# The Diagrammatic Soergel Category andsl(N)-Foams, for N≥4

Authors: Marco Mackaay; Pedro Vaz
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

For eachN≥4, we define a monoidal functor from Elias and Khovanov's diagrammatic
version of Soergel's category of bimodules to the category of sl(N) foams defined by Mackaay, Stošić, and Vaz. We show that through these functors Soergel's category can be obtained
from the sl(N) foams.


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## Body

## 1. Introduction

In [1] Soergel categorified the Hecke algebra using bimodules. Just as the Hecke algebra is important for the construction of the HOMFLY-PT link polynomial, so is Soergel's category for the construction of Khovanov and Rozansky's HOMFLY-PT link homology [2], as explained by Khovanov in  [3]. Elias and Khovanov [4] constructed a diagrammatic version of the Soergel category with generators and relations, which Elias and Krasner [5] used for a diagrammatic construction of Rouquier's complexes associated to braids.In [6] Bar-Natan gave a new version of Khovanov's [7] original link homology, also called the sl(2) link homology, using 2d-cobordisms modulo certain relations, which we will call sl(2) foams. Using 2d-cobordisms with a particular sort of singularity modulo certain relations, which we will call sl(3) foams, Khovanov constructed the sl(3) link homology [8]. Khovanov and Rozansky [9] then constructed the sl(N) link homologies, for any N≥1, using matrix factorizations. These link homologies are closely related to the HOMFLY-PT link homology by Rasmussen's spectral sequences [10], with E1-page isomorphic to the HOMFLY-PT homology and converging to the sl(N) homology, for any N≥1. In [11] Mackaay et al. gave an alternative construction of these sl(N) link homologies, for N≥4, using sl(N) foams, which are 2d-cobordisms with two types of singularities satisfying relations determined by a formula from quantum field theory, originally obtained by Kapustin and Li [12] and later adapted by Khovanov and Rozansky [13].Khovanov and Rozansky in [2, 9] and Rasmussen in [10] used matrix factorizations for their constructions. Therefore, the question arises whether their results can be understood in diagrammatic terms and what could be learned from that. In [14] Vaz constructed functors from Elias and Khovanov's diagrammatic version of Soergel’s category to the categories of sl(2) and sl(3) foams. In this paper we construct the analogous functors from the same version of Soergel's category to the category of sl(N) foams for N≥4. To complete the picture, one would like to construct  the analogues of Rasmussen's spectral sequences in this setting.  However for this, one would first have to understand the Hochschild homology of bimodules in diagrammatic terms, which has not been accomplished yet. Hochschild homology plays an integral part of the construction. Nevertheless, there is an interesting result which can already be shown using the functors in this paper. In a certain technical sense, which we will make precise in Proposition 4.2, Soergel's category can be obtained from the sl(N) foams, and therefore from the Kapustin-Li formula, using our functors. This result should be compared to Rasmussen's Theorem 1 in [10].We thank Catharina Stroppel for pointing out the connection of our work to results in [15]. We quote her directly: In [15] a categorification of "special  trivalent" graphs modulo the MOY  relations was constructed by exact functors acting between certain blocks of parabolic category 𝒪. Using Soergel's functor passing from Lie theory to the combinatorial bimodule category the construction in [15] in fact produces an action of the diagrammatic Soergel category on these various category 𝒪s.We have tried to make the paper as self-contained as possible, but the reader should definitely leaf through [4, 5, 11, 14] before reading the rest of this paper.In Section2 we recall Elias and Khovanov's version of Soergel's category. In Section 3 we review sl(N) foams, as defined by Mackaay, Stošic', and Vaz. Section 4 contains the new results: the definition of our functors, the proof that they are indeed monoidal, and a statement on faithfulness in Proposition 4.2.


## 2. Elias and Khovanov's Version of Soergel's Category

This section is a reminder of the diagrammatics for Soergel categories introduced by Elias and Khovanov in [4]. Actually we give the version which they explained in   [4, Section 4.5] and which can be found in detail in  [5].Fix a positive integern. The category 𝒮𝒞1 is the category whose objects are finite length sequences of points on the real line, where each point is colored by an integer between 1 and n. We read sequences of points from left to right. Two colors i and j are called adjacent if |i-j|=1 and distant if |i-j|>1. The morphisms of 𝒮𝒞1 are given by generators modulo relations. A morphism of 𝒮𝒞1 is a ℂ-linear combination of planar diagrams constructed by horizontal and vertical gluings of the following generators (by convention no label means a generic color j)  (i)
Generators involving only one color are(2.1)It is useful to define the cap and cup as(2.2)(ii)
Generators involving two colors are–The 4-valent vertex, with distant colors,(2.3)and the 6-valent vertex, with adjacent colorsi and j,  (2.4)read from bottom to top. In this setting a diagram represents a morphism from the bottom boundary to the top. We can add a new colored point to a sequence and this endows𝒮𝒞1 with a monoidal structure on objects, which is extended to morphisms in the obvious way. Composition of morphisms consists of stacking one diagram on top of the other.We consider our diagrams modulo the following relations.“Isotopy” relations are (2.5)(2.6)(2.7)(2.8)(2.9)The relations are presented in terms of diagrams with generic colorings. Because of isotopy invariance, one may draw a diagram with a boundary on the side, and view it as a morphism in𝒮𝒞1 by either bending the line up or down. By the same reasoning, a horizontal line corresponds to a sequence of cups and caps.One color relations are(2.10)(2.11)(2.12)Relations involvingtwo distant colors are(2.13)(2.14)(2.15)Relations involvingtwo adjacent colors are(2.16)(2.17)(2.18)(2.19)Relations involving three colors  are (adjacency is determined by the vertices which appear) (2.20)(2.21)(2.22)Furthermore, we also have a useful implication of relation (2.12) as follows:(2.23)Introduce aq-grading on 𝒮𝒞1 declaring that dots have degree 1, trivalent vertices have degree -1, and 4- and 6-valent vertices have degree 0.Definition 2.1.
The category𝒮𝒞2 is the category containing all direct sums and grading shifts of objects in 𝒮𝒞1 and whose morphisms are the grading-preserving morphisms from 𝒮𝒞1.Definition 2.2.
The category𝒮𝒞 is the Karoubi envelope of the category 𝒮𝒞2.Elias and Khovanov's main result in [4] is the following theorem.Theorem 2.3 (Elias-Khovanov).
The category𝒮𝒞 is equivalent to the Soergel category in [1].From Soergel's results from [1] we have the following corollary.Corollary 2.4.
The Grothendieck algebra of𝒮𝒞 is isomorphic to the Hecke algebra.Notice that𝒮𝒞 is an additive category but not abelian and we are using the (additive) split Grothendieck algebra.In  Section4 we will define a a family of functors from 𝒮𝒞1,n to the category of sl(N) foams, one for each N≥4. These functors are grading preserving, so they obviously extend uniquely to 𝒮𝒞2,n. By the universality of the Karoubi envelope, they also extend uniquely to functors between the respective Karoubi envelopes.


## 3. Foams

### 3.1. Prefoams

In this section we recall the basic facts about foams. For the definition of the Kapustin-Li formula, for proofs of the relations between foams, and for other details, see [11, 16]. The foams in this paper are composed of three types of facets: simple, double, and triple facets. The double facets are coloured and the triple facets are marked to show the difference.  Intersecting such a foam with a generic plane results in a web, as long as the plane avoids the singularities where six facets meet, such as on the right in Figure 1.Figure 1
Some elementary prefoams.Definition 3.1.
Letsγ be a finite oriented closed 4-valent graph, which may contain disjoint circles and loose endpoints. We assume that all edges of sγ are oriented. A cycle in sγ is defined to be a circle or a closed sequence of edges which form a piecewise linear circle. Let Σ be a compact orientable possibly disconnected surface, whose connected components are simple, double, or triple, denoted by white, coloured, or marked. Each component can have a boundary consisting of several disjoint circles and can have additional decorations which we discuss below. A closed  prefoamu is the identification space Σ/sγ obtained by gluing boundary circles of Σ to cycles in sγ such that every edge and circle in sγ are glued to exactly three boundary circles of Σ and such that for any point p∈sγ,  (1)
ifp is an interior point of an edge, then p has a neighborhood homeomorphic to the letter Y times an interval with exactly one of the facets being double, and at most one of them being triple; for an example see Figure 1,(2)
ifp is a vertex of sγ, then it has a neighborhood as shown in Figure 1.We call sγ the singular graph, its edges and vertices singular arcs and singular vertices, and the connected components of u-sγ the facets.
Furthermore the facets can be decorated with dots. A simple facet can only have black dots (•), a double facet can also have white dots (∘), and a triple facet besides black and white dots can have double dots (⊙). Dots can move freely on a facet but are not allowed to cross singular arcs.
Note that the cycles to which the boundaries of the simple and the triple facets are glued are always oriented, whereas the ones to which the boundaries of the double facets are glued are not, as can be seen in Figure1. Note also that there are two types of singular vertices. Given a singular vertex v, there are precisely two singular edges which meet at v and bound a triple facet: one oriented toward v, denoted as e1, and one oriented away from v, denoted as e2. If we use the “left-hand rule”, then the cyclic ordering of the facets incident to e1 and e2 is either (3,2,1) or (3,1,2), respectively, or the other way around. We say that v is of type I in the first case and of type II in the second case. When we go around a triple facet, we see that there have to be as many singular vertices of type I as there are of type II for the cyclic orderings of the facets to match up. This shows that for a closed prefoam the number of singular vertices of type I is equal to the number of singular vertices of type II.
We can intersect a prefoamu generically by a plane W in order to get a closed web, as long as the plane avoids the vertices of sγ. The orientation of sγ determines the orientation of the simple edges of the web according to the convention in Figure 2.
Suppose that, for all but a finite number of valuesi∈]0,1[, the plane W×i intersects u generically. Suppose also that W×0 and W×1 intersect u generically and outside the vertices of sγ. Furthermore, suppose that D⊂W is a disc in W and C⊂D its boundary circle such that C×[0,1]∩u is a disjoint union of vertical line segments. This means that we are assuming that sγ does not intersect C×[0,1]. We call D×[0,1]∩u an open prefoam between the open webs D×{0}∩u and D×{1}∩u. Interpreted as morphisms, we read open prefoams from bottom to top, and their composition consists of placing one prefoam on top of the other, as long as their boundaries are isotopic and the orientations of the simple edges coincide.Figure 2
Orientations near a singular arc.Definition 3.2.
LetPfoam be the category whose objects are webs and whose morphisms are ℚ-linear combinations of isotopy classes of prefoams with the obvious identity prefoams and composition rule.
We now define theq-degree of a prefoam. Let u be a prefoam, u1, u2, and u3 the disjoint union of its simple, double, and marked facets, respectively, and sγ(u) its singular graph. Furthermore, let b1, b2, and b3 be the number of simple, double, and marked vertical boundary edges of u, respectively. Define the partial q-gradings of u as
(3.1)qi(u)=χ(ui)-12χ(∂ui∩∂u)-12bi,i=1,2,3,qsγ(u)=χ(sγ(u))-12χ(∂sγ(u)),
where χ is the Euler characteristic and ∂ denotes the boundary.Definition 3.3.
Letu be a prefoam with d• dots of type •, d∘ dots of type ∘,  and d⊙ dots of type ⊙. The q-grading of u is given by
(3.2)q(u)=-∑i=13i(N-i)qi(u)-2(N-2)qsγ(u)+2d•+4d∘+6d⊙.The following result is a direct consequence of the definitions.Lemma 3.4.
q(u) is additive under the gluing of prefoams.We denote a simple facet withi dots by (3.3)Recall that the two-variable Schur polynomialπk,m can be expressed in terms of the elementary symmetric polynomials π1,0 and π1,1. By convention, the latter correspond to •  and ∘  on a double facet, respectively, so that (3.4)is defined to be the linear combination of dotted double facets corresponding to the expression ofπk,m in terms of π1,0 and π1,1. Analogously we can express the three-variable Schur polynomial πp,q,r in terms of the elementary symmetric polynomials π1,0,0, π1,1,0, and π1,1,1. By convention, the latter correspond to •, ∘, and ⊙ on a triple facet, respectively, so we can make sense of (3.5)


### 3.2. Foams

In [11, 16] we gave a precise definition of the Kapustin-Li formula, following Khovanov and Rozansky's work [13]. We will not repeat that definition here, since it is complicated and unnecessary for our purposes in this paper. The only thing one needs to remember is that the Kapustin-Li formula associates a number to any closed prefoam and that those numbers have very special properties, some of which we will recall below. By 〈u〉KL, we denote the Kapustin-Li evaluation of a closed prefoam u.Definition 3.5.
The categoryFoamN is the quotient of the category Pfoam by the kernel of 〈〉KL, that is, by the following identifications: for any webs Γ, Γ′ and finite sets fi∈HomPfoam(Γ,Γ′) and ci∈ℚ we impose the relations
(3.6)∑icifi=0⇔∑ici〈fi¯〉KL=0,
for any fixed way of closing the fi, denoted by fi¯. By “fixed” we mean that all the fi are closed in the same way. The morphisms of FoamN are called foams.In the next proposition we recall those relations inFoamN that we need in the sequel. For their proofs and other relations we refer to [11].Proposition 3.6.
The following identities hold inFoamN
The  dot conversion  relations are(3.7)(3.8)(3.9)
The  dot migration  relations are(3.10)(3.11)(3.12)(3.13)(3.14)
The neck cutting  relations  are (these were called cutting neck  relations in [11, 16])
(3.15)(3.16)
The  sphere  relations are(3.17)
TheΘ-foam  relations are
(3.18)
Inverting the orientation of the singular circle of (⊖*) inverts the sign of the corresponding foam. A theta-foam with dots on the double facet can be transformed into  a theta-foam with dots only on the other two facets, using the dot migration relations.
The   Matveev-Piergalini relation  is(MP)
The  disc removal  relations are(RD1)(RD2)
The  digon removal  relations  are(DR1)(DR31)(DR32)
The  square removal  relations are(SqR1)(SqR2)(3.19)(3.20)


## 3.1. Prefoams

In this section we recall the basic facts about foams. For the definition of the Kapustin-Li formula, for proofs of the relations between foams, and for other details, see [11, 16]. The foams in this paper are composed of three types of facets: simple, double, and triple facets. The double facets are coloured and the triple facets are marked to show the difference.  Intersecting such a foam with a generic plane results in a web, as long as the plane avoids the singularities where six facets meet, such as on the right in Figure 1.Figure 1
Some elementary prefoams.Definition 3.1.
Letsγ be a finite oriented closed 4-valent graph, which may contain disjoint circles and loose endpoints. We assume that all edges of sγ are oriented. A cycle in sγ is defined to be a circle or a closed sequence of edges which form a piecewise linear circle. Let Σ be a compact orientable possibly disconnected surface, whose connected components are simple, double, or triple, denoted by white, coloured, or marked. Each component can have a boundary consisting of several disjoint circles and can have additional decorations which we discuss below. A closed  prefoamu is the identification space Σ/sγ obtained by gluing boundary circles of Σ to cycles in sγ such that every edge and circle in sγ are glued to exactly three boundary circles of Σ and such that for any point p∈sγ,  (1)
ifp is an interior point of an edge, then p has a neighborhood homeomorphic to the letter Y times an interval with exactly one of the facets being double, and at most one of them being triple; for an example see Figure 1,(2)
ifp is a vertex of sγ, then it has a neighborhood as shown in Figure 1.We call sγ the singular graph, its edges and vertices singular arcs and singular vertices, and the connected components of u-sγ the facets.
Furthermore the facets can be decorated with dots. A simple facet can only have black dots (•), a double facet can also have white dots (∘), and a triple facet besides black and white dots can have double dots (⊙). Dots can move freely on a facet but are not allowed to cross singular arcs.
Note that the cycles to which the boundaries of the simple and the triple facets are glued are always oriented, whereas the ones to which the boundaries of the double facets are glued are not, as can be seen in Figure1. Note also that there are two types of singular vertices. Given a singular vertex v, there are precisely two singular edges which meet at v and bound a triple facet: one oriented toward v, denoted as e1, and one oriented away from v, denoted as e2. If we use the “left-hand rule”, then the cyclic ordering of the facets incident to e1 and e2 is either (3,2,1) or (3,1,2), respectively, or the other way around. We say that v is of type I in the first case and of type II in the second case. When we go around a triple facet, we see that there have to be as many singular vertices of type I as there are of type II for the cyclic orderings of the facets to match up. This shows that for a closed prefoam the number of singular vertices of type I is equal to the number of singular vertices of type II.
We can intersect a prefoamu generically by a plane W in order to get a closed web, as long as the plane avoids the vertices of sγ. The orientation of sγ determines the orientation of the simple edges of the web according to the convention in Figure 2.
Suppose that, for all but a finite number of valuesi∈]0,1[, the plane W×i intersects u generically. Suppose also that W×0 and W×1 intersect u generically and outside the vertices of sγ. Furthermore, suppose that D⊂W is a disc in W and C⊂D its boundary circle such that C×[0,1]∩u is a disjoint union of vertical line segments. This means that we are assuming that sγ does not intersect C×[0,1]. We call D×[0,1]∩u an open prefoam between the open webs D×{0}∩u and D×{1}∩u. Interpreted as morphisms, we read open prefoams from bottom to top, and their composition consists of placing one prefoam on top of the other, as long as their boundaries are isotopic and the orientations of the simple edges coincide.Figure 2
Orientations near a singular arc.Definition 3.2.
LetPfoam be the category whose objects are webs and whose morphisms are ℚ-linear combinations of isotopy classes of prefoams with the obvious identity prefoams and composition rule.
We now define theq-degree of a prefoam. Let u be a prefoam, u1, u2, and u3 the disjoint union of its simple, double, and marked facets, respectively, and sγ(u) its singular graph. Furthermore, let b1, b2, and b3 be the number of simple, double, and marked vertical boundary edges of u, respectively. Define the partial q-gradings of u as
(3.1)qi(u)=χ(ui)-12χ(∂ui∩∂u)-12bi,i=1,2,3,qsγ(u)=χ(sγ(u))-12χ(∂sγ(u)),
where χ is the Euler characteristic and ∂ denotes the boundary.Definition 3.3.
Letu be a prefoam with d• dots of type •, d∘ dots of type ∘,  and d⊙ dots of type ⊙. The q-grading of u is given by
(3.2)q(u)=-∑i=13i(N-i)qi(u)-2(N-2)qsγ(u)+2d•+4d∘+6d⊙.The following result is a direct consequence of the definitions.Lemma 3.4.
q(u) is additive under the gluing of prefoams.We denote a simple facet withi dots by (3.3)Recall that the two-variable Schur polynomialπk,m can be expressed in terms of the elementary symmetric polynomials π1,0 and π1,1. By convention, the latter correspond to •  and ∘  on a double facet, respectively, so that (3.4)is defined to be the linear combination of dotted double facets corresponding to the expression ofπk,m in terms of π1,0 and π1,1. Analogously we can express the three-variable Schur polynomial πp,q,r in terms of the elementary symmetric polynomials π1,0,0, π1,1,0, and π1,1,1. By convention, the latter correspond to •, ∘, and ⊙ on a triple facet, respectively, so we can make sense of (3.5)


## 3.2. Foams

In [11, 16] we gave a precise definition of the Kapustin-Li formula, following Khovanov and Rozansky's work [13]. We will not repeat that definition here, since it is complicated and unnecessary for our purposes in this paper. The only thing one needs to remember is that the Kapustin-Li formula associates a number to any closed prefoam and that those numbers have very special properties, some of which we will recall below. By 〈u〉KL, we denote the Kapustin-Li evaluation of a closed prefoam u.Definition 3.5.
The categoryFoamN is the quotient of the category Pfoam by the kernel of 〈〉KL, that is, by the following identifications: for any webs Γ, Γ′ and finite sets fi∈HomPfoam(Γ,Γ′) and ci∈ℚ we impose the relations
(3.6)∑icifi=0⇔∑ici〈fi¯〉KL=0,
for any fixed way of closing the fi, denoted by fi¯. By “fixed” we mean that all the fi are closed in the same way. The morphisms of FoamN are called foams.In the next proposition we recall those relations inFoamN that we need in the sequel. For their proofs and other relations we refer to [11].Proposition 3.6.
The following identities hold inFoamN
The  dot conversion  relations are(3.7)(3.8)(3.9)
The  dot migration  relations are(3.10)(3.11)(3.12)(3.13)(3.14)
The neck cutting  relations  are (these were called cutting neck  relations in [11, 16])
(3.15)(3.16)
The  sphere  relations are(3.17)
TheΘ-foam  relations are
(3.18)
Inverting the orientation of the singular circle of (⊖*) inverts the sign of the corresponding foam. A theta-foam with dots on the double facet can be transformed into  a theta-foam with dots only on the other two facets, using the dot migration relations.
The   Matveev-Piergalini relation  is(MP)
The  disc removal  relations are(RD1)(RD2)
The  digon removal  relations  are(DR1)(DR31)(DR32)
The  square removal  relations are(SqR1)(SqR2)(3.19)(3.20)


## 4. The FunctorsℱN,n

Letn≥1 and N≥4 be arbitrary but fixed. In this section we define a monoidal functor ℱN,n between the categories 𝒮𝒞1,n and FoamN.On Objects.ℱN,n sends the empty sequence to 1n and the one-term sequence (j) to wj: (4.1)withℱN,n(jk) given by the vertical composite wjwk.On Morphisms.(i)
The empty diagram is sent ton parallel vertical sheets:(4.2)(ii)
The vertical line colouredj is sent to the identity cobordism of wj:(4.3)The remainingn-2 vertical parallel sheets on the r.h.s. are not shown for simplicity, a convention that we will follow from now on.(iii)
TheStartDot and EndDot morphisms are sent to the zip and the unzip, respectively:(4.4)(iv)
Merge and Split are sent to cup and cap cobordisms:(4.5)(v)
The4-valent vertex with distant colors. For j+1<k we have(4.6)The casej>k+1 is given by reflexion around a horizontal plane.(vi)
For the6-valent vertices we have(4.7)The case with the colors switched is given by reflection in a vertical plane. Notice thatℱN,n respects the gradings of the morphisms.Proposition 4.1.
ℱN,n is a monoidal functor.Proof.
The assignment given byℱN,n clearly respects the monoidal structures of 𝒮𝒞1,n and FoamN. So we only need to show that ℱN,n is a functor, that is, it respects the relations (2.5) to (2.22) of Section 2.
“Isotopy Relations’’. Relations (2.5) to (2.9) are straightforward to check and correspond to isotopies of their images under ℱN,n. For the sake of completeness we show the first equality in (2.5). We have
(4.8)One Color Relations. For relation (2.10) we have
(4.9)
where the first equivalence follows from relations (2.5) and (2.7) and the second from isotopy of the foams involved.
For relation (2.11) we have
(4.10)
Relation (2.12) requires some more work. We have
(4.11)where the second equality follows from the (DR1) relation. We also have
(4.12)
Using (3.12), we obtain
(4.13)(4.14)
and, therefore, we have that
(4.15)Two Distant Colors. Relations (2.13) to (2.15) correspond to isotopies of the foams involved and are straightforward to check.
Adjacent Colors. We prove the case where “blue” corresponds to j and “red” corresponds to j+1. The relations with colors reversed are proved the same way. To prove relation (2.16) we first notice that using the (MP) move we get
(4.16)
Apply (SqR1) to the simple-double square tube perpendicular to the triple facet to obtain two terms. The first term contains a double-triple digon tube which is the left-hand side of the (DR32) relation rotated by 1800 around a vertical axis. Next apply the (DR32) relation and use (MP) to remove the four singular vertices, which results in simple-triple bubbles (with dots) in the double facets. Using   (3.19) to remove these bubbles gives
(4.17)
which is ℱN,n. The second term contains
(4.18)
behind a simple facet with d dots (notice that all dots are on simple facets). Using the (MP) relation to get a simple-triple bubble in the double facet, followed by (RD2) and (S1) we obtain
(4.19)which equals ℱN,n.
We now prove relation (2.17). We have an isotopy equivalence
(4.20)
Notice that ℱN,n is the l.h.s. of the  (SqR2) relation. The first term on the r.h.s. of (SqR2) is isotopic to -ℱN,n. For the second term on the r.h.s. of (SqR2) we notice that ℱN,n contains
(4.21)
Applying (DR31) followed by (MP) to remove the singular vertices creating simple-simple bubbles on the two double facets and using   (3.20) to remove these bubbles, we conclude that ℱN,n is the second term on the r.h.s. of (SqR2).
We now prove relation (2.18) in the form (4.22)
The image of the l.h.s. also contains a bit like the one in   (4.21). Simplifying it like we did in the proof of (2.17), we obtain that ℱN,n reduces to
(4.23)
For the r.h.s. we have
(4.24)
Using (DR31) on the vertical digon, followed by (MP) and the Bubble relation (3.20), we obtain (4.23).
Relation (2.19) follows from straightforward computation and is left to the reader.
Relations Involving Three Colors. Relations (2.20) and (2.21) follow from isotopies of the foams involved. To show that ℱN,n respects relation (2.22), we use a different type of argument. First of all, we note that the images under ℱN,n of both sides of relation (2.22) are multiples of each other, because the graded vector space of morphisms in FoamN between the bottom and top webs has dimension one in degree zero. Verifying this only requires computing the coefficient of q-(4N-4) (this includes the necessary shift!) in the MOY polynomial associated to the web
(4.25)
which is a standard calculation left to the reader. To see that the multiplicity coefficient is equal to one, we close both sides of relation (2.22) simply by putting a dot on each open end. Using relations (2.14) and (2.16) to reduce these closed diagrams, we see that both sides give the same nonzero sum of disjoint unions of coloured StartDot-EndDot diagrams. Note that we have already proved that ℱN,n respects relations (2.14) and (2.16). By applying foam relation (4.12) to the images of all nonzero terms in the sum, we obtain a nonzero sum of dotted sheets. This implies that both sides of (2.22) have the same image under ℱN,n.We have now proved thatℱN,n is a monoidal functor for all N≥4. Our main result about the whole family of these functors, that is, for all N≥4 together, is the proposition below. It implies that all the defining relations in Soergel's category can be obtained from the corresponding relations between sl(N) foams, when all N≥4 are considered, and that there are no other independent relations in Soergel's category corresponding to relations between foams.Proposition 4.2.
Leti̲, j̲ be two arbitrary objects in 𝒮𝒞1,n and let f∈Hom(i̲,j̲) be arbitrary. If ℱN,n(f)=0 for all N≥4, then f=0.Proof.
Let us first suppose thati̲=j̲=∅. Suppose also that f has degree 2d and that N≥max{4,d+1}. Recall that, as shown  in  [5, Corollary 3], we know that Hom(∅,∅) is the free commutative polynomial ring generated by the StartDot-EndDots of all possible colors. So f is a polynomial in StartDot-EndDots, and therefore a sum of monomials. Let m be one of these monomials, no matter which one, and let mj denote the power of the StartDot-EndDot with color j in m. Close ℱN,n(f) by gluing disjoint discs to the boundaries of all open simple facets (i.e., the vertical ones with corners in the pictures). For each color j, put N-1-mj dots on the left simple open facet corresponding to j and also put N-1 dots on the rightmost simple open facet. Note that, after applying (RD1), we get a linear combination of dotted simple spheres. Only one term survives and is equal to ±1, because only in that term each sphere has exactly N-1 dots. This shows that ℱN,n(f)≠0, because it admits a nonzero closure.
Now let us suppose thati̲=∅ and j̲ is arbitrary. By   [4, Corollaries 4.11 and 4.12], we know that Hom(∅,j̲) is the free Hom(∅,∅)-module of rank one generated by the disjoint union of StartDots coloured by j̲. Closing off the StartDots by putting dots on all open ends gives an element of Hom(∅,∅), whose image under ℱN,n is nonzero for N big enough by the above. This shows that the generator of Hom(∅,j̲) has nonzero image under ℱN,n for N big enough, because ℱN,n is a functor.
Finally, the general case, fori̲ and j̲ arbitrary, can be reduced to the previous case by   [4, Corollary 4.12].



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*Source: 468968-2010-07-29.xml*