The diagrammatic Soergel category and sl(N)-foams, for N>3

For each N>3, 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, Stosic and Vaz. We show that through these functors Soergel's category can be obtained from the sl(N) foams.

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 [12].
We have tried to make the paper as self-contained as possible, but the reader should definitely leaf through [2], [3], [11] and [15] before reading the rest of this paper.
In Section 2 we recall Elias and Khovanov's version of Soergel's category. In Section 3 we review sl(N ) foams, as defined by Mackaay, Stošić 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.

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 [2]. Actually we give the version which they explained in Section 4.5 of [2] and which can be found in detail in [3].
Fix a positive integer n. The category SC 1 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 SC 1 are given by generators modulo relations. A morphism of SC 1 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): • Generators involving only one color: It is useful to define the cap and cup as ≡ ≡ • Generators involving two colors: -The 4-valent vertex, with distant colors, . 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 endows SC 1 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: 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 in SC 1 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: Two distant colors: Two adjacent colors:  (Elias-Khovanov). The category SC is equivalent to the Soergel category in [13]. From Soergel's results from [13] we have the following corollary. Notice that SC is an additive category but not abelian and we are using the (additive) split Grothendieck algebra.
In Subsection 4 we will define a a family of functors from SC 1,n to the category of sl(N ) foams, one for each N ≥ 4. These functors are grading preserving, so they obviously extend uniquely to SC 2,n . By the universality of the Karoubi envelope, they also extend uniquely to functors between the respective Karoubi envelopes.

FOAMS
3.1. Pre-foams. 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] and [14]. 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 * * Figure 1: Some elementary pre-foams singularities where six facets meet, such as on the right in Figure 1.
Definition 3.1. Let s γ be a finite oriented closed 4-valent graph, which may contain disjoint circles and loose endpoints. We assume that all edges of s γ are oriented. A cycle in s γ is defined to be a circle or a closed sequence of edges which form a piece-wise 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 pre-foam u is the identification space Σ/ s γ obtained by glueing boundary circles of Σ to cycles in s γ such that every edge and circle in s γ is glued to exactly three boundary circles of Σ and such that for any point p ∈ s γ : (1) if p 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) if p is a vertex of s γ , then it has a neighborhood as shown in Figure 1. We call s γ the singular graph, its edges and vertices singular arcs and singular vertices, and the connected components of u − s γ 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 Figure 1. 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 e 1 , and one oriented away from v, denoted e 2 . If we use the "left hand rule", then the cyclic ordering of the facets incident to e 1 and e 2 is either (3, 2, 1) and (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 pre-foam the number of singular vertices of type I is equal to the number of singular vertices of type II.
We can intersect a pre-foam u generically by a plane W in order to get a closed web, as long as the plane avoids the vertices of s γ . The orientation of s γ 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 values i ∈]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 s γ . 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 s γ does not intersect C ×

Definition 3.2. Let
Pfoam be the category whose objects are webs and whose morphisms are Q-linear combinations of isotopy classes of pre-foams with the obvious identity pre-foams and composition rule. We now define the q-degree of a pre-foam. Let u be a pre-foam, u 1 , u 2 and u 3 the disjoint union of its simple and double and marked facets respectively and s γ (u) its singular graph. Furthermore, let b 1 , b 2 and b 3 be the number of simple, double and marked vertical boundary edges of u, respectively. Define the partial q-gradings of u as where χ is the Euler characteristic and ∂ denotes the boundary.
The following result is a direct consequence of the definitions.

Lemma 3.4. q(u) is additive under the glueing of pre-foams.
We denote a simple facet with i dots by i .
Recall that the two-variable Schur polynomial π k,m can be expressed in terms of the elementary symmetric polynomials π 1,0 and π 1,1 . By convention, the latter correspond to ℄ and on a double facet respectively, so that (k,m) is defined to be the linear combination of dotted double facets corresponding to the expression of π k,m in terms of π 1,0 and π 1,1 . Analogously we can express the three-variable Schur polynomial π p,q,r in terms of the elementary symmetric polynomials π 1,0,0 , π 1,1,0 and π 1,1,1 By convention, the latter correspond to ℄, and on a triple facet respectively, so we can make sense of * (p,q,r) .

Foams.
In [11,14] we gave a precise definition of the Kapustin-Li formula, following Khovanov and Rozansky's work [8]. 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 pre-foam u. In the next proposition we recall those relations in Foam N that we need in the sequel. For their proofs and other relations we refer to [11].  (The disc removal relations) These were called cutting neck relations in [11,14].
(The digon removal relations)

THE FUNCTORS F N,n
Let n ≥ 1 and N ≥ 4 be arbitrary but fixed. In this section we define a monoidal functor F N,n between the categories SC 1,n and Foam N . On objects: F N,n sends the empty sequence to 1 n and the one-term sequence (j) to w j : with F N,n (jk) given by the vertical composite w j w k .
On morphisms: • The empty diagram is sent to n parallel vertical sheets: • The vertical line coloured j is sent to the identity cobordism of w j : The remaining n − 2 vertical parallel sheets on the r.h.s are not shown for simplicity, a convention that we will follow from now on. • The StartDot and EndDot morphisms are sent to the zip and the unzip respectively: The case j > k + 1 is given by reflexion around a horizontal plane.
• For the 6-valent vertices we have: The case with the colors switched is given by reflection in a vertical plane. Notice that F N,n respects the gradings of the morphisms. Proof. The assignment given by F N,n clearly respects the monoidal structures of SC 1,n and Foam N . So we only need to show that F N,n is a functor, i.e. it respects the relations (1) to (18) of Section 2.
"Isotopy relations": Relations (1) to (5) are straightforward to check and correspond to isotopies of their images under F N,n . For the sake of completeness we show the first equality in (1). We have One color relations: For relation (6) we have where the first equivalence follows from relations (1) and (3) and the second from isotopy of the foams involved. For relation (7) we have and, therefore, we have that Two distant colors: Relations (9) to (11) 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 (12) we first notice that using the (MP) move we get Apply (SqR 1 ) 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 (DR which equals F N,n . We now prove relation (13). We have an isotopy equivalence Notice that F N,n is the l.h.s. of the (SqR 2 ) relation. The first term on the r.h.s. of (SqR 2 ) is isotopic to −F N,n . For the second term on the r.h.s. of (SqR 2 ) we notice that F N,n contains (27) * * Applying (DR 3 1 ) followed by (MP) to remove the singular vertices creating simple-simple bubbles on the two double facets and using Equation (23) to remove these bubbles we get that F N,n is the second term on the r.h.s. of (SqR 2 ). We now prove relation (14) in the form = .
The image of the l.h.s. also contains a bit like the one in Equation (27). Simplifying it like we did in the proof of (13) we obtain the F N,n reduces to For the r.h.s. we have * * j j + 1 j + 2 Using (DR 3 1 ) on the vertical digon, followed by (MP) and the Bubble relation (23), we obtain (28). Relation (15) follows from straightforward computation and is left to the reader. (16) and (17) (10) and (12) to reduce these closed diagrams, we see that both sides give the same non-zero sum of disjoint unions of coloured StartDot-EndDot diagrams. Note that we have already proved that F N,n respects relations (10) and (12). By applying foam relation (24) to the images of all non-zero terms in the sum, we obtain a non-zero sum of dotted sheets. This implies that both sides of (18) have the same image under F N,n .

Relations involving three colors: Relations
We have now proved that F N,n is a monoidal functor for all N ≥ 4. Our main result about the whole family of these functors, i.e. 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. Let i, j be two arbitrary objects in SC 1,n and let f ∈ Hom(i, j) be arbitrary. If F N,n (f ) = 0 for all N ≥ 4, then f = 0.
Proof. Let us first suppose that i = j = ∅. Suppose also that f has degree 2d and that N ≥ max{4, d + 1}. Recall that, as shown in Corollary 3 in [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 m j denote the power of the StartDot-EndDot with color j in m. Close Note that, after applying (RD 1 ), 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 F N,n (f ) = 0, because it admits a non-zero closure. Now let us suppose that i = ∅ and j is arbitrary. By Corollaries 4.11 and 4.12 in [2], we know that Hom(∅, j) is the free Hom(∅, ∅)-module of rank one generated by the disjoint union of StartDots coloured by j. Closing off the StartDots by putting dots on all open ends gives an element of Hom(∅, ∅), whose image under F N,n is non-zero for N big enough by the above. This shows that the generator of Hom(∅, j) has non-zero image under F N,n for N big enough, because F N,n is a functor.
Finally, the general case, for i and j arbitrary, can be reduced to the previous case by Corollary 4.12 in [2].