Twin TQFTs and Frobenius algebras

We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on a twin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra (C, W, z, z^*) consists of a commutative Frobenius algebra C, a symmetric Frobenius algebra W, and an algebra homomorphism z: C ->W with dual z^*: W ->C, satisfying some extra conditions. We also introduce a generalized 2-dimensional Topological Quantum Field Theory defined on singular 2-dimensional cobordisms and show that it is equivalent to a twin Frobenius algebra in a symmetric monoidal category.


Introduction
A 2-dimensional Topological Quantum Field Theory (TQFT) is a symmetric monoidal functor from the category 2Cob of 2-dimensional cobordisms to the category Vect k of vector spaces over a field k. The objects in 2Cob are smooth compact 1-manifolds without boundary, and the morphisms are the equivalence classes of smooth compact oriented cobordisms between them, modulo diffeomorphisms that restrict to the identity on the boundary. The category 2Cob of 2-cobordisms and that of 2D TQFTs are well understood, and it is known that 2D TQFTs are characterized by commutative Frobenius algebras, in the sense that the category of 2D TQFTs is equivalent as a symmetric monoidal category to the category of commutative Frobenius algebras. For the classic results involving these concepts, we refer to [1,6,12] and the book [10].
Lauda and Pfeiffer studied in [11] a special type of extended TQFTs defined on openclosed cobordisms. These cobordisms are certain smooth oriented 2-manifolds with corners that can be viewed as cobordisms between compact 1-manifolds with boundary, that is, between disjoint unions of circles S 1 and unit intervals I = [0, 1]. An openclosed TQFT is a symmetric monoidal functor Z : 2Cob ext → Vect k , where 2Cob ext denotes the category of open-closed cobordisms. Lauda and Pfeiffer showed that openclosed TQFTs are characterized by what they call knowledgeable Frobenius algebras (A, C, ι, ι * ), where the vector space C := Z(S 1 ) associated with the circle has the structure of a commutative Frobenius algebra, the vector space A := Z(I) associated with the interval has the structure of a symmetric Frobenius algebra, and there are linear maps ι : C → A and ι * : A → C satisfying certain conditions. This result was obtained by providing a description of the category of open-closed cobordisms in terms of generators and the Moore-Segal relations. They defined a normal form for such cobordisms, characterized by topological invariants, and then proved the sufficiency of the relations by constructing a sequence of moves which transforms the given cobordism into the normal form. They also showed that the category 2Cob ext of open-closed cobordisms is equivalent to the symmetric monoidal category freely generated by a knowledgeable Frobenius algebra. We remark that the entire construction in [11] was given for an arbitrary symmetric monoidal category, and not only for Vect k .
In [4] the author constructed the universal sl(2) link cohomology that uses dotted foams modulo local relations. This theory corresponds to a certain Frobenius algebra structure defined on Z[i, a, h, X]/(X 2 − hX − a), and for the case of a = h = 0 it gives rise to an isomorphic version of the sl(2) Khovanov homology [2,8]. One of the features of the construction in [4] is its functoriality property with respect to link cobordisms relative to boundaries, with no sign indeterminacy. In particular, it brings a properly functorial method for computing the Khovanov homology groups. In [5] the author described a method that computes fast and efficient the foam cohomology groups, and also provided a purely topological version of the foam theory in which no dots are required on cobordisms.
We give below a short review of the universal sl(2) foam link cohomology. Given a link diagram L, one associates to it a formal complex [L] whose objects are formally graded resolutions of L, called webs, and whose morphisms are formal linear combinations of singular cobordisms, also called foams, whose source and target boundaries are resolutions. These singular cobordisms are considered modulo certain local relations.
Each crossing of L is resolved in one of the following ways:

−→ and
A diagram obtained by resolving all crossings of L is a disjoint union of webs. A web is a planar oriented graph with 2k, k ≥ 0 bivalent vertices near which the two incident edges are oriented either towards the vertex or away from it. The simplest closed webs are depicted in (3.1).
A foam is an abstract cobordisms between such webs-regarded up to boundarypreserving isotopy-and has singular arcs (and/or singular circles) where orientations disagree, and near which the facets incident with a certain singular arc are compatibly oriented, inducing an orientation on that arc. The presence of these singular arcs (circles) is the reason for the second name of foams, namely singular cobordisms. Examples of such cobordisms are given in (3.2). The category of foams is denoted by Foams.
The construction of the universal sl(2) foam link cohomology is in the spirit of Bar-Natan's approach [3] to local Khovanov homology, with the difference that it uses webs and foams modulo a finite set of relations . From these relations, one obtains certain isomorphisms of webs in Foams / (see [4,Corollary 1]) allowing the removal of adjacent bivalent vertices satisfying some extra condition, and replacing each web by an isomorphic web containing exactly two vertices or, better, no vertices at all. Hence vertices are just some artifacts whose presence yields a homology theory for links which is functorial under link cobordisms with no sign ambiguity.
The original Khovanov homology relies on a 2D TQFT, but the sl(2) foam link cohomology uses a "tautological functor", as the 1-manifolds in the underlying theory are not oriented circles but piecewise oriented circles. Therefore, some type of extended 2D TQFT defined on foams would be quite desirable to have at hand for the foam cohomology theory for links/knots.
In this paper we make the first steps in achieving this goal. The cobordisms considered here are a particular case of those used in [4,5], in the sense that the 1-manifolds are disjoint unions of oriented circles and bi-webs, which are webs with exactly two vertices. The corresponding cobordisms between such 1-manifolds are called singular 2-cobordisms. We introduce the category Sing-2Cob of singular 2-cobordisms and show that it is equivalent as a symmetric monoidal category to the category freely generated by what we call a twin Frobenius algebra. A twin Frobenius algebra and knowledgeable Frobenius algebra resemble each other, in the sense that almost all properties of the second are satisfied by the first, except for the "Cardy relation", which is replaced by the so-called "isomorphism condition" for a twin Frobenius algebra. The definition of twin Frobenius algebras and their category is given in Section 2.
A question arises now: how can one address the more general case in which webs are allowed to have an arbitrary even number of bivalent vertices as in [4,5]? One way to do this is by imposing-in Foams-the isomorphisms of webs mentioned earlier, in particular, by imposing relations (CI) and (SR) obtained in [4]. Then the category Sing-2Cob can be regarded as a quotient of the category Foams.
We present in Section 3 a normal form for an arbitrary singular 2-cobordism and characterize the category Sing-2Cob in terms of generators and relations. We obtain that this category is equivalent, as a symmetric monoidal category, to the category freely generated by a twin Frobenius algebra. In Section 4 we define 2-dimensional twin TQFTs in C as symmetric monoidal functors Sing-2Cob → C, where C is a symmetric monoidal category, and prove that the category of twin TQFTs in C is equivalent as a symmetric monoidal category to the category of twin Frobenius algebras in C. For our purpose, the category C will be either Vect k or R-Mod, where k is a field and R a commutative ring, both containing the primitive fourth root of unity i.
In Section 5 we provide the example of twin Frobenius algebra and its corresponding 2-dimensional twin TQFT that, we believe, can be used in describing the sl(2) foam link cohomology theory with no dots.
As examples for the category C, we are interested in Vect k , the category of vector spaces over a field k and k -linear maps, and R-Mod, the category of modules over a commutative ring R and module homomorphisms. The ground field k or ring R is required to contain i, where i 2 = −1. Note that the unit object 1 ∈ C is then k or R.
For reader's convenience, we recall first a few definitions.
An algebra object (C, m, ι) in C consists of an object C and morphisms m : C ⊗C → C and ι : 1 → C in C such that: A coalgebra object (C, ∆, ) in C is an object C and morphisms ∆ : C → C ⊗ C and : C → 1 such that: A homomorphism of algebras f : C → C between two algebra objects (C, m, ι) and (C , m , ι ) in C is a morphism f of C such that: A homomorphism of coalgebras f : C → C between two coalgebra objects (C, ∆, ) and (C , ∆ , ) in C is a morphism f of C such that: A Frobenius algebra object (C, m, ι, ∆, ) in C consists of an object C together with morphisms m, ι, ∆, such that: • (C, m, ι) is an algebra object and (C, ∆, ) is a coalgebra object in C, A Frobenius object (C, m, ι, ∆, ) in C is called commutative if m•τ = m, and is called symmetric if • m = • m • τ. Given two Frobenius algebra objects (C, m, ι, ∆, ) and (C , m , ι , ∆ , ), a homomorhism of Frobenius algebras f : C → C is a morphism f in C which is both a homomorphism of algebra and coalgebra objects.

Definition 1.
A twin Frobenius algebra T := (C, W, z, z * ) in C consists of • two morphisms z : C → W and z * : such that z is a homomorphism of algebra objects in C and The first equality says that z * is the morphism dual to z, which implies that z * is a homomorphism of coalgebra objects in C. The second equality says that z(C) is contained in the center of the algebra W, and the last condition states that z and iz * are mutually inverse isomorphisms in C. Hence z (respectively z * ) is an isomorphism of algebra (respectively coalgebra) objects.

Definition 2.
A homomorphism of twin Frobenius algebras ) in a symmetric monoidal category C consists of a pair (f, g) of Frobenius algebra homomorphisms f : C 1 → C 2 and g : We denote by T-Frob(C) the category whose objects are twin Frobenius algebras in C and whose morphisms are twin Frobenius algebra homomorphisms. Proposition 1. The category T-Frob(C) forms a symmetric monoidal category in the following sense: • The tensor product of two twin Frobenius algebra objects T 1 = (C 1 , W 1 , z 1 , z * 1 ) and The unit object is given by 1 := (1, 1, id 1 , id 1 ); • The associativity and unit laws and the symmetric braiding are induced by those of C; • The tensor product of two homomorphisms f = (f 1 , g 1 ) and g = (f 2 , g 2 ) of twin Frobenius algebras is defined as f ⊗ g := (f 1 ⊗ f 2 , g 1 ⊗ g 2 ).

Singular cobordisms and the category Sing-2Cob
In this section we define the category of a singular 2-cobordisms and give a presentation of it in terms of generators and relations.

Description and topological invariants.
Definition 4. A singular 2-cobordism is an abstract piecewise oriented smooth 2dimensional manifold Σ with boundary ∂Σ = ∂ − Σ ∪ ∂ + Σ, where ∂ − Σ is ∂ − Σ with opposite orientation. Both ∂ − Σ and ∂ + Σ are embedded closed 1-manifolds and we call them the source and target boundary, respectively. A singular 2-cobordism has singular arcs and/or singular circles where orientations disagree. There are exactly two compatibly oriented 2-cells of the underlying 2-dimensional CW-complex Σ that meet at a singular arc/circle, and orientations of two neighboring 2-cells induce an orientation on the singular arc/circle that they share.
The objects in the category of singular 2-cobordisms are diffeomorphism classes of compact 1-manifolds. Specifically, such a manifold is diffeomorphic to a disjoint union of clockwise oriented circles and bi-webs; a bi-web is a closed graph with two bivalent vertices, as depicted below. The order of these manifolds is important, and there is a one-to-one correspondence between the objects of Sing-2Cob and sequences of zeros and ones.
(3.1) 0 = 1 = Definition 5. An object in the category Sing-2Cob consists of a finite sequence n = (n 1 , n 2 , . . . , n k ), where n j ∈ {0, 1}. The length of the sequence, denoted by |n| = k, can be any nonnegative integer, and equals the number of disjoint connected components of the corresponding object. Each such tuple stands for a 1-dimensional submanifold is a clockwise oriented circle if n j = 0 and a bi-web if n j = 1.
A morphism Σ : n → m in Sing-2Cob is an equivalence class [Σ]) (induced by ∼ =) of singular 2-cobordisms with source boundary n and target boundary m. Given Σ 1 a morphism from n to m and Σ 2 a morphism from m to k, their composition Σ 2 • Σ 1 is the singular 2-cobordism obtained by placing Σ 1 on top Σ 2 (by using rigid shifts along the z -axis). All our surfaces are assumed to have collars so that composition yields a smooth surface.
Examples of singular 2-cobordisms are given below. The source of our cobordisms is at the top and the target at the bottom of drawings, in other words, we read morphisms as cobordisms from top to bottom, by convention (note that this is the opposite convention of that used in [4,5]). The concatenation n m := (n 1 , n 2 , . . . , n |n| , m 1 , m 2 , . . . , m |m| ) of sequences together with the free union of singular 2-cobordisms, which we denote also by , endows the category Sing-2Cob with the structure of a symmetric monoidal category.
For each k ∈ N, there is an action of the symmetric group S k on the subset of objects n in Sing-2Cob for which |n| = k, defined by Given any object n in Sing-2Cob and any permutation σ ∈ S |n| , there is an obvious induced cobordism σ n : n → σ * n. For example, if n = (0, 1, 1, 0, 1) and σ = (12)(354) ∈ S 5 , the corresponding morphism σ n is the singular cobordism given in (3.3).
We remark that as morphisms of Sing-2Cob, these cobordisms satisfy τ σ * n • σ n = (τ • σ) n , for any object n and σ, τ ∈ S |n| . Definition 6. Let Σ : n → m be a morphism in Sing-2Cob and let l be the number of its boundary components that are diffeomorphic to the bi-web. In other words, l is the number of 1 entries of n m. Number these components by 1, 2, . . . , l. The orientation of Σ induces an orientation on all singular arcs of Σ and defines a permutation σ(Σ) ∈ S l , called the singular boundary permutation of Σ.
For exemplification, we consider the morphism Σ depicted in (3.4) and we number the bi-webs in its boundary by 1, 2, 3 and 4 from left to right, starting with those in the source and followed by those in the target. The singular boundary permutation of this morphism is σ( We remark that two equivalent singular 2-cobordisms are required to have the same singular boundary permutation (but two singular cobordisms having the same boundary permutation might not be equivalent). With this in mind, we need to refine the definition of the morphisms in Sing-2Cob.

3.2.
Generators. We use Morse theory to provide a generators and relations description of the category Sing-2Cob. Specifically, we decompose each singular 2-cobordism into cobordisms each of which contains exactly one critical point, and the components of such a decomposition are the generators for the morphisms.
Every singular 2-cobordism Σ ∈ Sing-2Cob admits a Morse function f : Σ → R. The set of critical points of f is finite, and they are all non-degenerate and isolated.
Proposition 2. Let Σ ∈ Sing-2Cob be a connected singular 2-cobordism and f : Σ → R a Morse function such that f has precisely one critical point. Then Σ is equivalent to one of the following singular 2-cobordisms: or to one of the following compositions of singular 2-cobordisms: These singular cobordisms are embedded in R 3 and have the source at the top and the target at the bottom of the diagram. The vertical axis of the drawing plane is −f.
Proof. It is well known that the critical point p is characterized by its index i(p), the number of negative eigenvalues of the Hessian of f at p.
(1) If i(p) = 2, the Morse function f has a local maximum. If p does not lie on a singular arc or singular circle, then Σ is diffeomorphic to C or to the first composition depicted in (3.7). If p lies on a singular arc or singular circle, then Σ is diffeomorphic to W or to the second composition in (3.7).
(2) If i(p) = 1, then we have a saddle point. If p does not lie on a singular arc, then Σ is of the form m C or ∆ C , or is equivalent to one of the compositions given in (3.9) or (3.10). If the saddle point p lies on a singular arc, then Σ is equivalent to m W or ∆ W , or to one of the compositions depicted in (3.11) or (3.12).
(3) If i(p) = 0, then f has a local minimum. If p does not belong to a singular arc or singular circle, then Σ is of the form ι C or is equivalent to the first composition displayed in (3.8). If p does belong to a singular arc or singular circle, then Σ is diffeomorphic to ι W or to the second composition given in (3.8).
Corollary 1. Every connected singular 2-cobordism can be obtained by gluing the cobordisms depicted in (3.5), (3.6) and those given in (3.13) below: 3.3. Non-connected singular 2-cobordisms. We treat the case of non-connected cobordisms via disjoint unions and permutations of the factors of disjoint unions, following Kock's work [10] for the case of ordinary 2-cobordisms. Since every permutation can be written as a product of transpositions, the following singular 2-cobordisms are sufficient to do this: There is no need to talk about crossing over or under, since our cobordisms are abstract manifolds, thus not embedded anywhere.
Without loss of generality, we assume that Σ : n → m has two connected components, Σ 1 and Σ 2 , and that n = (n 1 , n 2 , · · · , n |n| ). The source boundary of Σ 1 is a tuple p whose components form a subset of {n 1 , n 2 , · · · , n |n| }, and the source boundary of Σ 2 is the tuple q, which is the complement of p in {n 1 , n 2 , · · · , n |n| }.
We can permute the components of n by applying a diffeomorphism n → n, so that the components of p come before those of q. This diffeomorphism induces a cobordism S, and we can consider the singular 2-cobordism SΣ. Applying the same method to the target boundary of Σ, which is also the target boundary of SΣ, there is a permutation singular 2-cobordism T : m → m so that Σ = SΣT : n → m is a singular 2-cobordism which is the disjoint union (as a cobordism) of Σ 1 and Σ 2 . Then Σ ∼ = S −1 Σ T −1 , where S −1 and T −1 are the permutation cobordisms which are the inverses of S and T, respectively. For example, S −1 is the diffeomorphism that permutes the components of n such that the components of p come after those of q.
As an example, we consider the following singular cobordism: For the given cobordism we don't need to permute the source boundary of Σ, thus S is the disjoint union (as a cobordism) of two cylinders, but we do permute the target boundary of Σ by composing with a cobordism T. The composed cobordism SΣT is the disjoint union of its connected components (SΣT ) 1 and (SΣT ) 2 : We have proved the following: Lemma 1. Every singular 2-cobordism is equivalent to a composition of a permutation cobordism with a disjoint union of connected cobordisms, followed by a permutation cobordism.
Putting together the results of this subsection, we obtain: Proposition 3. The symmetric monoidal category Sing-2Cob is generated under composition and disjoint union by the following singular 2-cobordisms: 3.4. Relations. In this section we give a list of relations satisfied by the generators of Sing-2Cob. In Section 3.5 we define a normal form for singular 2-cobordisms with a given topological structure, namely the genus and singular boundary permutation. In Section 3.6 we prove that the relations given in Proposition 4 below are sufficient to completely describe the category Sing-2Cob; the techniques used are similar in spirit to those in [11].
Imposed local relations. We allow formal linear combinations of morphisms, with coefficients in the ground field/ring, and we extend the composition maps in the natural bilinear way. Then we impose the following local relations on the set of morphisms: We remark that the singular 2-cobordisms in the left equality above do not have the same singular boundary permutation.

Sufficient relations.
Proposition 4. The following relations hold in the symmetric monoidal category Sing-2Cob : (1) The object n = (0) forms a commutative Frobenius algebra object.

Consequences of relations.
We provide now additional relations that are implied by those described in Proposition 4, and which will be useful for the remaining of the paper.
Proposition 5. The cozipper is a coalgebra homomorphism, that is, the following singular cobordisms are equivalent: It will be useful to define the following singular cobordisms called singular pairing and singular copairing: Similarly, we define the cobordisms which we call the ordinary pairing and ordinary copairing: These cobordisms satisfy the zig-zag identities: 3.5. The normal form of a singular 2-cobordism. We give in this section the normal form of an arbitrary connected singular 2-cobordism Σ. The normal form is characterized by the singular boundary permutation σ(Σ) and genus g(Σ).

Particular case.
We first describe the normal form of a connected singular cobordism whose source consists entirely of bi-webs and whose target consists entirely of circles. Thus, we consider singular cobordisms Σ : n → m for which n = (1, 1, . . . , 1) and m = (0, 0, . . . , 0), and denote the set of all such cobordisms by Sing-2Cob W →C (n, m). Then we give the normal form for an arbitrary connected singular cobordism by using the zig-zag identities (3.28) and (3.29).
Notice that relations of the form (Σ id m ) • (id n Σ) = Σ Σ hold in Sing-2Cob for any Σ : n → m and Σ : n → m , and we will make use of them in order to have small heights for diagrams. Definition 8. Let Σ ∈ Sing-2Cob W →C (n, m) be a connected cobordism with singular boundary permutation σ(Σ) and genus g(Σ), and write the singular boundary permutation as a product of disjoint cycles σ(Σ) = σ 1 σ 2 . . . σ r , r ∈ N ∪ {0}, where σ k has length q k ∈ N, 1 ≤ k ≤ r. The normal form of Σ is the composition of the following singular 2-cobordisms: 1. For each cycle σ k , the singular cobordism A(q k ) consists of q k − 1 singular multiplications followed by a cozipper, as depicted below: A(q k ) := . . .
The normal form contains the free union of such cobordisms for each cycle σ k , 1 ≤ k ≤ r. If q k = 1 then A(q k ) is a cozipper, and if |n| = 0 then r = 0, and the free union r k=1 A(q k ) is replaced by the empty set.
Then σ(Σ) is the permutation that satisfies In Figure 1 we show a cobordism of the form (3.46), that is, the normal form of a cobordism in Sing-2Cob W →C (n, m), without precomposition with Σ σ(Σ) .
The following two results say that a cobordism given in its normal form is invariant, up to equivalence, under composition with certain permutation morphisms. • σ n k ] for any σ ∈ S |m| and for all cycles σ k ∈ S |n| , 1 ≤ k ≤ r, that appear in the decomposition of σ(Σ) = σ 1 σ 2 . . . σ r into disjoint cycles.
3.5.2. General case. We use the normal form for a connected singular cobordism in Sing-2Cob W →C (n, m) and the duality property for the bi-web and circle to obtain the normal form of a generic connected morphism [Σ] ∈ Sing-2Cob(n, m).
Let Σ be a representative of the equivalence class [Σ] and let n 0 n 1 be the permutation of n such that n 0 = (0, 0, . . . , 0) and n 1 = (1, 1, . . . , 1). Similarly, let m 0 m 1 be the permutation of m such that m 0 = (0, 0, . . . , 0) and m 1 = (1, 1, . . . , 1). In order to use the normal form described above, we need to associate to [Σ] a singular cobordism whose source contains only bi-webs and whose target contains only circles. We define the map where the singular cobordism f ([Σ]) is defined as follows. Let σ 1 be the permutation cobordism corresponding to σ 1 ∈ S |n| that sends n to n 1 n 0 . Similarly, denote byσ 2 the cobordism corresponding to the permutation σ 2 ∈ S |m| that sends m to m 1 m 0 .  We also remark that f ([Σ]) has a certain structure, in the sense that its source n and target m can be decomposed into free unions n = n t n s and m = m t m s , such that the bi-webs in n t (or n s ) and the circles in m t (or m s ) correspond to the bi-webs Figure 2. The image of [Σ] under the map f and circles coming from the target (or source) of σ 2 • Σ • σ −1 1 . The permutation σ 1 is an element of S |n s |+|m s | while σ 2 is an element of S |n t |+|m t | .
We define an inverse mapping f −1 that associates to is obtained by gluing singular copairings to the bi-webs in n t and ordinary pairings to the circles in m s , and then by precomposing the resulting cobordism with the cobordism corresponding to σ 1 and by postcomposing it with the cobordism corresponding to σ 2 .  3.6. Sufficiency of the relations. We show now that the relations described in Proposition 4 are sufficient in order to relate any connected singular 2-cobordism [Σ] ∈ Sing-2Cob W →C (n, m) to its normal form NF W →C (Σ).

This map is well-defined as well, and defines a bijection between the morphisms in
Recall that any connected singular 2-cobordism can be decomposed into the cobordisms (generators) given in (3.5), (3.6) and (3.13) (note that we ignore identity morphisms).
We use the notation X (or X ) for an arbitrary singular 2-cobordism X whose target (or source) is not glued to any other cobordism in the decomposition of Σ.
The following terminology is borrowed from [11,Definition 3.21].
Definition 10. Let [Σ] ∈ Sing-2Cob(n, m) be connected. The height of a generator G in the decomposition of Σ is the following number defined inductively: where X and Y are arbitrary cobordisms in the decomposition of Σ.
Then Σ is equivalent to its normal form, namely we have Proof. The proof is similar in spirit to that of [11,Theorem 3.22], with the difference that it uses our cobordisms and relations. We consider Σ be given in an arbitrary decomposition and construct a step by step diffeomorphism (relative to boundary) from this decomposition to the normal form NF W →C (Σ).
I. The decomposition of Σ is equivalent to one without singular cups and singular caps , by applying the following diffeomorphism: c) The following diffeomorhisms reduce the height of the singular comultiplication: i) −→ e) Iterate steps (IIa)-(IId). Since each step either removes the singular comultiplication or reduces its height, and since the target of Σ does not contain bi-webs, this process terminates with every singular comultiplication in one of the situations described above.
III. We look now at the possible cases in which the source and target of the singular multiplication may appear.
a) The decomposition of Σ is equivalent to one in which the source of the singular multiplication appears in one of the following situations: or those depicted in the statement of step (II). One can see that this claim holds by considering all possible situations of the singular comultiplication and then applying the following diffeomorphisms. Each iteration of the following steps either removes the singular multiplication or increases its height. As a result, each singular multiplication ends up into one of the situations above. i) iv) The diffeomorphisms of (IIc)i and (IId)i.
b) The decomposition of Σ is equivalent to one in which the target of the singular multiplication appears in one of the following situations: i) We first show that the source of every cozipper can be put so that it appears in one of the following situations: This claim is proved by applying the steps (I), (IId)ii and the following: ii) The singular genus-one operator can be removed by iterating the step (IIIa)iii, equation (3.44) and the following diffeomorphism:

−→
This process either reduces the height of the singular genus-one operator or removes it. Since the height of the operator cannot be zero, the process guarantees to remove the singular genus-one operator.
IV. In this step we show that there exists a sequence of diffeomorphisms that removes all singular comultiplications. Consider the set of all such comultiplications that appear in the decomposition of Σ and choose one of minimal height.
We can exclude the case since the singular comultiplication is of minimal height. From steps (II) and (IIIb)ii we know that the remaining situations to consider are: ? ?
where "?" may be any singular cobordism that contains no singular comultiplication. Since the above cobordisms are symmetric, it is enough to consider only one case, say the first one. Using step IIIb and the assumption that the singular comultiplication is of minimal height, there are exactly two possible situations for the first generator in the decomposition of "?", namely: ? ?
Iteratively applying the diffeomorphism (3.20) −→ and considering again the next two possible situations in the decomposition of "?", we see that after all, there are the following two possible cases: ? a) In the first case, the singular comultiplication is eliminated by applying the following sequence of diffeomorphisms: To remove the singular comultiplication in the second case above, we apply the following sequence of diffeomorphisms: We reapply steps II and III if needed. d) We iterate steps IVa-b and IVc until the last singular comultiplication has been eliminated. We remark that this process will terminate after a finite number of iterations, since steps II and III do not increase the number of singular comultiplications.
V. After the first four steps of the proof, all singular cups, caps and comultiplications have been eliminated from the decomposition of Σ, and the resulting decomposition has the following properties: a) Every singular multiplication has its source in one of the following situations and its target in one of the following situations b) Every cozipper appears in one of the situations explained in (IIIb)i. c) Every singular genus-one operator has been eliminated from the decomposition.
VI. We show now that the zipper can be eliminated from the decomposition of Σ. The following situations: are excluded by steps I, IV, (IIIb)ii, and the second equality in (3.15), respectively. It remains to consider the following cases: The second case can be reduced to the first one by applying the following sequence of diffeomorphisms:  VII. The (resulting) decomposition of Σ is equivalent to one in which the ordinary multiplication has its source in one of the following situations: Def = a) We can exclude the cases since we assume that the source of Σ is a free union of bi-webs.
To prove the claim we iterate the following diffeomorphisms whenever possible: b) Each of the above diffeomorphisms either removes the ordinary multiplication or increases its height, therefore applying these moves whenever possible assures that the process ends with each ordinary multiplication in the decomposition of Σ in one of the claimed situations.
VIII. The resulting decomposition of Σ is equivalent to one in which each ordinary comultiplication is in one of the following situations: Def = a) Employing step VII, we can exclude the cases: Moreover, every zipper has been eliminated at step VI, thus we can also exclude: The claim follows by iterating whenever possible the following diffeomorphisms: b)

←−
Notice that each of the above diffeomorphisms either decreases the height of the ordinary comultiplication or removes it, thus the process must end after a finite number of iterations.
IX. We claim that the resulting decomposition of Σ is now in the normal form. This follows from steps Va, VI, VII, and VIII, and the following two remarks.
a) Whenever an ordinary cap appears in the resulting decomposition of Σ, then has its target in one of the following situations: The other situations are excluded by steps VI and VIIb.
b) Whenever an ordinary cup appears in the resulting decomposition of Σ, then the source of is in one of the following situations: The other situations are excluded by step VIIIb. This completes the proof.  Proof. This follows at once from the fact that the normal form of a singular 2-cobordism is characterized by the singular boundary permutation and genus of the cobordism.
Proposition 14. The category Sing-2Cob of singular 2-dimensional cobordisms is equivalent to the symmetric monoidal category freely generated by a twin Frobenius algebra.

Twin TQFTs
In this section we define the notion of 2-dimensional twin TQFTs and show that the category of twin TQFTs is equivalent to the category of twin Frobenius algebras.
Definition 11. Let C be a symmetric monoidal category as before. A 2-dimensional twin Topological Quantum Field Theory in C is a symmetric monoidal functor Sing-2Cob → C. A homomorphism of twin TQFTs is a monoidal natural transformation of such functors. We denote by T-TQFT(C) the category of twin TQFTs in C and their homomorphisms.
There is a canonical equivalence of categories

T-TQFT(C) T-Frob(C).
Proof. Let Λ : Sing-2Cob → C be a 2-dimensional twin TQFT. In general, a monoidal functor is determined completely by its values on the generators of the source category, so let C and W be objects in C such that: Since Λ is monoidal, it implies that given a general object n = (n 1 , n 2 , n 3 , · · · , n |n| ) in Sing-2Cob, the functor Λ associates the tensor product of copies of C and W, with all parenthesis to the left. That is, Λ(n) = (((Λ(n 1 ) ⊗ Λ(n 2 )) ⊗ Λ(n 3 )) · · · Λ(n |n| )) with n i ∈ {0, 1} and Λ(0) := C and Λ(1) := W. Moreover, the fact that Λ is a symmetric monoidal functor also imply the following: Let the images of the generating morphisms in Sing-2Cob be denoted like below: The relations that hold in Sing-2Cob translate into relations among these maps and imply that (C, W, z, z * ) is a twin Frobenius algebra in C.
Conversely, let (C, W, z, z * ) be a twin Frobenius algebra in C. Thus (C, m C , ι C , ∆ C , C ) is a commutative Frobenius algebra and (W, m W , ι W , ∆ W , W ) is a symmetric Frobenius algebra. We can construct then a 2-dimensional twin TQFT Λ by using the above description as definition. Since the relations in Sing-2Cob correspond precisely to the axioms for a twin Frobenius algebra, the symmetric monoidal functor Λ is well-defined.
It is clear that these two constructions are inverse to each other. So we have established a one-to-one correspondence between 2-dimensional twin TQFTs and twin Frobenius algebras.
This correspondence also works for arrows (homomorphisms). An arrow in T-TQFT(C) is a monoidal natural transformation. Given two twin TQFTs Λ 1 and Λ 2 , a natural transformation η between them consists of maps Λ 1 (n) → Λ 2 (n) for each n = (n 1 , n 2 , n 3 , · · · , n |n| ) in Sing-2Cob. Since η is monoidal, the map Λ 1 (n) → Λ 2 (n) is the tensor product of maps Λ 1 (n l ) → Λ 2 (n l ), for 1 ≤ l ≤ |n|, and the naturality of η means that all these maps are compatible with morphisms in Sing-2Cob. Every such morphism is a composition of generators, so naturality translates into a commutative diagram for each generator. If we denote the twin Frobenius algebras corresponding to Λ 1 and Λ 2 by T 1 = (C 1 , W 1 , z 1 , z * 1 ) and T 2 = (C 2 , W 2 , z 2 , z * 2 ), respectively, then the diagrams corresponding to and are and they imply that g : W 1 → W 2 is a homomorphism of algebra objects. Similarly, the commutative diagrams corresponding to and are which states that g : W 1 → W 2 is a homomorphism of coalgebra objects. Hence, g : W 1 → W 2 is a Frobenius algebra homomorphism. There are similar commutative diagrams for the generators , , and , which amounts to the statement that f : C 1 → C 2 is also a Frobenius algebra homomorphism. Finally, the diagrams corresponding to the zipper and cozipper look like which are precisely the remaining conditions which f and g must satisfy in order for (f, g) to be a homomorphism of twin Frobenius algebras.
Conversely, given a homomorphism of twin Frobenius algebras T 1 → T 2 , we can use the above arguments backwards to construct a monoidal natural transformation between the twin TQFTs corresponding to T 1 and T 2 . 1, X R with inclusion map ι : R → A, ι(1) = 1. We remark that we consider these two rings for our main example, because they play an important role in [4,5].
Both A C and A W are commutative (thus symmetric) Frobenius algebras in R-Mod. It is clear that the multiplication maps m C,W : A ⊗ A → A are defined by the rules: m C,W (1 ⊗ X) = m C,W (X ⊗ 1) = X m C,W (1 ⊗ 1) = 1, m C,W (X ⊗ X) = hX + a.
Note that A W is a twisting of A C ; that is, the comultiplication ∆ W and counit W are obtained from ∆ C and C by "twisting" them with invertible element −i ∈ A: The fact that the above Frobenius structures differ by a twist is not surprising. Kadison showed that twisting by invertible elements of A is the only way to modify the counit and comultiplication in Frobenius systems (see [7, Theorem 1.6] and [9]). Note that a Frobenius system is the term used in the literature for a Frobenius algebra in R-Mod, where R is some ring.
A straightforward computation shows that (A C , A W , z, z * ) satisfies the axioms of a twin Frobenius algebra in R-Mod.
We denote by T : Sing-2Cob → R-Mod the 2-dimensional twin TQFT corresponding to (A C , A W , z, z * ), which assigns the ground ring R to the empty 1-manifold and assigns A ⊗k to a generic object n = (n 1 , n 2 , . . . , n |n| ) in Sing-2Cob. The i-th factor of A ⊗k is endowed with the structure A C if n i = 0 = , and with the structure A W if n i = 1 = .
On the generating morphisms of the category Sing-2Cob, the functor T is defined as follows: T : It is well worth noting that this twin TQFT satisfies the local relations for the dot-free version of the universal sl(2) foam cohomology for links (see [5,Section 4]). To be precise, the following identities hold: The last two identities are the 'UFO' local relations used in [5] and depicted below: Motivated by the above remarks, we believe that the twin Frobenius algebra and its associated twin TQFT considered in this example will play the key role in describing the universal dot-free sl(2) foam link cohomology using twin TQFTs. We will consider this problem in a subsequent paper.
Then (C, W, z, z * ) is a twin Frobenius algebra in S-Mod (or in Vect S , if S is a field), where z(x k ) = y k and z * (y k ) = −ix k , for every 0 ≤ k ≤ n − 1.