: = 1

26

:Φ1: = A + A∗

:Φ2: = A2 + 2A∗A + A∗2

:Φ3: = A3 + 3A∗A2 + 3A∗2A + A∗3

and so on.In quantum field theory one spends a lot of time doing calculations with

products of Wick powers. In the categorified context, these are stuff operatorsof the form

:Φp1 : · · · :Φpk : .

What is such a stuff operator like? In general, it does not map structuretypes to structure types, so we really need to think of it as a groupoid Oequipped with a functor U : O → FinSet0 × FinSet0. An object of O is aFeynman diagram: a graph with k vertices of valence p1, . . . , pk respectively,together with univalent vertices labelled by the elements of two finite sets,say S and T . We draw Feynman diagrams like this:

We draw the vertices labelled by elements of S on the top and think ofthem as ‘outgoing’ particles. We draw the vertices labelled by elements ofT on bottom, and think of them as ‘incoming’ particles. We can think ofthe Feynman diagram as the pair of sets (S, T ) equipped with extra ‘stuff’,namely the body of the diagram. The functor U : O → FinSet0 × FinSet0

sends the Feynman diagram to the pair (S, T ).While Feynman diagrams began as a kind of bookkeeping device for quan-

tum field theory calculations involving Wick powers, they have increasinglytaken on a life of their own. Here we see that they arise naturally from thetheory of finite sets equipped with extra structure, or more generally, extra‘stuff’. Since categorification eliminates problems with divergent power se-ries (as we have already seen in the previous section), it is tempting to hopethat this viewpoint will help us understand the divergences that have always

27

plagued the theory of Feynman diagrams. Of course this may or may nothappen — it is difficult to predict, especially when it comes to the future.

Acknowledgements

We dedicate this paper to Gian-Carlo Rota and Irving Segal.

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