Categorifying Coloring Numbers

Contemporary Mathematics Categorifying Coloring Num b ers John Armstrong Abstract. Coloring num bers are one of the simplest combinatorial in v arian ts of knots and links to describ e. And with Joyce’s introduction of quandles, we can understand them more algebraically . But can w e extend these in v ariants to tangles – knots and links with free ends? Indeed we can, once we categorify . Starting from the definition of coloring num b ers, we will categorify them and establish this extension to tangles. Then, decategorifying will leav e us with matrix representations of the monoidal category of tangles. 1. In tro duction 1.1. T op ological Quan tum Computation and T angle Represen tations. The rise of topological quan tum computation as a method to pro vide fault-tolerance for quan tum computers[ 13, 9, 18 ] brings with it the need to turn knot theory into represen tation theory . Ev ery computation is actually approximating a top ological in v ariant of the knotted paths any ons follo w, and every knot inv arian t should giv e a quantum computer. But w e cannot simply consider these as inv ariants of knots. Computations take place through time, and we must b e able to understand what happ ens in the first half of a computation as separate from what happ ens in the second half. When we consider less than the complete run of a top ological quantum computer w e do not find neatly knotted paths of any ons, but rather a lo ose collection of tangled paths with free ends hanging out at the beginning and end of the computation. Th us we m ust consider tangles[ 10 ] as a natural generalization of knots and links, and a simpler one for the purp oses of top ological quantum computation. T o describe a top ological quan tum computer corresp onding to a tangle we m ust select one transition matrix for eac h of four simple generating tangles, sub ject to a short list of conditions. That is, we m ust define a matrix representation of the category T ang of tangles. This sp ecifies not only the evolution of the computer’s state as w e mo v e any ons around each other, but also the initial conditions and the measuremen ts to b e p erformed as we pair them off. 1991 Mathematics Subje ct Classific ation. Primary 57M27, 57M99; Secondary 18B99. Key wor ds and phr ases. T angles, quandles, categorification. I am deeply indebted to the input and advice of John Baez and J. Scott Carter on the preliminary versions of this paper, and to Sam Lomonaco and Louis Kauffman in the developmen t of these ideas. c  0000 (copyrigh t holder) 1 2 JOHN ARMSTRONG And so quantum computation requires us to consider the representation theory of tangles, and to think of knot in v ariants as restrictions of these representations. 1.2. Colorings and Quandles. In this pap er we will lay out this picture for a particularly simple combinatorial inv ariant of knots and links: the num b er of colorings of a link by a giv en inv olutory quandle. Colorings of knot and link diagrams go bac k to F ox[ 7 ], who asked if we can color the arcs of a diagram red, green, and blue, so that at each crossing either one color app ears or all three do. More generally , in how many w ays can we manage this? It turns out that this n umber of colorings depends only on the knot t ype and not on the particular diagram. Quandles were introduced by Joyce[ 12 ] and Matveev[ 14 ] (under the name, “distributiv e group oids”) as a to ol for studying oriented knots and links. The sp ecial case of inv olutory quandles first appeared as “keis”[ 23 ]. These fill the same role for unoriented links that general quandles do for oriented links. The connection b etw een quandles and colorings is that a coloring is essen tially a homomorphism of quandles[ 11 ]. First, the set of colors { red , green , blue } can b e giv en the structure of an in volutory quandle. Then, to an unorien ted link diagram w e can assign a “fundamental” inv olutory quandle encoding exactly those relations demanded by the diagram’s crossings. F ox’s colorings, then, are homomorphisms from this fundamental in volutory quandle to the quandle of colors. Replacing this target quandle with other inv olutory quandles gives a ric h sto c k of inv ariants to in vestigate. The framew ork of quandles has b een extended to include a cohomology theory analogous to that of groups[ 6, 5 ]. Link colorings b y v arious sorts of quandles hav e also been extensively studied[ 16, 19, 21, 20, 15, 17, 8 ]. How ev er, these inv ariants m ust b e extended to tangles for our purposes! Our first step will b e to “categorify” the coloring n umber in v ariants by consid- ering instead the set of colorings of a given diagram. It is essen tial at this point to note that this set is not inv ariant under the Reidemeister mo ves – only its cardi- nalit y is. This leads us in passing to define it as an example of a link (or tangle) “co v ariant”. Next w e extend our definition to cov er tangles b y in tro ducing the category of spans, as defined by B ´ enab ou[ 4 ]. W e find that defining the colorings of a tangle to b e a span of sets giv es us exactly the handles we need to compose them prop erly , and to define colorings as a functor on the category of tangles. Finally , w e “decategorify” our spans to find matrices[ 2 ]. This gives us our sough t-after matrix representation of the category of tangles. When we regard a link as a tangle, our represen tation will give us a 1 × 1 matrix whose single entry is the old num b er of colorings. Ac kno wledgements. I am deeply indebted to the input and advice of John Baez and J. Scott Carter on the preliminary v ersions of this pap er, and to Sam Lomonaco and Louis Kauffman in the dev elopmen t of these ideas. 2. Quandle Coloring Numbers 2.1. Quandles. A “quandle” is an algebraic structure consisting of a set Q and tw o binary operations . and / . These satisfy the three conditions Q1 . F or all a ∈ Q , a . a = a . CA TEGORIFYING COLORING NUMBERS 3 Q2 . F or all a, b ∈ Q , ( b . a ) / b = a = b . ( a / b ). Q3 . F or all a, b, c ∈ Q , a . ( b . c ) = ( a . b ) . ( a . c ). As is usual for algebraic structures, we ha ve a notion of a “quandle homomor- phism” f : Q 1 → Q 2 , whic h is simply a function from the underlying set of Q 1 to that of Q 2 whic h preserv es the t w o quandle op erations. W e then hav e the category Quan of quandles and quandle homomorphisms, which will feature prominen tly in our discussion. It is useful to keep the follo wing quandles in mind as examples. Giv en any group G , the conjugation Conj( G ) with the same underlying set as G . W e define the op erations by conjugation within the group: b . a = bab − 1 a / b = b − 1 ab If G is ab elian, then the op erations in Conj( G ) are trivial. But we do hav e another in teresting quandle structure. The dihedral quandle D ( G ) also has the same underlying set as G , but w e now define the t w o op erations: b . a = 2 b − a = a / b This quandle satisfies an additional condition QIn v . F or all a, b ∈ Q , b . a = a / b When this condition is satisfied, we sa y the quandle is “in v olutory”. 2.2. Colorings. Given a unoriented knot or link diagram and an inv olutory quandle X , we color the diagram by assigning an element of X to each arc of the diagram. When an arc with color a meets an o v ercrossing arc with color b , the arc on the other side must be colored b . a , as in figure 1. Figure 1. Coloring arcs at a crossing 4 JOHN ARMSTRONG Notice here that it do esn’t matter which undercrossing arc we regard as com- ing in and which we regard as going out of the crossing b ecause we are using an in volutory quandle. The axioms QInv and Q2 tell us that b . ( b . a ) = b . ( b / a ) = a As it turns out, the num b er of colorings of a diagram for a given link b y a given in volutory quandle is indep endent of which diagram of the link we use. Indeed, giv en a coloring of a link diagram, w e get a unique coloring of any link diagram related to it b y a Reidemeister mov e. In fact, the three quandle axioms exactly correspond to the three Reidemeister mov es, as indicated in figure 2. Th us we ha v e the Theorem 2.1. F or any involutory quand le X , the numb er of c olorings of an un- oriente d link diagr am by X is an invariant of unoriente d links. 2.3. The F undamen tal In v olutory Quandle. Given an unoriented link diagram, w e can define its fundamental inv olutory quandle[ 24 ]. This is a quandle whic h contains exactly the relations forced by the crossings in the diagram. It is, in a sense, “universal” for colorings. W e generate a free quandle[ 12 ] on the set of arcs in the diagram K . W e then imp ose a relation for eac h crossing. If generators a and c meet at the ov ercrossing generator b , we add the relation c = b . a . Once these relations are added, the result is the fundamental in v olutory quandle Q ( K ). A coloring of the diagram K by the quandle X assigns to each arc of K an elemen t of X . But these arcs are the generators of Q ( K ). F urther, the rela- tions defining Q ( K ) are enforced by the definition of an X -coloring. Thus an X -coloring of the link diagram K is exactly the same as a quandle homomorphism hom Quan ( Q ( K ) , X ). When w e apply a Reidemeister mo v e to turn the diagram K 1 in to the diagram K 2 , the fundamental inv olutory quandle do esn ’t stay the same. The set of arcs in K 2 is not the same as the set of arcs in K 1 , and there are differen t relations imposed b y the different crossings. How ever, w e do ha ve the Theorem 2.2. If link diagr ams K 1 and K 2 ar e r elate d by a R eidemeister move, then ther e is an isomorphism Q ( K 1 ) ∼ = Q ( K 2 ) . Proof. If w e refer to figure 2 w e can see the proof. F or example, let’s sa y that K 1 is on the left side of a Reidemeister I I mo ve, while K 2 is on the right. The lab els in the middle row of figure 2 describ e a coloring of K 2 using the quandle Q ( K 1 ), or equiv alently a coloring of K 1 using the quandle Q ( K 2 ). Thus w ecan define tw o homomorphisms of quandles: f ∈ hom Quan ( Q ( K 2 ) , Q ( K 1 )) and g ∈ hom Quan ( Q ( K 1 ) , Q ( K 2 )). These are clearly in v erses of each other, establishing the isomorphism.  In particular, this isomorphism gives a bijection b et ween the sets of colorings hom Quan ( Q ( K 1 ) , X ) and hom Quan ( Q ( K 2 ) , X ), which reestablishes the in v ariance of coloring num b ers. It is imp ortant to note at this p oin t that these sets of colorings are not the same set. They are merely isomorphic as sets, rather than iden tical. Therefore the set of colorings is not an in v ariant of the knot t yp e. Only its cardinalit y is inv ariant. W e must no w lay out a language in whic h to talk about exactly these details. CA TEGORIFYING COLORING NUMBERS 5 Figure 2. The quandle axioms correspond to the Reidemeister mov es 6 JOHN ARMSTRONG 3. Categorification Categorification is, simply put ... the pro cess of finding category-theoretic analogues of set- theoretic concepts b y replacing sets with categories, functions with functors, and equations b etw een functions by natural iso- morphisms b etw een functors, whic h in turn should satisfy certain equations of their own, called ‘coherence la ws’.[ 3 ] More to the point, we w an t to tak e things we’d called “identical” and see them as merely “equiv alent”. In the case at hand, w e’re considering a knot to b e an equiv alence class of knot diagrams under the Reidemeister mov es. In stead, we’d like to think of link diagrams as the ob jects of a category KDiag . The morphisms will b e sequences of Reidemeister mov es. Since any suc h mov e can b e reversed, this category of link diagrams forms a group oid. No w we can recast theorem 2.1 as follows: Theorem 3.1. F or a ny involutory quand le X we have a functor Col X fr om the gr oup oid KDiag to the set of natur al numb ers, c onsider e d as a c ate gory with no non-identity morphisms. Proof. T o an y diagram we asso ciate the num b er of X -colorings. This defines the functor on ob jects. Since ev ery morphism is a comp osite of Reidemeister mov es, we just need to define the functor on the Reidemeister mov es to define it on all morphisms. But we kno w that under a Reidemeister mo ve the n um b er of X -colorings remains the same, so to an y mov e b etw een tw o diagrams w e can asso ciate the iden tit y morphism on the (common) num b er of colorings.  W e can also categorify the v alue of our inv ariant. Instead of considering how man y colorings a giv en diagram has, w e should instead consider the set of colorings itself. W e further refine theorem 3.1 to state: Theorem 3.2. F or any involutory quand le X we have a functor Col X : KDiag → Set which asso ciates to any link diagr am K the set of X -c olorings of K . Proof. Indeed, we can no w see 2.2 as asserting the functoriality of the fun- damen tal in volutory quandle construction. That is, to a sequence of Reidemeister mo ves connecting t wo link diagrams we get an isomorphism of fundamen tal inv o- lutory quandles. Then we can define Col X ( K ) = hom Quan ( Q ( K ) , X )  Th us a sequence of Reidemeister mov es connecting t wo link diagrams giv es an explicit bijection b et ween the sets of X -colorings. Since the sets are changing as w e change the diagram, it no longer seems appropriate to call our functor a “link in v ariant”. Instead, w e will make the follo wing definition Definition 3.3. A link c ovariant is a functor from the group oid KDiag to any other category . If the image of each morphism is an iden tity morphism, w e call the functor a link invariant . Th us the fundamental in volutory quandle of a knot diagram is a co v arian t, as is the set of X -colorings for any inv olutory quandle X . Man y other well-kno wn CA TEGORIFYING COLORING NUMBERS 7 “in v ariants” are actually co v ariants under this definition, like the knot group given b y the Wirtinger presentation[ 22 ]. 4. T angles 4.1. The 2-category of T angles. No w that we’v e categorified our link in- v ariant, we hav e enough breathing ro om to truly extend its domain of definition. Sp ecifically , we w an t to color tangle diagrams. T op ologically , a tangle is lik e a knot or a link em b edded in a cub e, but we now allo w arc comp onen ts with their edges running to mark ed p oints on the top and b ottom of the cub e. These tangles are known to form a monoidal category T ang . The ob jects of this category are the natural n umbers, and a morphism from m to n is a tangle with m p oin ts on the b ottom of its cub e, and n endp oin ts on the top. If we ha ve a tangle from n 1 to n 2 , and another tangle from n 2 to n 3 , we can stac k the second cub e on top of the first and splice together the n 2 endp oin ts in the middle. This defines our comp osition. The monoidal pro duct of t w o ob jects is their sum as natural num b ers, while the monoidal product of tw o tangles is giv en b y stacking their cubes side-b y-side. Just as for knots and links, tangles can b e describ ed by tangle diagrams. Am- bien t isotopies of tangles are again equiv alent to sequences of Reidemeister mov es. This leads to a well-kno wn presentation of T ang as a monoidal category[ 10 ]: Theorem 4.1. The c ate gory T ang of tangle diagr ams is gener ate d by the tangle diagr ams { X + , X − , ∪ , ∩} with r elations T 0 . ( ∪ ⊗ I 1 ) ◦ ( I 1 ⊗ ∩ ) = I 1 = ( I 1 ⊗ ∪ ) ◦ ( ∩ ⊗ I 1 ) T 0 0 . ( I 1 ⊗ ∪ ) ◦ ( X ± ⊗ I 1 ) = ( ∪ ⊗ I 1 ) ◦ ( I 1 ⊗ X ∓ ) T 1 . ∪ ◦ X ± = ∪ T 2 . X ± ◦ X ∓ = I 2 T 3 . ( X + ⊗ I 1 ) ◦ ( I 1 ⊗ X + ) ◦ ( X + ⊗ I 1 ) = ( I 1 ⊗ X + ) ◦ ( X + ⊗ I 1 ) ◦ ( I 1 ⊗ X + ) W e read the generator X + as a righ t-handed crossing, X − as a left-handed crossing, ∪ as a lo cal minimum in the tangle diagram, and ∩ as a local maximum. The relations T 1 , T 2 , and T 3 then enco de the three Reidemeister mov es, while T 0 and T 0 0 handle the interaction of lo cal maxima and minima with each other and with crossings. As we did before, let’s categorify this picture. Inste ad of identifying tw o tangle diagrams if they are related by a Reidemeister mo v e (or one of the new “top ological” tangle mov es), let’s jut consider them to b e equiv alent. That is, w e consider a (strict) monoidal 2-category whose ob jects are again the natural n um b ers, and whose morphisms are built from compositions and monoidal pro ducts of the four generating tangles. No w instead of imp osing the five relations, w e add 2-isomorphisms to relate any tangle diagrams that w ould b e identified b y the relations. It is this 2-category that we will refer to as T ang . In analogy with definition 3.3 for links, w e introduce the following Definition 4.2. A tangle c ovariant is a monoidal 2-functor from the monoidal 2-category T ang to any other 2-category . If the image of each 2-morphism is an iden tity 2-morphism, w e call the functor a tangle invariant . The straightforw ard approach no w is to define a coloring of an unorien ted tangle diagram by an inv olutory quandle X exactly as we did for link diagrams. 8 JOHN ARMSTRONG W e assign an elemen t of X to eac h arc and sub ject these assignmen ts to restrictions at crossings just as b efore. This indeed gives a set of X -colorings, but there is no w ay to comp ose tw o of these sets as morphisms in some category . W e need to extend our na iv e notion of the set of tangle colorings and give it “handles” that w e can use to comp ose them. 5. Spans 5.1. The 2-category of spans. Giv en a category C with pullbacks we define the 2-category Span ( C ) of spans on C . It will ha ve the same ob jects as C . A morphism f : A → B in Span ( C ) will b e a “span” in C : an ob ject F and a pair of morphisms in C : A f l ← − F f r − → B . Then, giv en spans f = A f l ← − F f r − → B and g = A g l ← − G g r − → B , a 2-morphism φ : f ⇒ g is an arrow φ : F → G so that the follo wing diagram commutes: A  f l F G g l 6 g r -  φ B f r ? The “vertical” composition of 2-morphisms is straigh tforward. The comp o- sition of morphisms (and the “horizontal” comp osition of 2-morphisms) in vok es the pullbac ks we assumed C to ha ve. If we hav e spans f = A f l ← − F f r − → B and g = B g l ← − G g r − → C w e form their comp osite b y pulling back the square in the diagram F ◦ G - G g r - C F ? f r - B g l ? A f l ? This comp osition is not quite asso ciativ e, but it’s easily verified to b e asso ciativ e up to a unique 2-isomorphism, which giv es the asso ciator for the 2-category . There are a few facts ab out the span construction which will b e useful to us.[ 1 ] Theorem 5.1. Given c ate gories C and D with pul lb acks and a functor F : C → D pr eserving them, ther e is a 2-functor Span ( F ) : Span ( C ) → Span ( D ) define d by applying F to al l p arts of a sp an diagr am. Theorem 5.2. If C is a monoidal c ate gory such that the monoidal pr o duct pr eserves pul lb acks, then Span ( C ) is a monoidal 2-c ate gory. CA TEGORIFYING COLORING NUMBERS 9 Dually , giv en a category C with pushouts we can define the 2-category CoSpan ( C ) of cospans. A cospan diagram is lik e a span diagram, but with the arrows pointing in instead of out, and w e compose them b y pushing out a square rather than pulling bac k, but otherwise everything w e’ve said about spans holds for cospans. 5.2. Coloring Spans. The category Set of sets has fib ered pro ducts, whic h act as pullbac ks, and so we ha ve a 2-category Span ( Set ). The tw o functions out to the side of the cen tral set in a span will provide us with exactly the handles w e need to comp ose sets of colorings. No w we can extend theorem 3.2 to: Theorem 5.3. F or any involutory quand le X we have a 2-functor Col X : T ang → Span ( Set ) On an obje ct n of T ang we define Col X ( n ) = X n the set of n -tuples of elements of X . F or a tangle diagr am T : m → n fr om m fr e e ends to n fr e e ends we define the sp an X m ← Col X ( T ) → X n wher e the arr ow on the left is the function sending a c oloring of T to the c oloring it induc es on the lower endp oints of the tangle, and the one on the right is the similar function for the upp er endp oints. The 2-functor is define d on 2-morphisms by the diagr ams in figur e 2, as in the or em 3.2. Proof. The main thing to chec k here is that comp osition of coloring spans really do es reflect comp osition of tangles. But giv en a comp osite tangle T 1 ◦ T 2 , a coloring in Col X ( T 1 ◦ T 2 ) is exactly a coloring of T 1 and a coloring of T 2 that agree on the endp oin ts we splice together to compose the tangles. This is exactly the definition of the fib ered pro duct Col X ( T 1 ◦ T 2 ) - Col X ( T 2 ) - X p Col X ( T 1 ) ? - X n ? X m ?  Notice what happ ens to this picture when we consider a link as a tangle from 0 to 0. Both sides of the span become empty products – singletons – and the functions in the span b ecome trivial. What remains is the old set of link colorings. 10 JOHN ARMSTRONG 5.3. The F undamenal In volutory Quandle Cospan. Earlier w e iden tified the fundamental inv olutory quandle Q ( K ) of a link diagram K as the quandle that captures coloring num b ers for all in v olutory quandles X : Col X ( K ) ∼ = hom Quan ( Q ( K ) , X ) The same construction can give us a quandle Q ( T ) from a tangle diagram T , whic h then giv es us the set of X -colorings of T . Can we get the sides of our span as well? Indeed, the free quandle on n generators Q n satisfies hom Quan ( Q n , X ) = X n . W e can c ho ose these generators to b e a collection of free ends of our tangle diagram, and the inclusion of those ends into the whole diagram giv es us a homomorphism Q n → Q ( T ). Theorem 5.4. Ther e is a 2-functor extending the fundamental involutory quand le to tangles: Q : T ang → CoSpan ( Quan ) On an obje ct n in T ang we define Q ( n ) = Q n , the fr e e quand le on n gener ators. F or a tangle diagr am T : m → n fr om m fr e e ends to n fr e e ends we let Q m b e the fr e e quand le on the inc oming ends and Q n b e the fr e e quand le on the outgoing ends. We define the c osp an Q m → Q ( T ) ← Q n wher e the arr ows ar e the quand le homomorphisms induc e d by including the end- p oints into the tangle diagr ams. F or a 2-morphism φ we define Q ( φ ) by r eferring to figur e 2, as in the or em 2.2. Proof. Again, the meat of the proof is in sho wing that comp osition of tangles really do es corresp ond to a pushout in Quan . Comp osition of tangle diagrams T 1 and T 2 consists of laying do wn b oth dia- grams and joining some arcs from T 1 to arcs from T 2 , as determined b y the lineup of the endp oin ts. But matching endpoints corresponds to adding relations sa ying that the image of a generator of Q n in Q ( T 1 ) equals its image as a generator in Q ( T 2 ). This amalgamated free pro duct is exactly the pushout construction in Quan .  Again, if we consider a link as a tangle from 0 to 0, the free quandle on zero generators is trivial, as are all homomorphisms from it. The only non trivial infor- mation in this cospan is the old fundamen tal inv olutory quandle of the link. No w we can use this fundamental in volutory quandle cospan to recov er the col- oring spans. The contra v ariant hom-functor hom Quan ( , X ) automatically takes all colimits to limits, so in particular it preserv es pullbacks as a functor Quan op → Set . Theorem 5.5. The c oloring sp an 2-functor Col X factors as the c omp osition of the sp an of the hom-functor Span (hom Quan ( , X )) and the fundamental involutory quand le 2-functor Q . 5.4. Monoidal structure. All of the 2-categories consid ered abov e also carry monoidal structures, and all the 2-functors preserv e them. This allo ws us to obtain tangle cov ariants, and to decategorify them to tangle inv ariants. The category Set has all finite pro ducts, so it has the Cartesian monoidal struc- ture. The direct pro duct of sets preserves pullbac ks, so Span ( Set ) is a monoidal 2-category . CA TEGORIFYING COLORING NUMBERS 11 Similarly , Quan has finite copro ducts given by the free pro duct of quandles, or equiv alently by the pushout ov er the free quandle on zero generators. These copro ducts preserv e pushouts, so CoSpan ( Quan ) is a monoidal 2-category . Theorem 5.6. The induc e d 2-functor Span (hom Quan ( , X )) : CoSpan ( Quan ) → Span ( Set ) is monoidal. Proof. This is a straigh tforward consequence of the fact that the hom-functor hom Quan ( , X ) : Quan op → Set preserves products.  Theorem 5.7. The fundamental involutory quand le c osp an 2-functor Q : T ang → CoSpan ( Quan ) is monoidal. Proof. Giv en tw o tangles T 1 : m 1 → n 1 and T 2 : m 2 → n 2 w e form their monoidal pro duct T 1 ⊗ T 2 b y laying them side-by-side. When w e calculate the fundamen tal in v olutory quandle of this diagram, we just use all the generators and relations that come from each of T 1 and T 2 , and none of them interact with each other. Thus the quandle of T 1 ⊗ T 2 is the free pro duct of the quandles of T 1 and T 2 . Similarly at the ends, Q m 1 + m 2 is the free pro duct of Q m 1 and Q m 2 , and Q n 1 + n 2 is the free product of Q n 1 and Q n 2 . So the monoidal pro duct of tangles corresp onds under Q to taking free pro ducts of cospan diagrams. But this is just the induced monoidal structure on CoSpan ( Quan ).  Theorem 5.8. F or any involutory quand le X the c oloring sp an 2-functor Col X : T ang → Span ( Set ) is monoidal. Proof. This is an immediate corollary of the preceding theorems and theorem 5.5  6. Decategorifying 6.1. Coloring Matrices. When we decategorify a coloring set we get a col- oring num b er. What happ ens when we decategorify a coloring span? A 2-isomorphism in the 2-category Span ( Set ) is a bijection φ : F → G in diagram A  f l F G g l 6 g r -  φ B f r ? The span functions f l and f r partition F in to its “double preimages” F = [ a ∈ A b ∈ B F a,b F a,b = { x ∈ F | f l ( x ) = a, f r ( x ) = b } 12 JOHN ARMSTRONG Similarly , the functions g l and g r partition G into its double preimages G a,b . Then for the diagram abov e to comm ute the function φ m ust decompose in to func- tions φ a,b : F a,b → G a,b . And then for φ to b e a bijection, eac h of the φ a,b m ust b e a bijection. So when we identify isomorphic spans of sets, we retain only the cardinality of each of the double preimages. W e are left with a matrix of cardinal num b ers indexed by the set A on the one side and the set B on the other. F or a coloring span, these index sets are the colorings of the endp oints. Thus when w e decategorify a coloring span w e get a matrix Col X ( T ) indexed b y colorings of the endp oints of the tangle. The en try Col X ( T ) µν is the num b er of colorings of the diagram T that agree with the coloring µ on the incoming ends and with the coloring ν on the outgoing ends. This interpretation as matrices is compatible with matrix multiplication. That is, given tangle diagrams T 1 : m → l and T 2 : l → n , the num b er of colorings Col X ( T 1 ◦ T 2 ) µν agreeing with the colorings µ and ν on the ends can be calculated as a sum of pro ducts of coloring num b ers: Col X ( T 1 ◦ T 2 ) µν = X λ ∈ X l Col X ( T 1 ) µλ Col X ( T 2 ) λν Decategorification also plays nice with the monoidal structure on spans induced b y the pro duct of sets. T ak e tw o diagrams T 1 : m 1 → n 1 and T 2 : m 2 → n 2 . A coloring µ 1 of the incoming ends of T 1 and a coloring µ 2 of the incoming ends of T 2 com bine to give a coloring ( µ 1 , µ 2 ) ∈ X m 1 + m 2 of the incoming ends of T 1 ⊗ T 2 . Similarly , we can combine colorings of the outgoing strands of eac h diagram to get a coloring ( ν 1 , ν 2 ) ∈ X n 1 + n 2 of the outgoing strands of T 1 ⊗ T 2 . Every coloring of the incoming or outgoing strands arises in this manner. No w when we coun t the colorings of T 1 ⊗ T 2 compatible with a given coloring of the incoming and outgoing ends, w e find Col X ( T 1 ⊗ T 2 ) ( µ 1 ,µ 2 )( ν 1 ,ν 2 ) = Col X ( T 1 ) µ 1 ν 1 Col X ( T 2 ) µ 2 ν 2 = (Col X ( T 1 )  Col X ( T 2 )) ( µ 1 ,µ 2 )( ν 1 ,ν 2 ) This follows since a coloring of T 1 ⊗ T 2 is simply a coloring of each of T 1 and T 2 with no particular relation b et ween them. This shows that the coloring matrix for the monoidal product T 1 ⊗ T 2 is the Kronec ker pro duct of the coloring matrices for T 1 and T 2 . Theorem 6.1. F or any finite involutory quand le X , ther e is a monoidal 2-functor Col X : T ang → Mat ( N ) wher e the tar get c ate gory is that of matric es with natur al numb er entries, and with identity 2-morphisms adde d. Proof. If we pic k d to b e the cardinality of X , then there are exactly d n colorings of a collection of n endpoints in a tangle. W e thus set Col X ( n ) = d n on ob jects. W e already ha ve a coloring span of sets for ev ery tangle. Ev en if w e disregard the coloring relations at crossings, w e can only pic k one color from X for each arc in the diagram, and so the sets in the coloring span are finite. T aking cardinalities, w e CA TEGORIFYING COLORING NUMBERS 13 get a matrix of natural n umbers. As described abov e, this assignmen t of a coloring matrix to a tangle preserves the comp osition and monoidal structure. Finally , if w e ha v e a 2-morphism φ : T 1 ⇒ T 2 in T ang we kno w that the coloring matrices for T 1 and T 2 will be the same, so w e can pic k Col X ( φ ) to be the iden tit y 2-morphism on that matrix.  Since every 2-morphism b ecomes an identit y 2-morphism under this functor, w e hav e a tangle inv ariant. In particular, when w e consider a link L as a tangle from 0 to 0, we can find the 1 × 1 matrix Col X ( L ). The single en try in this matrix is the n um b er of X -colorings of the link L . Instead of restricting our attention to links, w e ma y instead consider any n - strand braid as a tangle from n to n . In this case we find a matrix representation Col X of each braid group B n . 6.2. Computation. It turns out that not only do w e ha ve a tangle inv arian t in our coloring matrices, we ha ve a straightforw ard wa y of computing them. The category of tangles w as giv en b y generators and relations. 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