bruce w. westbury- presentations of spherical categories

8/3/2019 Bruce W. Westbury- Presentations of Spherical Categories

1/26

PRESENTATIONS OF SPHERICAL CATEGORIES

BRUCE W. WESTBURY

Abstract.

Shall we by Topology cast out Topology? [Tut79]

1. Introduction

In this paper we study the use of diagrams as a setting for ten-sor calculations. The first use of a graphical notation for algebraiccalculations that we are aware of is the graphical method developed

by Clifford for the description of invariants and covariants of binaryforms. A modern account of this method is given in [OS89] and [Olv99,Chapter 7]. However the use of graphical notation for tensor calcula-tions originated in mathematical physics and specifically in the workof Roger Penrose on spin networks. This is a calculus for recouplingtheory and 6j-symbols. This method was developed further by Cvi-tanovic in [Cvi77],[Cvi84],[Cvi] where the diagrams are referred to asbirdtracks; and also in [Ken82]. There is an account of this method in[PR87, Chapter 2] which refers to abstract indices. My impression isthat other people in this community may have made use of diagramsprivately but that in their publications they reverted to index nota-

tion for tensor calculations. More recent papers in the same vein are[Wes03], [Kim] and [MT06].The situation changed abruptly with the introduction of quantum

groups. The simplest quantum group is Uq(sl(2)). The representationtheory of this Hopf algebra can be understood using a q-analogue of thespin-network calculus. There are accounts of this in [KL94], [CFS95]and [FK97]. One application of this is the construction of a series oflink invariants the simplest of which is the Jones polynomial. Follow-ing this it was understood that there is an intimate relation betweenabstract tensor categories and topology in two (and three) dimensions;see [JS91], [FY89], [Shu94].

The next development was the work of Greg Kuperberg in [Kup96].This paper gives, for each of three simple Lie algebras of rank two,a graphical calculus for the invariant tensors of the fundamental rep-resentations. This gives a description of the categories of invarianttensors in terms of generators and relations. This raises the problem

Date: November 9, 2007.1


2/26

2 BRUCE W. WESTBURY

of extending this to higher rank simple Lie algebras. This problem issolved for the rank three simple Lie algebra so(7) in [Wes06].

A separate and independent development was the work by VaughanJones on planar algebras, see [Jon] and [Bis02]. The motivation for theintroduction of planar algebras is that a planar algebra is an algebraicobject and that there is a planar algebra associated to a finite indexsubfactor. This planar algebra is a powerful invariant of the subfactor.This raises the problem of describing the planar algebra of a subfactorin terms of generators and relations.

These two problems both involve working with generators and re-lations where instead of words we have planar graphs. The originalmotivation for this paper was the desire to develop algorithms for thesecalculations and to implement them. For example, the paper [SW06]gives a version of the Knuth-Bendix algorithm for planar graphs. TheKnuth-Bendix algorithm takes a finite set of rewrite rules and com-pletes it to a confluent set of rewrite rules. The accepted definition ofa planar graph is not suited to this.

The standard approach to defining a planar graph is that it is anisotopy equivalence class of a graph embedded in a surface. Usuallythis is all that is said. However it seems that most authors have inmind not an embedding of a graph but instead a closed subset of thesurface such that the surface has a finite open cover with a prescribedset of possibilities for each open set. Then there is a choice of thestructure of a surface; the usual choices are smooth or piecewise-linear.However most of the illustrations produced using computer graphicsare based on curves which are piecewise polynomial with continuousfirst derivative. The accepted view on this is that none of these choicesmatter since we are only considering graphs up to isotopy. This meansthat a graph has an infinite dimensional group of automorphisms andthat we take the orbits of the action of the connected component ofthis group. Then there are the operations. The basic operation is tosew (or glue) along two intervals or two circles in the boundary. Themeaning here is that we choose a representative of each isotopy classso that the boundaries match up (in the smooth category this meanseither using collars or else smoothing corners); then sew (or glue) theserepresentatives and take the isotopy class of the result. It takes somework to show that this is independent of the various choices and thisis invariably omitted. The analogous result in higher dimensions isfalse; for example, the naive generalisations of this to define weak n-categories of cobordisms or tangles fail because of this issue.

The original purpose of this paper was to provide an alternativedefinition of planar graph. The requirements were that the definitionshould be combinatorial; that the equivalence relation should be iso-morphism; and that the basic operation of sewing (or glueing) shouldbe defined. In this paper we define cyclic graphs and show that all


3/26

PRESENTATIONS OF SPHERICAL CATEGORIES 3

these requirements are met. This gives a simpler and more rigorousfoundation for the subject. However we still find the usual diagramsmore appealing and we will continue to make use of them.

The secondary purpose of this paper is to show that the algebraicstructure that is obtained by taking planar graphs with the sewing(or glueing) operation has been reinvented several times. Specifically,the following are all equivalent (in the sense that they are objects ofcategories which are all equivalent): spherical categories, in [BW99],spiders, in [Kup96], planar algebras, in [Jon]. This structure is calleda Clebsch-Gordan operad in [GK94] and is the operadic formulationin [FK97]. Furthermore this is an example of a generalized operad asdefined in [BM06]. Moreover this also makes it clear that the work inthis paper adheres to their maxim:

One can and must approach operadic constructions fromvarious directions and with various stocks of analogies.

There is a version of these constructions using abstract graphs andthis corresponds to abstract indices, in [PR87, Chapter 2]. There is abrief account of this in the final section.

2. Cyclic graphs

In this section we introduce cyclic graphs and sewing which is thebasic operation on cyclic graphs.

2.1. Closed surfaces. An elementary construction of a surface is tostart with a finite collection of disjoint polygons. Each polygon istaken to be the convex hull of its vertices in the Euclidean plane. Thenwe identify the edges in pairs using a linear homeomorphism. If eachidentification is orientation reversing then the surface is oriented.

Then we associate to the surface the following combinatorial data.Take the 1-complex of the edges of the polygons (after the identifica-tions). Let F be the set of pairs consisting of a vertex contained in anedge. Then there is an involution on F since each edge has two ver-tices. There is also a bijection which moves an edge clockwise arounda vertex. Then the oriented surface can be recovered from the set F

with these two maps.

2.2. Boundaries. A directed graph consists of two sets V and Eand two maps h, t : E V.

A homomorphism of directed graphs : consists of two setmaps V : V V and E : E E

such that the following two


4/26

4 BRUCE W. WESTBURY

diagrams commute:

VV

V

h

h

E

E E

VV

V

t

t

E

E E

For any directed graph the opposite graph op is defined by

Vop = V Eop = E hop = t top = h

Definition 2.1. A boundary B is a directed graph hB, tB : EB VBsuch that hB and tB are inclusions.

A boundary B is closed if hB and tB are both bijections.

Definition 2.2. The dual of a boundaryB is Bop, the opposite directedgraph.

A homomorphism of boundaries is a homomorphism of the underly-ing directed graphs.

Definition 2.3. If B is a boundary then we define B to be B+ B where B+ is the complement of the image of hB and B is thecomplement of the image of tB.

A directed graph has a geometric realisation which is a 1-complex.Each connected component of the geometric realisation of a boundary iseither a point, an interval or a circle. Usually we have B+ B = . Inthis case B is an oriented (or signed) set and the geometric realisationis a 1-manifold.

2.3. Cyclic graphs.

Definition 2.4. A cyclic graph consists of disjoint sets C, B, Eand a subset I (B E) together with :

Two inclusions h, t : B C with the same image. An involution e : E E. A bijection c : B E B E.

such that if x I then xe B or xce B or xe = xc and such thatc restricts to an involution on I.

A cyclic graph is closed if C = . In this case B = and I =

and so the set E has an involution e and a bijection c as in theIntroduction.

If is a cyclic graph then is a boundary. This is constructed asfollows. The set E is B and the set V is the common image of hand t. The map h is h and the map t is t.

In this paper we will only consider a restricted form of homomor-phism between cyclic graphs.


5/26


{emb}Definition 2.1. An embedding of cyclic graphs is an inclusion : E E such that c = c and xe = xe for all x for whichboth of these are defined.

Definition 2.5. Let be a cyclic graph. Then the dual cyclic graph is

op

. This is constructed as follows. The sets are the same sets so thatCop = C, Bop = B, Eop = E, Iop = I

The maps are defined by

hop = t, top = h, eop = e, cop = c1

There is the following compatibility between taking duals and theboundary. If is a cyclic graph then the boundaries ()op and (op)are canonically isomorphic.

2.4. Geometric realisation. In this section we explain the correspon-dence between cyclic graphs and graphs in a surface up to isotopy.

This is included to justify the proposal that cyclic graphs are a substi-tute for the usual notion of a graphs in a surface up to isotopy.

Definition 2.2. There is a directed graph with vertices the set

C B E

The arrows are defined as follows. For each x B we have edgesx xh and x xt. For each pair x, y E with xe = y andye = x we have an undirected edge x y. For each x B E wehave an edge x xc.

Next we define a cyclic ordering of the edges incident to each vertex.

This ordering is given by taking the clockwise ordering of the verticesin Figure 1. In this Figure a vertex labeled C is an element of C, avertex labeled B is an element ofB, a vertex labeled E is an elementof E, a vertex labeled I is an element of I, a vertex labeled V is anelement of E I, a vertex labeled U is an element of B E.

This is a cyclic graph and so corresponds to an oriented surface.The geometric realisationof is obtained from this oriented surface byomitting all polygons whose edges are all labeled h or t. This gives anoriented surface with boundary.

Next we draw a piecewise linear graph in this surface. This surfacecontains some polygons with all edges labeled c and some digons with

both edges labeled e. For each of these polygons choose a point in theinterior of the polygon and connect it to each vertex of the polygon bya straight line.

This gives a graph in a surface. Moreover the isotopy equivalenceclass of this graph in a surface is independent of the choices involved inthe construction. Hence we have a well-defined map from cyclic graphsto isotopy equivalence classes of graphs in surfaces.


6/26

6 BRUCE W. WESTBURY

B C Bh t

C B C

U U

t hc c

V

U

E

U

c

ce

V I Iec

c

Figure 1. Vertices {gr}

E E

E E

B

C

B

C

h

th

t

e e

c

c

c

c

cc

Figure 2. Example{bu}

Now suppose we are given a graph in a surface. Assume that everyloop contains a vertex and that no edge has both endpoints on theboundary. This assumption does not involve any loss of generalitysince we can put a vertex in the middle of any edge which fails eitherof these conditions. Let B be the set of boundary points and let O bethe set of interior vertices. Let E B be the set of pairs (e, v) wheree is an edge of the graph, v is an interior vertex and v is incident toe. This set has a bijection c which moves each pair clockwise aboutv. For each (e, v) if the other vertex incident to e is a boundary pointthen (e, v) B and if the the other vertex is u then (e, v) E and weput (e, v)e = (e, u).

This defines a cyclic graph. Moreover the cyclic graph only dependson the isotopy class of the graph in a surface. These two maps areinverse bijections between cyclic graphs and isotopy classes of graphsin a surface.

2.5. Normal form. Define a pre cyclic graph by the same data as fora cyclic graph but without the final condition. If is a pre cyclic graph


7/26


C

B

C

B

C

B

ht

h

t h

t

c

c c

Figure 3. Example

I Ie

c

c

I I

C C

C C

c

c

Figure 4. Exceptions {ex}

then the set R() is the set

R() = {x I|xe / B and xe = xc}

Then a pre cyclic graph is a cyclic graph if and only if R() = .Figure 4 shows two ways in which elements of R() can arise. In factthese are the only possibilities.

Next we describe a construction for simplifying a pre cyclic graph.Let be a pre cyclic graph with an element x R(). Then weconstruct a pre cyclic graph /x. Put y = xc. Then there are twocases; y E and y B.

Assume y E. The sets are defined by

C/x = C, B/x = B, E/x = E, I/x = I/{x, y}

The maps are defined as follows

h/x = h, t/x = t, c/x = c

The involution c/x : I/x I/x is the restriction of the map c : I

I. The involution e/x is defined by

ue/x =

ye if u = xexe if u = yeue otherwise

This is illustrated in Figure 5.


8/26

8 BRUCE W. WESTBURY

E x y Ee ec

cE Ee

Figure 5. Internal simplification {si}

x y

C

C

E ec

c

t

h

B

C

C

t

h

Figure 6. Boundary simplification{sb}

Assume y B. The sets are defined by

C/x = C, E/x = E, I/x = I/{x, y}

and the set B/x is obtained from B by deleting the element y andinserting the element x. The maps h/x, t/x are defined as follows

uh/x =

uh if u = x

yh if u = xut/x =

ut if u = x

yt if u = x

The map c/x is c. The involution c/x : I/x I/x is the restrictionof the map c : I I. The involution e/x is defined by

ue/x =

xe if u = ye

ue otherwise

This is illustrated in Figure 6.Ify R(/x) then the preimage is an element ofR(). This includes

R(/x) in R() as a proper subset and so |R(/x)| < |R()|. Thisshows that if we are given a pre cyclic graph then we obtain a cyclicgraph by a finite sequence of simplifications. Next we show that thiscyclic graph is independent of the choice of the sequence.

Construct a directed graph with vertices pre cyclic graphs and adirected edge /x for all and all x R(). We have alreadyobserved that the simplifications are terminal. This means that this


9/26


is the directed graph of a partially ordered set and that the partiallyordered set has no infinite descending sequence. Therefore in order toshow that this is globally confluent it is sufficient to show that this islocally confluent. This is the diamond lemma from [New42].

The statement that the simplifications are locally confluent is thefollowing statement. For any pre cyclic graph with two distinctelements x, y R(), there is a pre cyclic graph which can beobtained from both /x and /y by a finite sequence of simplifications.The typical situation is that the image of y is an element of R(/x)and x is an element of R(/y). In this situation the pre cyclic graphs(/x)/y and (/y)/x are the same so we take to be this pre cyclicgraph. Put x = xc and y

= yc. Then the exceptions to the typicalsituation are:

(1) y = xe, x = ye, y = xe, x

= ye.(2) y = xe, x = y

e, y = xe, x

= ye.(3) y = xe, x = ye, y

B, y B.

In each of these situations the pre cyclic graphs /x and /y are thesame so we take to be this pre cyclic graph.

2.6. Labels. In this construction we will consider edge labeled cyclicgraphs. It is also possible to label the vertices as well.

The usual notion of an edge labeling is given by taking a finite set with an involution . Then an edge labeling of a closed cyclic graphF is a function : F such that i = . Here we consider ageneralisation and take to be a directed graph. with an ant-involution. This means that : E E and : V V are involutions andthat they satisfy the conditions h = t and t = h.

An edge labelingof a pre cylic graph F consists of set maps E: F E and V : F V. The map V is required to be invariant under icso that it gives a labeling of the faces of F by V. The first conditionon E is that it is compatible with taking duals. This means thatiE = E. The other condition on E is that Et = V and Eh = iV.A less formal way to express this is that for each x F,

V(x)E(x) V(ix)

is an edge in .

2.7. Sewing. The data for sewingconsists of a cyclic graph , a bound-

ary B and an embedding : B Bop

. Given sewing data wewill construct a cyclic graph / together with an embedding (see Def-inition 2.1) /. Given sewing data then we have embeddings+ : B and : Bop such that the inclusions : VB Cand : EB B have disjoint images.

First we construct a pre cyclic graph. First we construct the sets.

Define C/ to be (C/(VBBop)) B.


10/26

10 BRUCE W. WESTBURY

Define B/ to be B/ ((EBBop)). Define E/ to be E EB EB. Define I/ to be I EB EB.

Next we construct the maps

The maps h/ and t/ are defined by

xh/ =

v if xh = (v)

xh if xh / (VBBop), xt/ =

v if xt = (v)

xt if xt / (VBBop)

The map e/ is defined as follows. For y EBBop put ye/ =(y) and for x E I define

xe/ =

y if (y) = x

xe otherwise

The map c/ is the restriction of c. Let r be the involution on EB EB which sends an element

in one subset EB to the same element in the other subset EB.Then c/ is defined to be c r.

Now simplify this pre cyclic graph to obtain a cyclic graph. This isthe required cyclic graph.

The canonical embedding / arises from the construction ofE/ from E.

{ass}2.8. Associativity. The basic property of sewing is the following as-sociativity condition. Assume that we are given a cyclic graph , twoboundaries B1 and B2 and embeddings i : Bi B

opi . for i = 1, 2.

Put B = B1 B2 and assume that = 1 2 : B Bop is also anembedding. Then there is an embedding 2 : B2 Bop2 /1 definedas the composite

2 : B2 Bop2 /1

Then the associativity condition is that (/1)/2 and / are iso-morphic and the isomorphism is compatible with the embeddings (/1)/2 and /. Then it follows that the cyclic graphs (/1)/2and (/2)/1 are isomorphic and that the isomorphism respects theembeddings of .

2.9. Cobordisms. In this section we construct a 2-category with du-

als. The intuition for this is that this is a 2-category of cobordisms ofsurfaces with corners. This is a digression since we will not make useof this construction in this paper.

Let be a directed graph with anti-involution which acts as theidentity on vertices. Then we have the free category on whose mor-phisms are directed paths in . Then the anti-involution on extendsuniquely to an anti-involution on this category. The intuition for this


11/26


is that taking the geometric realisations of morphisms gives a cobor-dism category whose objects are 0-manifolds and whose morphisms are1-manifolds and the identity maps.

Let B1 and B2 be 1-morphisms. Then a 2-morphism B1 B2is a labeled cyclic graph with boundary B1 B

op2 . Then the com-

position Hom(B1, B2) Hom(B2, B2) Hom(B1, B3) is constructedusing sewing. Let 1 Hom(B1, B2) and 2 Hom(B2, B3). Put = 1 2. Then = B1 B

op2 B2 B

op3 so we have an inclusion of

Bop2 B2 in . This gives sewing data and the boundary after sewing isB1 B

op3 so this gives the composite in Hom(B1, B3). The associativity

of this composition follows from the associativity of sewing discussedin 2.8.

Let B be a boundary so we have a directed graph hB, tB : EB VB such that hB and tB are inclusions. Assume that the geometricrealisation is an interval. Then we construct the boundary B Bop.Let VB be the set v0, . . . , vn and EB the set b1, . . . , bn. Then we takeB to be the union of the sets v1, . . . , vn and v

1, . . . , v

n . Take C to be

the union of the sets b0, . . . , bn and b0, . . . , bn with the identificationsb0 = b

n and bn = b

0. Then we put vih = bi, vit = bi1, v

ih = b

i1,

vit = bi. The set E is empty and I = B. Then the involution c on

E is defined by vic = vni, v

ic = vni.

There is a second composition of 2-morphisms. This is constructedusing sewing with boundaries with no edges. Let 1 Hom(B1, B1)and 2 Hom(B2, B2). Let B1, B

1 : C1 C

op2 and B2, B

2 : C2 C

op3 .

Put = 1 2. Then we have sewing data Cop2 C2 . The

composite is constructed by sewing.The proof that these compositions satisfy the conditions for a strict

2-category are omitted.

3. Spiders

In this section we give the definition of a linear spider. This is basedon the definition of spider in [Kup96]. Moreover the two definitions areequivalent in the case that the -category has one object.

{sp}3.1. Spiders. A -categoryis a category with an anti-involution such

that is the identity map on objects. In particular, ifa is any morphismthen the composite aa is defined.

A -category with one object gives a monoid with an anti-involutionand conversely any monoid with anti-involution gives a -category withone object.

Let (M, ) be a -category and let K be a commutative ring. Thedata is a K-module W(a) for each morphism a M such that the


12/26


source and target of a are equal. The operations are

W(ab) W(ba)(1) {rot1}

W(a) W(b) W(ab)(2) {prod1}

W(abbc) W(ac)(3) {st1}

W(ac) W(abb

c)(4) {un1}Then we have a number of compatibility conditions.The map W(ae) W(ea) is the identity map on W(a). The follow-

ing are identity maps

W(aeeb) W(ab)(5)

W(ab) W(aeeb)(6)

The following composite is the identity map

(7) W(abc) W(cab) W(bca) W(abc)

The following composites are identity maps

W(abc) W(abbbc) W(abc)(8)

W(abc) W(abbbc) W(abc)(9)

These are sometimes known as the two zig-zag identities. The followingtwo maps are equal

W(ad) W(abccbd)(10)

W(ad) W(abbd) W(abccbd)(11)

The following two maps are equal

W(abccbd) W(ad)(12)

W(abcc

b

d) W(abb

d) W(ad)(13)The following diagram commutes

(14)

W(abbcdde) W(acdde) W(abbce) W(ace)

The following diagram commutes

(15)

W(ace) W(acdde)

W(abbce) W(abbcdde)

The following diagram commutes

(16)

W(a) W(b) W(c) W(ab) W(c) W(a) W(bc) W(abc)


13/26


The following diagrams commute

(17)

W(abbcd) W(dabbc)

W(acd) W(dac)

(18) {rotc3}

W(acd) W(abbcd) W(dac) W(dabbc)

The definition of a plain spider is obtained by requiring each W(a)to be a set and replacing the tensor product in (2) and (16) by theproduct of sets.

Any plain spider gives a linear spider by replacing the set W(a) by thefree K-module on this set and extending the maps by (multi)-linearity.

A natural generalisation of the definition which includes both of thesecases is to take a monoidal category M and then to take W(a) to bean object of M and then to replace the tensor product of K-modulesby tensor product in M.

This generalisation can be taken further. Note that we can write thedefinition using multi-linear maps to avoid any tensor products. Thisshows that we can define spiders enriched in a multicategory (a.k.a.coloured operad). Further generalisations are possible using the generaltheory of enrichment in [Lei02].

3.2. Example. In this section we will construct a plain spider using

cyclic graphs. This is the main construction of the paper.Let W be the set of (isomorphism classes of) edge labeled cyclic

graphs whose geometric realisation is a disc. Then each element of W

has one boundary component and the boundary edges are a directedcycle in . Let W be the set of pairs (w, m) where w W and m is atotal order on the boundary edges ofw compatible with the cyclic order.In terms of the geometric realisation this means choosing a basepointon the boundary of the disc which is not one of the boundary pointsof the embedded graph. This choice also corresponds to choosing adirected path in whose start and end points are equal and such thatthe associated directed cycle is the boundary of w.

Then we have a map from W to the set of directed paths in thedirected graph . If a is a directed path then define W(a) to be thefibre of this map. Note that W(a) will be empty if the start and endpoints of a are distinct.

Next we have to define the structure maps. Let a and b be twodirected paths in such that ab and ba are both directed paths. Thenab and ba each have the same start and end points. Then ab and ba


14/26


I

C

I

C

h t

ht

c

c

Figure 7. Cup{cup}

both arise from the same directed cycle in and correspond to differentchoices of the total order. This map is referred to as rotation.

The maps W(a)W(b) W(ab) are constructed using sewing. Takean element of W(a) and an element of W(b) and form the disjointunion . This has sewing data consisting of two points of C onein each component. Then sewing gives an element of W(ab). andW(abbc) W(ac) are constructed using sewing.

The operations W(abbc) W(ac) and W(ac) W(abbc) are bothdefined using sewing and the cylic graph in Figure 7.

Let W(abb

c). Let B be the boundary of the cyclic graph inFigure 7 regarded as an interval. Then we have have an inclusion ofBop in . This gives sewing data in the disjoint union and sewinggives an element of W(ac).

Let W(ac). Let B be one of the points marked C in the boundaryof the cyclic graph in Figure 7. Then we have have an inclusion of Bop

in . This gives sewing data in the disjoint union and sewing givesan element of W(abbc).

Then the claim is that these maps satisfy the conditions for a plainspider. Several of the conditions follow from the associativity propertyin 2.8.

4. Spherical categories

In this section we give a definition of a strict 2-category with strictduals. This is a generalisation of the definition of a strict sphericalcategory given in [BW99] which is the special case in which the 2-category has one object.

4.1. Duals. Let x and y be objects in a monoidal category C. Theny is left dual to x and x is right dual to y means that we are givenmorphisms e : y x 1 and i : 1 x y such that the following twodiagrams commute

(19)

1 x x y x x x 1

x 1 x y x x 1 x

An alternative definition is that we have natural isomorphisms Hom(ax, b) = Hom(a, b y). Here natural means that the following diagrams


15/26


commute

(20)

Hom(a, b) Hom(b x, c) Hom(a, b) Hom(b, c y) Hom(a x, c) Hom(a, c y)

(21)

Hom(a x, b) Hom(b, c) Hom(a, b y) Hom(b, c) Hom(a x, c) Hom(a, c y)

The equivalence between is these definitions is as follows. Givennatural isomorphisms put a = 1 and b = x. Then we have naturalisomorphisms Hom(x, x) = Hom(1, xy) so we take i to be the elementcorresponding to the identity map. Put b = 1 and a = y then we havenatural isomorphisms Hom(y x, 1) = Hom(y, y) so we take e to bethe element corresponding to the identity map.

Conversely given elements i and e then we construct inverse naturalisomorphisms by

(a xf

y) (aai

a x yfy

b y)

(ag

b y) (a xgx

b y xbe

b)

The definition of a pivotal category is given in [FY89]. A pivotalcategory is a -category such that for all objects x, x is left and rightdual to x. Here we require that is an anti-involution of monoidalcategories. In addition we require that both of the following identitieshold for all x,

(ix) = ex (ex)

= ix{eqv1}

4.2. Spiders from spherical categories. First we assume we aregiven a spherical category and then we construct a spider. Let M bethe set of isomorphism classes of objects of the category. Then M isa monoid with product given by the tensor product. This monoid hasan anti-involution given by the -functor. Construct a vector space foreach object by W(a) = Hom(e, a). The map W(a) W(b) W(a b)is the tensor product. The map W(abbc) W(ac) is defined bycomposition with 1a eb 1c. The map W(ac) W(abbc) is definedto be the composite

W(ac) W(ca) W(cabb) W(abbc)

where the first and third arrows are rotations and the middle arrow istensoring with ib.

To define rotation. Note that we have inverse isomorphisms

Hom(e, b a) Hom(a, b) Hom(a, b) Hom(e, b a)


16/26


The first map sends f: e b a to the composite

afa

b a abe

b

The second map sends g : a b to the composite

ei

a a ga

b a

Similarly we have inverse isomorphisms

Hom(e, a b) Hom(a, b) Hom(a, b) Hom(e, a b)

The first map sends f: e a b to the composite

aai

a a beb

b

The second map sends g : a b to the composite

ei

a aga

b a

This gives isomorphisms W(ba) = Hom(a, b) = W(ab).

{eqv2}4.3. Spherical categories from spiders. Now we assume we aregiven a spider and we construct a spherical category. Given a spi-der we define Hom(a, b) to be W(ab). The composition Hom(a, b) Hom(b, c) Hom(a, c) is given by the composition

W(ab) W(bc) W(abbc) W(ac)

The unit in Hom(a, a) is given by the inclusion W(e) W(aa). Then

we can check that we have a category.The tensor product Hom(a, b) Hom(c, d) Hom(a c, b d) isdefined by the composite

W(ab) W(dc) W(abdc) W(cabd)

Then we check that this is a monoidal category.The functor is defined by either of the following composites

W(ab) W(abaa) W(aaba) W(ba)(22)

W(ab) W(bbab) W(babb) W(ba)(23)

Then we can check that this is an anti-involution of monoidal cate-gories.

The isomorphisms Hom(a x, b) = Hom(a, b x) are the isomor-phisms W(xab) = W(abx). Then we can check that these arenatural isomorphisms.

These are the conditions for a pivotal category. Then we can checkthat this is in fact a spherical category.


17/26


5. Planar algebras

In this section we discuss the planar algebras of [Jon]. The originalmotivation was the construction discussed in [Bis02] which associates aplanar algebra to a finite index subfactor. Planar algebras are algebrasfor a multicategory. Multicategories were introduced in [Lam69] and

are also known as non-symmetric coloured operads. A non-symmetricoperad is a multicategory with one colour. For more information onmulticategories see [Lei04, Chapters 2,4,6]. In [Jon] planar algebrasare defined as algebras over a multicategory with two colours. In thispaper we modify this definition to allow for more colours.

The data for a multicategory consists of a set C0 whose elements arecalled colours; for each n and a1, . . . , an, a C0 a set C(a1, . . . , an; a)whose elements are called arrows; for each n, k1, . . . kn and a, ai, a

ji C0

a function

C(a1, . . . , an; a) C(a11, . . . , a

k11 ; a1) . . . C (a

1n, . . . , a

knn ; an)

C(a11, . . . , ak11 , . . . , a1n, . . . , aknn ; a)

called composition and written

(, 1, . . . , n) (1, . . . , n)

For each a C0 an element 1a C(a; a).The conditions on this data are the associativity condition that

1 (11, . . . ,

k11 ), . . . , n (

1n, . . . ,

knn )

= ( (1, . . . , n)) (11, . . . ,

k11 , . . . ,

1n, . . . ,

knn )

whenever these composites are defined; and the unit condition that (1a1, . . . , 1an) = = 1a ()

for all arrows : a1, . . . , an a.An operad is a multicategory for which C0 has one element. Any

monoidal category gives a multicategory T by taking

T(b1 bn; a) = Hom(b1 bn, a)

Then we will construct a multicategory P. The set of colours is theset of directed paths in whose start and end points are the same.

The first step in the construction of the multicategory is the construc-

tion of a set P(a1, . . . , an; a) of cyclic graphs for all objects a1, . . . , anand a. It is more convenient to describe the set of geometric realisationsof these cyclic graphs.

First draw a large circle with centre on the x-axis. Then inside itput n disjoint discs each with centre on the x-axis. The outside circleand the boundaries of the n inside discs are all given a basepoint bytaking the leftmost of the two points of intersection of the circle with


18/26


the x-axis. Let be the large disc with the interiors of the inside discsremoved.

Each ai and a is a directed cycle in the directed graph. Let |a| bethe number of edges in the cycle a and similarly for ai. Mark |a| pointson the outside circle and mark |ai| points on the i-th inside circle.The marked points are required to be distinct and none of them is thebasepoint. Next draw arcs in with no intersections such that eacharc connects two marked points and each marked point is an endpointof precisely one arc. In particular if |ai| + + |an| + |a| is odd thenthis is not possible. The connected components of the complement in of these arcs are called regions. These diagrams are considered up toisotopy of preserving the basepoints. Next label the diagram. Eacharc is directed and labelled by an edge of the directed graph. Eachregion is labelled by a vertex of the directed graph. We require thatthe labels on the regions on the two sides of an edge are the source andtarget of the label on the edge. A diagram need not admit a colouringsatisfying these conditions.

Example 5.1. In [Jon] the regions are labelled black and white. Inthis case the directed graph has two vertices and one undirected edgeconnecting them.

There is a labelled cyclic graph associated to each isotopy class oflabelled diagrams. These are the elements of the set P(a1, . . . , an; a).These labeled cyclic graphs have closed boundary, are planar and theset E is empty. In fact they are characterised by these properties.

Then to construct the multicategory it remains to construct the com-positions. These are constructed using sewing. Assume we are given

P(a1, . . . , an; a) and i P(a1

i , . . . , a

ki

i ; ai) for 1 i n. Thentake the union 1 n. Then the composite is defined bysewing using the sewing data associated to a1 an. The identi-ties are cyclic graphs with E empty and whose geometric realisationsare cylinders or annuli. The proof that this composition and identitiessatisfy the conditions for a multicategory are omitted.

A plain planar algebrais a map of multicategories from P to the mul-ticategory of sets. Let K be a commutative ring. Then a linear planaralgebra over K is a map of multicategories from P to the multicategoryof K-modules.

Equivalently, a plain planar algebra can be defined as having a set

W(a) for each a and for each P(a1, . . . , an; a) a mapW() : W(a1) W(an) W(a)

which satisfy associativity conditions. We also require units whichsatisfy the unital conditions.

There is a planar algebra whose set W(a) is the set of labeled cyclicgraphs whose boundary is a and whose geometric realisation is a disc.


19/26


Figure 8. Composition {comp}

Figure 9. Tensor product {tensor}

Figure 10. Duals {duals}

The operations W() are constructed using sewing. The associativityconditions follow from the associativity in 2.8.

{p2s}5.1. Spherical categories from planar algebras. In this section weshow how to construct a spherical category and a spider given a planaralgebra. In this section we revert to using diagrams rather than cyclicgraphs.

First we show that the operations in a spherical category are partic-

ular cases of operations in a planar algebra. The composition is givenin Figure 8, the tensor product is given in Figure 9, the duality functoris given in Figure 10 and the adjunctions are given in Figure 11.

Next we show that the spider operations in 3.1 are particular casesof operations in a planar algebra. The operation (1) is shown in Figure12; the operation (2) is shown in Figure 13; the operation (3) is shownin Figure 14; and the operation (4) is shown in Figure 15.


20/26


Figure 11. Adjunctions {adj}

Figure 12. Rotate {rot}

Figure 13. Join {join}

Figure 14. Stitch {stitch}

Given a planar algebra we have constructed both a spider and aspherical category. These correspond under the equivalence in 4.2and 4.3.

{s2p}5.2. Planar algebras from spherical categories. In this section we

sketch the construction of a planar algebra from a spherical categoryor spider.

Here we show that given a spider or spherical category we can con-struct a planar algebra. This is an inverse to the construction of aspider or spherical category from a planar algebra and this this showsthat the categories of spiders, spherical categories and planar algebrasare all equivalent. A consequence of the proof is that the definition of a


21/26


Figure 15. Unnamed {unn}

Figure 16. Planar operation {plop}

spider can be streamlined by dropping the rotation operation and theconditions involving this operation.

First we discuss Temperley-Lieb diagrams. These are labeled cyclicgraphs with E the empty set and whose geometric realisation can beembedded in the plane. This gives both a spherical category and a pla-nar algebra. These are the simplest examples and have the universalproperty that they are included in any further example. Furthermorethe spherical category constructed from the Temperley-Lieb planar al-gebra is isomorphic to the Temperley-Lieb spherical category.

First of all any planar algebra operation can be constructed using thespherical category operations. Let P(a1, . . . , an; a). Put W(a) =

Hom(e, a). Then we can construct W() as a compositee b0 bn b0 a1 b1 an bn a

where the first and last maps are Temperley-Lieb morphisms. This isillustrated in the case n = 3 in Figure 16. In this Figure we inserta Temperley-Lieb diagram in the top and bottom rectangles. Thisdetermines .

This shows that we have enough operations in the spherical category.The issue is to show that the finite list of compatibility conditions implythe infinite list of conditions for a planar algebra.

{pls}Proposition 5.1. Any two choices of pairs of Temperley-Lieb diagrams

give the same if and only if the two maps are equal.

Proof. If we fix then the pair of Temperley-Lieb diagrams is not de-termined. However we can consider the condition that no arc in eitherTemperley-Lieb diagram connects two lines in the same bi. Then foreach there is a unique pair of Temperley-Lieb diagrams which sat-isfies this condition. Furthermore given any pair of Temperley-Lieb


22/26


Figure 17. Moves{mv1}

Figure 18. Moves{mv2}

diagrams we can convert it to a unique pair which satisfies this con-dition by a finite sequence of the two moves in Figure 17 and Figure18.

This shows that the operation W() is well-defined. The units in theplanar algebra are constructed from the identity maps in the categoryand it is clear that these satisfy the unit conditions. The associativityconditions are a direct consequence of Proposition 5.1.

6. Monads and algebras

In this section we fix a -category and consider plain spiders, spher-ical categories and planar algebras over this -category. In each casewe will consider these as the objects of a category. The results in theprevious sections construct equivalences between these categories. In

this section we give a different perspective on these equivalences ofcategories.

In each of these three cases we have a set W(a) for each morphisma M such that the source and target of a are equal. For spiders andplanar algebras this data is given explicitly. For spherical categorieswe take the set Hom(e, a). Let A be the set of morphisms of M suchthat the source and target are equal. Then this data can be regardedas the objects of Set/A, the category of sets over A. Then in each casetaking this data is a functor by construction. Then the adjoint func-tor applies and this says that these functors have left adjoints. ThenBecks monadicity theorem also applies and says that these functors

are monadic. However this is not particularly useful.Given an object X A of Set/A we define another object W(X)

A. The set W(X) is the set of all labeled cyclic graphs with labels in Xwhose geometric realisation is a disc. The map W(X) A is the mapto the boundary. Then it is clear that this is a functor on Set/A. Thereis an obvious natural transformation 1 W. There is also a naturaltransformation W W W. This just means that given a cyclic graph


23/26


labeled by W(X) we can construct a cyclic graph labeled by X. Thenthese natural transformations satisfy the defining relations for a monador triple. This is monadic and so we can also consider the category ofW-algebras.

There is a forgetful functor from planar algebras to Set/A and us-ing labeled cyclic graphs we also have a functor from Set/A to planaralgebras. The composite of these two functors is the functor W. Fur-thermore it is clear that the functor using labeled cyclic graphs is a leftadjoint to the forgetful functor. This is, for us, the main merit of theplanar algebra point of view. This shows that the category of planaralgebras is equivalent to the category of W-algebras.

Similarly we have a forgetful functor from spherical categories toSet/A and a functor from Set/A to spherical categories constructedusing labeled cyclic graphs. Again, the composite of these two functorsis the functor W. It is not clear that the functor which constructsthe spherical category is left adjoint to the forgetful functor. Howeverit is clear that if we construct the planar algebra using labeled cyclicgraphs and then take the spherical category then we construct the samespherical category. This shows that the statement that labeled cyclicgraphs construct a left adjoint is equivalent to the statement that thecategory of planar graphs is equivalent to the category of sphericalcategories. Therefore this follows from 5.1 and 5.2. Alternativelywe can construe the arguments of 5.2 as showing that the sphericalcategory constructed from labeled cyclic graphs is free. From this pointof view the Temperley-Lieb spherical category and planar algebra arethe free objects on the element W Set/A where W(a) is the emptyset for all a A. Then since we have a spherical category we havea functor from the free spherical category to the spherical category ofcyclic graphs. The result we want is that this is an isomorphism. It isclear that we have enough operations so that this functor is surjective.It remains to show that we have enough relations so that this functor isinjective. From this point of view the argument of 5.2 gives a normalform for each morphism.

There is an alternative approach. The set A can be graded by thelength of directed path. Then the set of directed paths of length nhas an action of the cyclic group of order n. Then the object Set/Aunderlying a spherical category, spider or planar algebra can be takento be a map of cyclic sets. Then we can construct a left adjoint tothis functor by taking labeled cyclic graphs where the vertices haveadditional labeling reflecting the symmetry. This approach is the onetaken in the literature.


24/26


7. Graphs

In this section we give a brief discussion of a variation in which thegeometric realisation is an abstract graph and is not embedded in asurface.

7.1. Symmetric graphs.

Definition 7.1. A symmetric graph consists of sets B, E, I andV together with maps f : BEI V and an involutione : EI E I.

A cyclic graph determines a symmetric graph by first forgetting theset C and the maps h and t. Then we take V to be the orbits ofc and define f to be the map sending each element to its orbit. Thisshows that both symmetric graphs and cyclic graphs give generalisedoperads in the sense of [BM06].

A symmetric graph has a geometric realisation which is a 1-complex.

Take the space [0, 1/2] B E I and make the following iden-tifications. First identify (1/2, x) with (1/2, xe) for all x E I.Secondly identify (0, x) with (0, y) for all x, y such that xf = yf.The boundary is the set of points (1/2, x) for x B.

Then there is a normal form whose construction follows the normalform for cyclic graphs. The boundary of a symmetric graph is the setB. An edge labelling is a map : B E I . The labeling set has an involution and we require that xe = x whenever xe isdefined.

The data for sewing is just an inclusion of B Bop in B for someset B. Then the construction for sewing follows the construction for

cyclic graphs.7.2. Symmetric spiders. Let (, ) be a set with an involution. Thedata for a symmetric spideris a set W(a) for each sequence of elementsof , a. Then the operations are as follows.

Ifa has length n and is a permutation ofn symbols then we have asecond sequence a. The first operation is a bijection W() : W(a) W(a) for each sequence a and each permutation . The other opera-tions are:

W(a) W(b) W(ab)(24)

W(abbc) W(ac)(25)

W(ac) W(abbc)(26)

The compatibility relations which involve rotation are (7), (17), (18).These conditions are modified. The remaining conditions are retained.

The basic example is constructed from symmetric graphs labeled by. The set W(a) is a symmetric graph together with a total order onthe boundary points which gives an identification with a. The maps


25/26


W() are constructed by changing the total order. The other opera-tions are constructed using sewing.

7.3. Symmetric categories. Instead of spherical categories we haverigid symmetric monoidal categories. We can construct a rigid sym-metric monopidal category from a symmetric spider and vice versa.

This is [JS91, Theorem 2.3].

7.4. Operads. This section is an analogue of the discussion of planaralgebras. First we construct a multicategory. Let a1, a2, . . . an and abe sequences of elements of . Then let P(a1, a2, . . . an; a) be the setof all labeled symmetric graphs whose set E is empty together with atotal order on the boundary points which gives an identification with[a1, a2, . . . , an, a

op]. Then we can define the compositions using sewingmuch as before. This multicategory has some additional structure sincewe can permute the sequence a1, a2, . . . an.

7.5. Monads. Given a symmetric spider, rigid symmetric monoidal

category or algebra over P then we have an underlying functor toSet/A. Here A is the set of sequences of elements of . Each of thesefunctors has a left adjoint which can be constructed using symmetricgraphs. This gives rise to the same three monads and so the threecategories are equivalent.

There is a variation of this where the set A is graded by the lengthof a sequence. Then the set of sequences of length n has an actionof the symmetric group S(n). Then we have underlying functors tothe category whose objects are symmetric sets together with a map ofsymmetric sets to A.

References

[Bis02] D. Bisch. Subfactors and planar algebras. In Proceedings of the Interna-tional Congress of Mathematicians, Vol. II (Beijing, 2002), pages 775785,Beijing, 2002. Higher Ed. Press.

[BM06] D. Borisov and Yu I. Manin. Generalized operads and their inner coho-momorphisms, 2006, arXiv:math/0609748.

[BW99] John W. Barrett and Bruce W. Westbury. Spherical categories. Adv.Math., 143(2):357375, 1999.

[CFS95] J. Scott Carter, Daniel E. Flath, and Masahico Saito. The classical andquantum 6j-symbols, volume 43 ofMathematical Notes. Princeton Univer-sity Press, Princeton, NJ, 1995.

[Cvi] Predrag Cvitanovic. Group theory. http://www.nbi.dk/GroupTheory/.

[Cvi77] Predrag Cvitanovic. Group theory. preprint, Oxford University, 1977.[Cvi84] Predrag Cvitanovic. Group Theory. Nordita, 1984.[FK97] Igor B. Frenkel and Mikhail G. Khovanov. Canonical bases in tensor prod-

ucts and graphical calculus for Uq(sl2). Duke Math. J., 87(3):409480,1997.

[FY89] Peter J. Freyd and David N. Yetter. Braided compact closed categorieswith applications to low-dimensional topology. Adv. Math., 77(2):156182,1989.


26/26


[GK94] Victor Ginzburg and Mikhail Kapranov. Koszul duality for operads. DukeMath. J., 76(1):203272, 1994.

[Jon] Vaughan F. R. Jones. Planar algebras, I, arXiv:math.QA/9909027.[JS91] Andre Joyal and Ross Street. The geometry of tensor calculus. I. Adv.

Math., 88(1):55112, 1991.[Ken82] A. D. Kennedy. Diagrammatic methods for spinors in Feynman diagrams.

Phys. Rev. D (3), 26(8):19361955, 1982.[Kim] Dongseok Kim. Graphical calculus on representations of quantum lie al-

gebras, arXiv:math.QA/0310143.[KL94] Louis H. Kauffman and Sostenes L. Lins. Temperley-Lieb recoupling the-

ory and invariants of 3-manifolds, volume 134 of Annals of MathematicsStudies. Princeton University Press, Princeton, NJ, 1994.

[Kup96] Greg Kuperberg. Spiders for rank 2 Lie algebras. Comm. Math. Phys.,180(1):109151, 1996.

[Lam69] Joachim Lambek. Deductive systems and categories. II. Standard con-structions and closed categories. In Category Theory, Homology Theoryand their Applications, I (Battelle Institute Conference, Seattle, Wash.,1968, Vol. One), pages 76122. Springer, Berlin, 1969.

[Lei02] Tom Leinster. Generalized enrichment of categories. J. Pure Appl. Algebra,

168(2-3):391406, 2002, arXiv:math.CT/9901139. Category theory 1999(Coimbra).[Lei04] Tom Leinster. Higher operads, higher categories, volume 298 of London

Mathematical Society Lecture Note Series. Cambridge University Press,Cambridge, 2004, arXiv:math/0305049.

[MT06] N. J. MacKay and A. Taylor. Rational R-matrices and exceptional Liealgebras. Czechoslovak J. Phys., 56(10-11):12311236, 2006.

[New42] M. H. A. Newman. On theories with a combinatorial definition of equiv-alence.. Ann. of Math. (2), 43:223243, 1942.

[Olv99] Peter J. Olver. Classical invariant theory, volume 44 of London Mathe-matical Society Student Texts. Cambridge University Press, Cambridge,1999.

[OS89] Peter J. Olver and Chehrzad Shakiban. Graph theory and classical invari-

ant theory. Adv. Math., 75(2):212245, 1989.[PR87] Roger Penrose and Wolfgang Rindler. Spinors and space-time. Vol. 1.

Cambridge Monographs on Mathematical Physics. Cambridge UniversityPress, Cambridge, 1987. Two-spinor calculus and relativistic fields.

[Shu94] Mei Chee Shum. Tortile tensor categories. J. Pure Appl. Algebra, 93(1):57110, 1994.

[SW06] Adam S. Sikora and Bruce W. Westbury. Confluence theory for graphs,2006, arXiv:math/0609832.

[Tut79] W. T. Tutte. Combinatorial oriented maps. Canad. J. Math., 31(5):9861004, 1979.

[Wes03] B. W. Westbury. Invariant tensors and diagrams. Internat. J. ModernPhys. A, 18(Supplement, October):4982, 2003.

[Wes06] Bruce W. Westbury. Invariant tensors for the spin representation of so(7),

2006, arXiv:math/0601209.

Mathematics Institute, University of Warwick, Coventry CV4 7AL

E-mail address: [email protected]

bruce w. westbury- presentations of spherical categories

Documents