Twin TQFTs and Frobenius Algebras

Abstract

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.

1. 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 [1–3] and the book [4].

Lauda and Pfeiffer studied in [5] a special type of extended TQFTs defined on open-closed 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 𝑆¹ and unit intervals 𝐼=[0,1]. An open-closed 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 open-closed TQFTs are characterized by what they call knowledgeable Frobenius algebras (𝐴,𝐶,𝜄,𝜄^∗), where the vector space 𝐶∶=𝑍(𝑆¹) 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 [5] was given for an arbitrary symmetric monoidal category, and not only for 𝐕𝐞𝐜𝐭_𝑘.

In [6], the author constructed a bigraded tangle cohomology theory which depends on one parameter, via a setup with webs and dotted foams (singular cobordisms) modulo a finite set of local relations. This work is a blend of Bar-Natan’s [7] approach to “local” Khovanov homology and Khovanov’s framework [8] using webs and foams. In [9], the author generalized the construction given in [6] to a two-parameter theory for tangles, which we call the universal 𝔰𝔩(2)  foam cohomology (it is “universal” in the sense of [10]). This two-parameter theory corresponds to a certain Frobenius algebra structure defined on ℤ[𝑖,𝑎,ℎ,𝑋]/(𝑋²−ℎ𝑋−𝑎), and which, for the case of of links, is a categorification of the quantum 𝔰𝔩(2)-link invariant. Adding the relation 𝑎=ℎ=0 yields an isomorphic version of the 𝔰𝔩(2) Khovanov homology [11, 12], while imposing 𝑎=1, ℎ=0 recovers Lee’s theory [13]. The advantage of working with foams instead of classical 2-cobordisms is that the construction in [9] (as well as its particular case introduced in [6]) yields a theory that satisfies an honest functoriality property with respect to tangle or link cobordisms, relative boundary, that is, with no sign indeterminacy. In particular, it resolves the sign ambiguity residing in the functoriality property of the Khovanov homology. We note that there is also the work by Clark et al. [14] that fixes the functoriality property of Khovanov’s invariant through the use of singular cobordisms (they called these “disoriented cobordisms”).

In [15], the author described a method that computes fast and efficient the 𝔰𝔩(2) foam cohomology groups, and also provided a purely topological version of the 𝔰𝔩(2) foam theory in which no dots are required on cobordisms.

We briefly review below the gadgets used in [6, 9]. A web is a planar graph with bivalent vertices near which the two incident edges are oriented either towards the vertex or away from it. Webs without vertices, thus oriented circles, are also allowed. Examples of webs are depicted in (11). A foam (also called singular cobordism) is an abstract cobordism between webs—regarded up to boundary-preserving isotopies—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. Examples of such cobordisms are given in (12) (the red curves in a singular cobordism diagram are singular arcs/circles).

The author arrives at webs and foams by considering a link diagram 𝐿 (we talk here about links instead of tangles just for simplicity) and resolving each crossing in 𝐿 in one of the following ways: (1) Then she associates to 𝐿 a “formal” chain complex [𝐿], whose objects are formally graded webs, and whose morphisms are formal linear combinations of foams. When considered as an object in the category of complexes of foams modulo a certain set of local relations, [𝐿] is an up-to-homotopy invariant of 𝐿. The jump from the geometric setup to an algebraic one, allowing to obtain a cohomology theory, is done via a “tautological functor” similar to the one used in [7].

There are many similarities between the algebraic structure of the category of open-closed cobordisms and certain relations satisfied by the singular cobordisms, although the two types of cobordisms are topologically different. The original Khovanov homology relies on a 2D TQFT, and it would be quite desirable and refreshing to have some kind of TQFT defined on foams/singular cobordisms and use it to obtain another method for defining the universal 𝔰𝔩(2) foam cohomology theory (specifically, the purely topological one—with no dots on cobordisms—discussed in [15], Section 4), and a generalization of the Khovanov homology. In particular, this would provide us with knowledge of the algebraic structure that governs this cohomology theory that provides a properly functorial Khovanov homology theory.

In this paper we make the first step in achieving this goal. The singular cobordisms considered here are a particular case of those used in [6, 9, 15], in the sense that the 1-manifolds are disjoint unions of oriented circles and biwebs (webs with exactly two bivalent vertices). The second step in reaching our goal will be treated in a subsequent paper, where we also show that it suffices to work with biwebs, as opposed to arbitrary webs (webs with an even number of bivalent vertices).

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 is almost the same as a knowledgeable Frobenius algebra; specifically, all properties of the latter one are satisfied by the first one, except for the “Cardy relation" which is replaced by what we call the “genus-one relation.” The definition of twin Frobenius algebras and their category is given in Section 2.

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. In Section 4 we define twin TQFTs as symmetric monoidal functors 𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛→𝒞, where 𝒞 is an arbitrary 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.

In Section 5 we provide examples of twin Frobenius algebras and thus twin TQFTs in 𝒞=𝐕𝐞𝐜𝐭_𝑘 and/or 𝒞=𝐑-𝐌𝐨𝐝, where 𝑘 is a field and 𝑅 a commutative ring.

2. Twin Frobenius Algebras

2.1. Definitions

Throughout the paper, one considers an arbitrary symmetric monoidal (tensor) category (𝒞,⊗,𝟏,𝛼,𝜆,𝜌,𝜏) with unit object 𝟏∈𝒞, associativity law 𝛼_{𝑋,𝑌,𝑍}∶(𝑋⊗𝑌)⊗𝑍→𝑋⊗(𝑌⊗𝑍), left-unit and right-unit laws 𝜆_𝑋∶𝟏⊗𝑋→𝑋 and 𝜌_𝑋∶𝑋⊗𝟏→𝑋, and with symmetric braiding 𝜏_𝑋,𝑌∶𝑋⊗𝑌→𝑌⊗𝑋, for 𝑋,𝑌 and 𝑍 objects in 𝒞.

For reader’s convenience, we recall 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 𝑚∘id_𝐶⊗𝑚∘𝛼_{𝐶,𝐶,𝐶}=𝑚∘𝑚⊗id_𝐶,𝑚∘id_𝐶⊗𝜄=𝜌_𝐶,𝑚∘𝜄⊗id_𝐶=𝜆_𝐶. (2) A coalgebra object (𝐶,Δ,𝜖) in 𝒞 is an object 𝐶 and morphisms Δ∶𝐶→𝐶⊗𝐶 and 𝜖∶𝐶→𝟏 such that id_𝐶⊗Δ∘Δ=𝛼_{𝐶,𝐶,𝐶}∘Δ⊗id_𝐶∘Δ,id_𝐶⊗𝜖∘Δ=𝜌_𝐶⁻¹,𝜖⊗id_𝐶∘Δ=𝜆_𝐶⁻¹. (3) 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 𝑓∘𝑚=𝑚^′∘𝑓⊗𝑓,𝑓∘𝜄=𝜄^′. (4) A homomorphism of coalgebras 𝑓∶𝐶→𝐶^′ between two coalgebra objects (𝐶,Δ,𝜖) and (𝐶^′,Δ^′,𝜖^′) in 𝒞 is a morphism 𝑓 of 𝒞 such that 𝑓⊗𝑓∘Δ=Δ^′∘𝑓,𝜖^′∘𝑓=𝜖. (5) A Frobenius algebra object (𝐶,𝑚,𝜄,Δ,𝜖) in 𝒞 consists of an object 𝐶 together with morphisms 𝑚,𝜄,Δ,𝜖 such that:(i)(𝐶,𝑚,𝜄) is an algebra object and (𝐶,Δ,𝜖) is a coalgebra object in 𝒞,(ii)(𝑚⊗id_𝐶)∘𝛼⁻¹_{𝐶,𝐶,𝐶}∘(id_𝐶⊗Δ)=Δ∘𝑚=(id_𝐶⊗𝑚)∘𝛼_{𝐶,𝐶,𝐶}∘(Δ⊗id_𝐶).

A Frobenius object (𝐶,𝑚,𝜄,Δ,𝜖) in 𝒞 is called commutative if 𝑚∘𝜏=𝑚, and it is called symmetric if 𝜖∘𝑚=𝜖∘𝑚∘𝜏.

Given two Frobenius algebra objects (𝐶,𝑚,𝜄,Δ,𝜖) and (𝐶^′,𝑚^′,𝜄^′,Δ^′,𝜖^′), a homomorphism 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 (i)a commutative Frobenius algebra 𝐶=(𝐶,𝑚_𝐶,𝜄_𝐶,Δ_𝐶,𝜖_𝐶),(ii)a symmetric Frobenius algebra 𝑊=(𝑊,𝑚_𝑊,𝜄_𝑊,Δ_𝑊,𝜖_𝑊),(iii)two morphisms 𝑧∶𝐶→𝑊 and 𝑧^∗∶𝑊→𝐶 of 𝒞

such that 𝑧 is a homomorphism of algebra objects in 𝒞 and 𝜖_𝐶∘𝑚_𝐶∘id_𝐶⊗𝑧^∗=𝜖_𝑊∘𝑚_𝑊∘𝑧⊗id_𝑊,,𝑚duality_𝑊∘id_𝑊⊗𝑧=𝑚_𝑊∘𝜏_𝑊,𝑊∘id_𝑊,,⊗𝑧centralitycondition𝑧∘𝑚_𝐶∘Δ_𝐶∘𝑧^∗=𝑚_𝑊∘𝜏_𝑊,𝑊∘Δ_𝑊,genus-onecondition. (6) The first equality says that 𝑧^∗ is the morphism dual to 𝑧 (which implies that 𝑧^∗ is a homomorphism of coalgebras in 𝒞). If 𝒞=𝐕𝐞𝐜𝐭_𝑘, the second equality says that 𝑧(𝐶) is contained in the center of the algebra 𝑊.

The reader will notice the similarities between twin and knowledgeable Frobenius algebras: their properties are almost the same, except that the Cardy condition for a knowledgeable Frobenius algebra is replaced by the genus-one condition in the definition of a twin Frobenius algebra.

Definition 2. A homomorphism of twin Frobenius algebras𝑓∶𝐶₁,𝑊₁,𝑧₁,𝑧^∗₁⟶𝐶₂,𝑊₂,𝑧₂,𝑧^∗₂ (7) in a symmetric monoidal category 𝒞 consists of a pair 𝑓=(𝑓₁,𝑓₂) of Frobenius algebra homomorphisms 𝑓₁∶𝐶₁→𝐶₂ and 𝑓₂∶𝑊₁→𝑊₂ such that 𝑧₂∘𝑓₁=𝑓₂∘𝑧₁ and 𝑧^∗₂∘𝑓₂=𝑓₁∘𝑧^∗₁.

Definition 3. 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 4. The category 𝐓−𝐅𝐫𝐨𝐛(𝒞) forms a symmetric monoidal category in the following sense:(i)the tensor product of two twin Frobenius algebra objects 𝐓_𝟏=(𝐶₁,𝑊₁,𝑧₁,𝑧^∗₁) and 𝐓_𝟐=(𝐶₂,𝑊₂,𝑧₂,𝑧^∗₂)  is defined as𝐓_𝟏⊗𝐓_𝟐∶=𝐶₁⊗𝐶₂,𝑊₁⊗𝑊₂,𝑧₁⊗𝑧₂,𝑧^∗₁⊗𝑧^∗₂; (8)(ii)the unit object is given by 𝟏∶=(𝟏,𝟏,id_𝟏,id_𝟏);(iii)the associativity and unit laws and the symmetric braiding are induced by those of 𝒞;(iv)the tensor product of two homomorphisms 𝑓=(𝑓₁,𝑓₂)  and 𝑔=(𝑔₁,𝑔₂)  of twin Frobenius algebras is defined as 𝑓⊗𝑔∶=(𝑓₁⊗𝑔₁,𝑓₂⊗𝑔₂).

2.2. The Category Th(T-Frob)

The definition of the category called the theory of twin Frobenius algebras, denoted by Th(T-Frob), follows Laplaza’s [16] construction of the “free category with group structure.” The objects of this category are elements of the free {𝟏,⊗}-algebra over the two element set {𝐶,𝑊} and is the analogue of the category Th(K-Frob) introduced in [5, Section 2.2].

The objects of Th(T-Frob) are words generated by the symbols 𝟏,𝐶, and 𝑊, which are objects by themselves. If 𝑋 and 𝑌 are objects of Th(T-Frob) then (𝑋⊗𝑌) is also an object.

Consider a graph 𝒢 whose vertices are the objects of Th(T-Frob). Then, there are the following edges: 𝑚_𝐶∶𝐶⊗𝐶⟶𝐶,𝜄_𝐶Δ∶𝟏⟶𝐶,_𝐶∶𝐶⟶𝐶⊗𝐶,𝜖_𝐶𝑚∶𝐶⟶𝟏,_𝑊∶𝑊⊗𝑊⟶𝑊,𝜄_𝑊Δ∶𝟏⟶𝑊,_𝑊∶𝑊⟶𝑊⊗𝑊,𝜖_𝑊∶𝑊⟶𝟏,𝑧∶𝐶⟶𝑊,𝑧^∗∶𝑊⟶𝐶. (9) For all objects 𝑋,𝑌,𝑍 there are the following edges: 𝛼_{𝑋,𝑌,𝑍}∶(𝑋⊗𝑌)⊗𝑍⟶𝑋⊗(𝑌⊗𝑍),𝛼_{𝑋,𝑌,𝑍}()⟶()𝜏∶𝑋⊗𝑌⊗𝑍𝑋⊗𝑌⊗𝑍,_𝑋,𝑌∶𝑋⊗𝑌⟶𝑌⊗𝑋,𝜏_𝑋,𝑌𝜆∶𝑌⊗𝑋⟶𝑋⊗𝑌,_𝑋∶𝟏⊗𝑋⟶𝑋,𝜆_𝑋𝜌∶𝑋⊗𝟏⟶𝑋,_𝑋∶𝑋⊗𝟏⟶𝑋,𝜌_𝑋∶𝑋⟶𝑋⊗𝟏. (10) For every edge 𝑓∶𝑋→𝑌 and for every object 𝑍, there are edges 𝑓⊗𝑍∶𝑋⊗𝑍→𝑌⊗𝑍 and 𝑍⊗𝑓∶𝑍⊗𝑋→𝑍⊗𝑌.

We denote by ℋ the category freely generated by the graph 𝒢, and define a relation ∼ on ℋ by requiring the following:(i)that each pair of edges 𝑒 and 𝑒 are inverses of each other;(ii)the relations that make (𝐶,𝑚_𝐶,𝜄_𝐶,Δ_𝐶,𝜖_𝐶) a commutative Frobenius algebra and (𝑊,𝑚_𝑊,𝜄_𝑊,Δ_𝑊,𝜖_𝑊) a symmetric Frobenius algebra;(iii)the relations that make 𝑧∶𝐶→𝑊 an algebra homomorphism and those given in (6);(iv)the relations that make 𝛼_{𝑋,𝑌,𝑍},𝜆_𝑋, and 𝜌_𝑋 satisfy the pentagon and triangle axioms of a monoidal category and those that make 𝜏_𝑋,𝑌 a symmetric braiding, for all objects 𝑋,𝑌,𝑍;(v)the relations that state the naturality of 𝛼,𝜆,𝜌,𝜏,𝛼,𝜆,𝜌,𝜏 and those that make ⊗ a functor.

We also require that for each relation 𝑎∼𝑏, we have as well the relations 𝑎⊗𝑋∼𝑏⊗𝑋 and 𝑋⊗𝑎∼𝑋⊗𝑏 for all objects 𝑋 in ℋ, as well as the relations obtained from these by applying this process a finite number of times.

We define the category 𝐓𝐡(𝐓-𝐅𝐫𝐨𝐛)∶=ℋ_/∼, that is, the category ℋ modulo the category congruence generated by ∼ defined above. The category Th(T-Frob) contains a twin Frobenius algebra object 𝐓=(𝐶,𝑊,𝑧,𝑧^∗) and is the symmetric monoidal category freely generated by 𝐓. For each twin Frobenius algebra 𝐓^′=(𝐶^′,𝑊^′,𝑧^′,𝑧^′∗) in 𝒞, there is exactly one strict symmetric monoidal functor 𝐹_𝐓′∶𝐓𝐡(𝐓-𝐅𝐫𝐨𝐛)→𝒞 that maps 𝐓 to 𝐓^′ and 𝟏∈𝐓𝐡(𝐓-𝐅𝐫𝐨𝐛) to 𝟏∈𝒞.

Proposition 5. The category 𝐓-𝐅𝐫𝐨𝐛(𝒞) is equivalent as a symmetric monoidal category to the category of symmetric monoidal functors 𝐓𝐡(𝐓-𝐅𝐫𝐨𝐛)→𝒞 and their monoidal natural transformations.

Proof. The monoidal equivalence of the two categories is constructed identical to that in the proof of [5, Proposition 2.8]. One has only to replace the category Th(K-Frob) used in [5] with our category Th(T-Frob).

3. Singular Cobordisms and the Category Sing-2Cob

In this section we define the category of singular 2-cobordisms and give a presentation of it in terms of generators and relations. Singular 2-cobordisms form a special type of compact, globally oriented, smooth 2-manifolds. What we call Sing-2Cob in the following is in fact a skeleton of the category of singular 2-cobordisms; we choose particular embedded 1-manifolds as the objects of this category.

3.1. Description and Topological Invariants

Definition 6. A singular 2-cobordism (or shortly, singular cobordisms) is an abstract, piecewise oriented (but globally orientable) smooth 2-manifold Σ with boundary 𝜕Σ=𝜕⁻Σ∪𝜕⁺Σ, where 𝜕⁻Σ is 𝜕⁻Σ with opposite orientation. Both 𝜕⁻Σ and 𝜕⁺Σ are embedded, closed 1-manifolds, called the source and target boundary, respectively. A singular 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. In our diagrams we draw the singular arcs/circles using red oriented curves.

Two singular cobordisms Σ₁ and Σ₂ are considered equivalent, and we write Σ₁≅Σ₂, if there exists an orientation-preserving diffeomorphism Σ₁→Σ₂ which restricts to the identity on the boundary and which preserves the singular arcs and circles.

The boundary 𝜕Σ of a singular cobordism Σ is a disjoint union of (clockwise) oriented circles and biwebs. In this paper, a biweb is a closed oriented graph with two bivalent vertices, such that each vertex is either a source or a sink. Since we want to work with the skeleton of the category of singular cobordisms, we fix one specific oriented circle, and a specific biweb. We denote the circle by 0 and the biweb by 1: (11)

Definition 7. An object in the category Sing-2Cob consists of a finite sequence 𝐧=(𝑛₁,𝑛₂,…,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. Hence 𝐧 is the disjoint union of 𝑠 copies of the fixed circle and 𝑡 copies of the fixed biweb, for some nonnegative integers 𝑠 and 𝑡 with 𝑠+𝑡=|𝐧|. If |𝐧|=0, then 𝐧 is the empty 1-manifold.

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. The composition of morphisms is obtained in the standard way, namely, by gluing along the common boundary.

Although the objects of Sing-2Cob are embedded 1-manifolds, the morphisms are not embedded. Examples of morphisms of Sing-2Cob 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. (12)

The concatenation 𝐧∐𝐦∶=(𝑛₁,𝑛₂,…,𝑛_|𝐧|,𝑚₁,𝑚₂,…,𝑚_|𝐦|) of sequences together with the free union of singular cobordisms, which we also denote 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 𝜎∗𝐧∶=𝑛_𝜎−1(1),𝑛_𝜎−1(2),…,𝑛_{𝜎−1(𝑘)}. (13) Given any object 𝐧 in Sing-2Cob and any permutation 𝜎∈𝑆_|𝑛|, there is an obvious induced permutation cobordism𝜎^𝐧∶𝐧⟶𝜎∗𝐧. (14) For example, if 𝐧=(0,1,1,0,1) and 𝜎=(12)(354)∈𝑆₅, the corresponding morphism 𝜎^𝐧 is the singular cobordism given in (15) 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 8. Let Σ∶𝐧→𝐦 be a morphism in Sing-2Cob and let 𝑙 be the number of its boundary components representing the biweb. 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 (16) and we number the biwebs 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(Σ)=(1)(234)=(234)∈𝑆₄(16) Since a singular cobordism determines a boundary permutation, we need to refine the definition of the morphisms in Sing-2Cob.

Definition 9. A morphism Σ∶𝐧→𝐦 is a pair Σ=([Σ],𝜎) consisting of an equivalence class [Σ] of singular cobordisms with source boundary 𝐧 and target boundary 𝐦, and with singular boundary permutation 𝜎(Σ)=𝜎.

Definition 10. Let Σ∶𝐧→𝐦 be a morphism of Sing-2Cob.(1)The genus 𝑔(Σ) is the genus of the topological 2-manifold underlying Σ.(2)The singular number 𝑠(Σ) is the number of singular circles that Σ contains. (Note that each such circle is homotopic to a point within Σ.)

3.2. Structural Relations

In this subsection we provide a list of diffeomorphisms which describe the algebraic structure of the category Sing-2Cob.

Proposition 11. The following diffeomorphisms hold in the symmetric monoidal category 𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛.(1)The object  𝐧=(0)  forms a commutative Frobenius algebra object(17)

(18)

(19)(20)(2)The object  𝐧=(1)  forms a symmetric Frobenius algebra object (21)(22)(23)(24)(3)The zipper    forms an algebra homomorphism (25)(4)The cozipper    is dual to the zipper(26)(5)Centrality relation(27)(6)The genus-one relation(28)

Proof. It is well known that the first set of equivalences of cobordisms depicted in (17)–(20) hold, and it is not hard to see that also the remaining diffeomorphisms hold. We prove these using nested discs.
We can use punctured discs to represent singular cobordisms. For example, we have the following graphical representations: (29) Then the singular cobordism is interpreted as sewing in an annulus and a disc     
Thus we have
The associativity property depicted in (21) is obtained by making the following different decompositions of a disc with three punctures: (30) We explain now why the biweb = (1) forms a symmetric Frobenius algebra object, instead of a commutative Frobenius algebra object. For this, let us consider the following singular cobordisms and their graphical representation: (31) We observe that , since the vertices of the biwebs forming the boundary of the two discs have different connection types. However, connecting the vertices of the outside biweb, thus gluing in the disc (corresponding to the singular “cup” cobordism ) along the outside biweb, we obtain the diffeomorphism in (24).
The first relation in (25) is obtained from the following two different decompositions of a disc with two punctures: (32) We verify below the centrality relation given in (27) as follows: (33) Finally, we prove the genus-one relation depicted in (28). The graphical representations of the two involved cobordisms are hard to draw. However, we have the following: (34) Since the two compositions of nested disks above are diffeomorphic, the genus-one relation follows.
We leave to the interested reader the proof of the remaining diffeomorphisms.

3.3. Consequences of Relations

We provide now additional diffeomorphisms implied by those described in Proposition 11, and which will be useful for the remaining of the paper.

Proposition 12. The cozipper is a coalgebra homomorphism, that is, the following singular cobordisms are equivalent: (35) It will be useful to define the following singular cobordisms called singular pairing and singular copairing: (36) Similarly, one defines the cobordisms which one calls the ordinary pairing and ordinary copairing: (37) These cobordisms satisfy the zig-zag identities: (38)(39) It follows from (24) and (21) that the singular pairing is invariant and symmetric: (40) Similarly, it follows from (20) and (17) that the ordinary pairing is invariant and symmetric: (41) It is easy to see that similar results hold for both singular and ordinary copairings.

Proposition 13. The following singular cobordisms are equivalent: (42)(43)

Proof. The first equivalence of cobordisms in (42) is the same as the equivalence in (26). The proof of the second diffeomorphism in (42) is given below, where by “Nat” we denote the diffeomorphisms which express the natural behavior of the symmetric twist:(44)The proof of (43) is given below:(45)

Proposition 14. The following singular cobordisms are equivalent:(46)(47)

Proof. The diffeomorphisms given in (46) follow from the following sequences of diffeomorphisms:(48)The proof of (47) is done similarly by replacing the biweb with an oriented circle, thus replacing the singular cobordisms above with their ordinary cobordisms counterparts.

Proposition 15. The following singular cobordisms are equivalent: (49)(50)

Proof. The proof of these equivalences is done in a similar manner as in the previous proposition.

Proposition 16. The following singular cobordisms are equivalent: (51)(52)

Proof. The first diffeomorphism in (51) is obtained from the following sequence of diffeomorphisms:(53)The first diffeomorphism depicted in (52) follows from the following sequence of diffeomorphisms:(54) The second equivalences of singular cobordisms in (51) and (52) are obtained in a similar manner.

It is easy to see that the next three propositions hold.

Proposition 17. The singular genus-one operator can be moved around freely in any diagram. Specifically, the following cobordisms are equivalent: (55)(56)

Proposition 18. The genus-one operator can be moved around freely in any diagram. Specifically, the following cobordisms are equivalent: (57)(58)

Proposition 19. Singular cobordisms of the form can be moved freely in any diagram. That is, the following singular cobordisms are equivalent: (59)(60)(61)

3.4. The Normal Form of a Connected Singular 2-Cobordisms

In this subsection we define the normal form of a connected singular cobordism Σ with a given topological structure, namely, genus, singular number, and singular boundary permutation. The reader will find many similarities with the description of the normal form of an open-closed cobordism given in [5].

3.4.1. Particular Case

We define first the normal form of a connected singular cobordism whose source consists entirely of copies of the biweb 1=, and whose target consists entirely of copies of the circle 0=. Specifically, 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 (38) and (39).

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 20. Let Σ∈𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛_𝑊→𝐶(𝐧,𝐦) be a connected cobordism with singular boundary permutation 𝜎(Σ), genus 𝑔(Σ), and singular number 𝑠(Σ) and write the singular boundary permutation as a product of disjoint cycles 𝜎(Σ)=𝜎₁𝜎₂⋯𝜎_𝑟, 𝑟∈ℕ∪{0}, where 𝜎_𝑘 has length 𝑞_𝑘∈ℕ, 1≤𝑘≤𝑟. The normal form of Σ is the composition NF_𝑊→𝐶(Σ)=𝐸_|𝐦|∘𝐷_𝑔(Σ)∘𝐶_𝑠(Σ)∘𝐵_𝑟∘^𝑟_𝑘=1𝐴𝑞_𝑘∘Σ_𝜎(Σ) (62) of the following singular 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: (63)  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=1𝐴(𝑞_𝑘) is replaced by the empty set.(2) If 𝑟≥1, then the singular cobordism 𝐵_𝑟 consists of 𝑟−1 multiplications (64)  If 𝑟=0 then 𝐵₀∶=.(3) If the singular number 𝑠(𝜎)≥1, the singular cobordism 𝐶_𝑠(𝜎) is the composite (65)  If 𝑠(Σ)=0 then 𝐶_𝑠(Σ)=∅.(4) If 𝑔(Σ)≥1, the singular cobordism 𝐷_𝑔(Σ) is the composite (66)  If 𝑔(Σ)=0 then 𝐷_𝑠(Σ)=∅.(5) If |𝐦|≥1, then the singular cobordism 𝐸_|𝐦| consists of |𝐦|−1 comultiplications, as depicted below: (67)  If |𝐦|=0 then 𝐸₀∶=.(6)Σ_𝜎(Σ) represents the permutation singular cobordism induced by the permutation 𝜎(Σ) given below (permutation cobordisms were defined in (14)). Let 𝜏(Σ) be the singular boundary permutation of the cobordism 𝐸_|𝐦|∘𝐷_𝑔(Σ)∘𝐶_𝑠(Σ)∘𝐵_𝑟∘^𝑟_𝑘=1𝐴𝑞_𝑘. (68)  Then 𝜎(Σ) is the permutation that satisfies 𝜎(Σ)=𝜎(Σ)⁻¹⋅𝜏(Σ)⋅𝜎(Σ). (69)

Note that precomposing 𝐸_|𝐦|∘𝐷_𝑔(Σ)∘𝐶_𝑠(Σ)∘𝐵_𝑟∘(∐^𝑟_𝑘=1𝐴(𝑞_𝑘)) with Σ_𝜎(Σ)  yields a singular cobordism whose singular boundary permutation is 𝜎(Σ).

In Figure 1 we show a cobordism of the form (68), that is, the normal form of a cobordism in 𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛_𝑊→𝐶(𝐧,𝐦), without precomposition with Σ_𝜎(Σ).

Figure 1 

Normal form of a cobordism in 𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛𝑊→𝐶(𝐧,𝐦), without precomposition with a permutation.

The following two results say that a cobordism given in its normal form is invariant, up to equivalence, under composition with certain permutation morphisms.

Proposition 21. Let [Σ]∈𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛_𝑊→𝐶(𝐧,𝐦). Then 𝜎^𝐦∘NF_𝑊→𝐶(Σ)=NF_𝑊→𝐶(Σ=)NF_𝑊→𝐶(Σ)∘𝜎^𝐧_𝑘 (70) for any 𝜎∈𝑆_|𝐦| and for all cycles 𝜎_𝑘∈𝑆_|𝐧|,1≤𝑘≤𝑟, that appear in the decomposition of 𝜎(Σ)=𝜎₁𝜎₂⋯𝜎_𝑟 into disjoint cycles.

3.4.2. 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 biweb and the 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 𝐧₀∐𝐧₁ be the permutation of 𝐧 such that 𝐧₀=(0,0,…,0) and 𝐧₁=(1,1,…,1). Similarly, let 𝐦₀∐𝐦₁ be the permutation of 𝐦 such that 𝐦₀=(0,0,…,0) and 𝐦₁=(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 copies of the biweb and whose target contains only copies of the circle. We define the map 𝑓∶𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛(𝐧,𝐦)⟶𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛_𝑊→𝐶𝐦₁𝐧₁,𝐦₀𝐧₀,[Σ]↦𝑓([Σ]), (71) where the singular cobordism 𝑓([Σ]) is defined as follows. Let 𝜎₁ be the permutation cobordism corresponding to 𝜎₁∈𝑆_|𝐧| that sends 𝐧 to 𝐧₁∐𝐧₀. Similarly, denote by 𝜎₂ the cobordism corresponding to the permutation 𝜎₂∈𝑆_|𝐦| that sends 𝐦 to 𝐦₁∐𝐦₀. We define 𝑓([Σ]) as the singular cobordism obtained from [Σ] by precomposing with 𝜎₁⁻¹, postcomposing with 𝜎₂ and gluing copairings on every circle that 𝐧₀ contains, and singular pairings on every biweb that 𝐦₁ contains.

For exemplification, consider 𝐧=(0,1,1),𝐦=(0,1,0) and [Σ] an arbitrary singular cobordism from 𝐧 to 𝐦 (note that |𝐧| and |𝐦| do not have to be equal). The corresponding permutation cobordisms 𝜎₁,𝜎₂ and the image of [Σ] under 𝑓 are given in (72).(72)Notice that the mapping 𝑓 is well defined; namely, 𝑓([Σ]) is a morphism in the category 𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛_𝑊→𝐶(𝐦₁∐𝐧₁,𝐦₀∐𝐧₀), and [𝑓([Σ])]=[𝑓([Σ^′])] whenever [Σ]=[Σ^′]. Therefore it makes sense to consider the normal form NF_𝑊→𝐶(𝑓([Σ])).

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 copies of the biweb in 𝐧^′_𝑡 (or 𝐧^′_𝑠) and the copies of the circle in 𝐦^′_𝑡 (or 𝐦^′_𝑠) correspond to the copies of the biweb and of the circle coming from the target (or source) of 𝜎₂∘Σ∘𝜎₁⁻¹. The permutation 𝜎₁ is an element of 𝑆_{|𝐧′𝑠|+|𝐦′𝑠|}, while 𝜎₂ is an element of 𝑆_{|𝐧′𝑡|+|𝐦′𝑡|}.

We define an inverse mapping 𝑓⁻¹ that associates to [Φ]∈𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛_𝑊→𝐶(𝐧^′,𝐦^′) the singular cobordism 𝑓⁻¹([Φ])∈𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛_𝑊→𝐶(𝜎₂(𝐧^′_𝑡∐𝐦^′_𝑡),𝜎₁(𝐧^′_𝑠∐𝐦^′_𝑠)). The cobordism 𝑓⁻¹([Φ]) is obtained by gluing singular copairings to the biwebs in 𝐧^′_𝑡 and ordinary pairings to the circles in 𝐦^′_𝑠, and then by precomposing the resulting cobordism with the cobordism corresponding to 𝜎₁ and by postcomposing it with the cobordism corresponding to 𝜎₂⁻¹. This map is well defined and provides a bijection between the morphisms in 𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛(𝐧,𝐦) and those in 𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛_𝑊→𝐶(𝐦₁∐𝐧₁,𝐦₀∐𝐧₀).

Going back to the example in (72), we give in Figure 2 the singular cobordism [𝑓⁻¹([𝑓([Σ])])]≅[Σ].

Figure 2 

Figure 3 

Assignments of Λ on the generating morphisms.

Definition 22. Let [Σ]∈𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛(𝐧,𝐦) where Σ is a connected cobordism. One defines the normal form of [Σ] by [NF(Σ)]∶=𝑓⁻¹NF_𝑊→𝐶𝑓([Σ]). (73)

3.5. Nonconnected Singular 2-Cobordisms

We treat the case of nonconnected cobordisms via disjoint unions and permutations of the factors of disjoint unions, following Kock’s work [4] for the case of ordinary 2-cobordisms. Since every permutation can be written as a product of transpositions, the following singular cobordisms are sufficient to do this: (74) 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, Σ₁ and Σ₂, and that 𝐧=(𝑛₁,𝑛₂,…,𝑛_|𝑛|). The source boundary of Σ₁ is a tuple 𝐩 whose components form a subset of {𝑛₁,𝑛₂,…,𝑛_|𝑛|}, and the source boundary of Σ₂ is the tuple 𝐪, which is the complement of 𝐩 in {𝑛₁,𝑛₂,…,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 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 cobordism 𝑇∶𝐦→𝐦 so that Σ^′=𝑆Σ𝑇∶𝐧→𝐦 is a singular cobordism which is the disjoint union (as a cobordism) of Σ₁ and Σ₂. Then Σ≅𝑆⁻¹Σ^′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: (75) For the given cobordism we do not 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: (76) We have proved the following.

Lemma 23. Every singular cobordism is equivalent to a composition of a permutation cobordism with a disjoint union of connected cobordisms, followed by a permutation cobordism.

3.6. Sufficiency of the Relations

In this subsection, we show that the relations described in Proposition 11 are sufficient in order to relate any connected singular cobordism [Σ]∈𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛_𝑊→𝐶(𝐧,𝐦) to its normal form NF_𝑊→𝐶(Σ).

We use the notation for an arbitrary singular cobordism 𝑋 whose target (or source) is not glued to any other cobordism in the decomposition of Σ.

The following terminology is borrowed from [5, Definition 3.21].

Definition 24. 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: (77) where 𝑋 and 𝑌 are arbitrary cobordisms in the decomposition of Σ.

Theorem 25. Let [Σ]∈𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛_𝑊→𝐶(𝐧,𝐦) be a connected singular 2-cobordism. Then Σ is equivalent to its normal form; namely, one has [Σ]=NF_𝑊→𝐶(Σ). (78)

Proof. The proof is similar in spirit to that of [5, Theorem 3.22], with the difference that it uses our cobordisms and their topological structure. 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: (79)(II) The decomposition of Σ is equivalent to one in which every singular comultiplication has its target in one of the following situations: (80)  or (81)  where the symbol “?” is any singular cobordism which may or may not be connected to the singular multiplication at the bottom of the diagram. This is proved by considering every possible situation in which the singular comultiplication may appear.(a) We can exclude the cases and by step (I).(b) Apply the diffeomorphism whenever possible.(c) The following diffeomorphisms reduce the height of the singular comultiplication: (82)(d) The following diffeomorphisms eliminate the singular comultiplication: (83)(e) Iterate steps (II)(a)–(II)(d). Since each step either removes the singular comultiplication or reduces its height, and since the target of Σ does not contain biwebs, 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.  We remark first that we do not need to consider the case , since the part Σ_𝜎(Σ) in the definition of the normal form of Σ (see (62)) is taking care of the twist cobordism . Similarly, the cases (84)
  can be excluded by moving upwards the twist past the singular multiplication(s) (i.e., by the naturality of the twist). In each case, the corresponding diffeomorphism yields a cobordism equivalent to Σ, whose decomposition starts with a permutation cobordism, which will be reflected in the Σ_𝜎(Σ) part of the normal form.  More generally, we use this method whenever the source of has only singular multiplications and identities above it. If there are also singular comultiplications, we apply step (IV) below. Therefore, we have the following.(a) The decomposition of Σ is equivalent to one in which the source of the singular multiplication appears in one of the following situations:(85)  or that depicted in (81). 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)(ii)(iii)(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: (86)(i) We first show that the source of every cozipper can be put so that it appears in one of the following situations: (87)  This claim is proved by applying the steps (I), (IId)(ii), and the diffeomorphism: (88)  whenever it is possible.(ii) The singular genus-one operator can be removed by iterating the step (III)(a)(iii), the diffeomorphism given in (88) and the following diffeomorphism:(89)  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 (III)(b)(ii) we know that the remaining situations to consider are (90)  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 (III)(b) 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, (91)  Iteratively applying the diffeomorphism and considering again the next two possible situations in the decomposition of “?," we see that after all, there are the following two possible cases: (92)(a) In the first case, the singular comultiplication is eliminated by applying the following sequence of diffeomorphisms:(93)(b) To remove the singular comultiplication in the second case above, we apply the following sequence of diffeomorphisms:(94)(c) We reapply steps (II) and (III) if needed.(d) We iterate steps (IV)(a)-(b) and (IV)(c) 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:(95)  and its target in one of the following situations: (96)(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 or that its target can be put in the situation , thus eliminated completely by defining .  The following situations: (97)

  are excluded by steps (I), (IV), and (IIIb)(ii), respectively. It remains to consider the cases: (98)  The second case can be reduced to the first one by applying the following sequence of diffeomorphisms: (99)  Then, by taking into account the second part of step (V)(a), we need to consider either .(a) For the first possibility we apply the diffeomorphism which reduces the height of the zipper.(b)   For the second possibility we employ a sequence of diffeomorphisms that removes the zipper:(100)  By repeating these steps if necessary and applying , we guarantee that the zipper has been eliminated from the decomposition of Σ.(VII) The (resulting) decomposition of Σ is equivalent to one in which the ordinary multiplication has its source in one of the following situations: (101)(a) We can exclude the cases (102)  since we assume that the source of Σ is a free union of biwebs.  To prove the claim we iterate the following diffeomorphisms whenever possible:(b)(c)(d)(e)(f)  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: (103)(a) Employing step (VII), we can exclude the cases: (104)  Moreover, every zipper has been eliminated at step (VI), thus we can also exclude (105)  The claim follows by iterating whenever possible the following diffeomorphisms:(b)(c)(d)(e)  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) The cobordism with a singular circle can be put in an equivalent decomposition of Σ, so that it has above it one of the following cobordisms: (106)  The case in which has above it the ordinary comultiplication is excluded by step (VIII)(d), while the case in which it has above it is eliminated by iterating the diffeomorphism (107)(X) We claim that the resulting decomposition of Σ is now in the normal form. This follows from steps (V)(a), (VI), (VII), (VIII), and (IX), 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: (108)  The other situations are excluded by steps (VI) and (VII)(b). (b) Whenever an ordinary cup appears in the resulting decomposition of Σ, then the source of is in one of the following situations: (109)  The other situations are excluded by step (VIII)(c). This completes the proof.

Corollary 26. Let [Σ]∈𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛(𝐧,𝐦) be connected. Then [Σ]=[NF(Σ)].

Proof. [NF(Σ)]=𝑓⁻¹NF_𝑊→𝐶𝑓Σ=𝑓([])₋₁𝑓([Σ])=[Σ]. (110)

Corollary 27. If [Σ],[Σ^′]∈𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛(𝐧,𝐦) are connected cobordisms with the same singular boundary permutation, genus, and singular number, then [Σ]=[Σ^′].

Proof. This follows at once from the fact that the normal form of a singular cobordism is characterized by the singular boundary permutation, genus, and singular number of the cobordism.

Putting together the results of this section, we obtain the following.

Theorem 28. The symmetric monoidal category 𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛 is generated (under composition and disjoint union) by the following singular 2-cobordisms: (111) with relations given in Proposition 11.

4. Twin TQFTs

In this section we define the notion of twin TQFTs and show that the category of twin TQFTs is equivalent to the category of twin Frobenius algebras.

Definition 29. Let 𝒞 be a symmetric monoidal category. A twin Topological Quantum Field Theory (TQFT) in 𝒞 is a symmetric monoidal functor 𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛→𝒞. A homomorphism of twin TQFTs is a monoidal natural transformation of such functors. One denotes by T-TQFT(𝐶) the category of twin TQFTs in 𝒞.

Theorem 30. The category 𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛 is equivalent as a symmetric monoidal category to the category 𝐓𝐡(𝐓-𝐅𝐫𝐨𝐛).

Proof. We need to construct a functor Λ∶𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛→𝐓𝐡(𝐓-𝐅𝐫𝐨𝐛). In general, a monoidal functor is completely determined, up to equivalence, by its values on the generators of the source category. On the generating objects of Sing-2Cob, Λ is defined as follows: (112) Given a general object 𝐧=(𝑛₁,𝑛₂,𝑛₃,…,𝑛_𝑘) in Sing-2Cob, the functor Λ associates the tensor product in Th(T-Frob) of copies of 𝐶 and 𝑊, with all parenthesis to the left. That is, Λ(𝐧)=(((Λ(𝑛₁)⊗Λ(𝑛₂))⊗Λ(𝑛₃))⋯Λ(𝑛_𝑘)) with 𝑛_𝑖∈{0,1} and Λ(0)∶=𝐶 and Λ(1)∶=𝑊.
On the generating morphisms in Sing-2Cob, Λ is defined as explained in Figure 3.
There is an obvious way to extend inductively Λ to a map defined on all morphisms of Sing-2Cob. From the coherence theorems for symmetric monoidal categories, it follows that this assignment is well defined and extends to all general morphisms in Sing-2Cob. Moreover, the relations given in Proposition 11 and the proof that these are all the required relations in Sing-2Cob imply that the image of Λ is a twin Frobenius algebra in Th(T-Frob) and, in particular, that Λ defines a functor 𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛→𝐓𝐡(𝐓-𝐅𝐫𝐨𝐛).
Given 𝐧=(𝑛₁,𝑛₂,𝑛₃,…,𝑛_𝑘), 𝐦=(𝑚₁,𝑚₂,𝑚₃,…,𝑚_𝑙)∈𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛 we construct a natural isomorphism Λ₂∶Λ(𝐧)⊗Λ(𝐦)→Λ(𝐧∐𝐦) as follows: Λ(𝐧)=Λ𝑛₁⊗Λ𝑛₂⊗Λ𝑛₃⋯Λ𝑛_𝑘,Λ(𝐦)=Λ𝑚₁⊗Λ𝑚₂⊗Λ𝑚₃⋯Λ𝑚_𝑙,Λ𝐧𝐦=Λ𝑛₁⊗Λ𝑛₂⊗Λ𝑛₃⋯Λ𝑛_𝑘⊗Λ𝑚₁⊗Λ𝑚₂⋯Λ𝑚_𝑙. (113) Define Λ₀∶=1_𝟏. It can be easily verified that the triple (Λ,Λ₂,Λ₀) defines a symmetric monoidal functor.
On the other hand, by reversing the arrows in the assignments of Λ on the generating objects and morphisms in Sing-2Cob, we see that the singular cobordisms define a twin Frobenius algebra structure on the oriented circle and bi-web (thus we obtain the relations provided in Proposition 11). Therefore, from the results given in Section 2.2, we obtain a strict symmetric monoidal functor Λ∶𝐓𝐡(𝐓-𝐅𝐫𝐨𝐛)→𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛. If two objects in 𝐓𝐡(𝐓-𝐅𝐫𝐨𝐛) are related by a sequence of associators and unit constraints, then they are mapped to the same object in Sing-2Cob.
Given a general object 𝐧∈𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛 we have that ΛΛ(𝐧)=𝐧. Thus ΛΛ=1_{𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛}. If 𝑋 is an object of Th(T-Frob), then 𝑋 is a parenthesized word made up of symbols 𝟏,𝐶,𝑊, and ⊗. It is not hard to see that ΛΛ(𝑋) is isomorphic to 𝑋 by a sequence of associators and unit constraints. In conclusion, Λ and Λ define an equivalence of categories.

Corollary 31. The category 𝐓-𝐅𝐫𝐨𝐛(𝒞) of twin Frobenius algebras in 𝒞 is equivalent, as a symmetric monoidal category, to the category 𝐓-𝐓𝐐𝐅𝐓(𝒞) of twin TQFTs in 𝒞.

5. Examples of Twin Frobenius Algebras

Example 1. Let 𝑖 be the primitive fourth root of unity and let 𝑅=ℤ[𝑖][𝑎,ℎ] be the ring of polynomials in indeterminates 𝑎 and ℎ and with Gaussian integer coefficients. Consider also the ring 𝒜=𝑅[𝑋]/(𝑋²−ℎ𝑋−𝑎)=⟨1,𝑋⟩_𝑅 with inclusion map 𝜄∶𝑅→𝒜,𝜄(1)=1. We remark that we consider these two rings for our first example, because they play an important role in [6, 9, 15].

It is well known (see [17, Section 4]) that 𝜄 is a Frobenius extension if and only if there exists an 𝒜-bimodule map Δ∶𝒜→𝒜⊗_𝑅𝒜 and an 𝑅-module map 𝜖∶𝒜→𝑅 such that Δ is coassociative and cocommutative, and (𝜖⊗id)∘Δ=id.

A Frobenius system is a Frobenius extension together with a choice of the maps 𝜖 and Δ. We denote a Frobenius system by ℱ=(𝑅,𝒜,𝜖,Δ) (following [10, 17]).

We consider two Frobenius systems ℱ_𝐶=(𝑅,𝒜,𝜖_𝐶,Δ_𝐶) and ℱ_𝑊=(𝑅,𝒜,𝜖_𝑊,Δ_𝑊), with 𝜖_𝐶(1)𝜖=0,_𝐶(𝑋)𝜖=1,_𝑊(1)𝜖=0,_𝑊(𝑋)Δ=−𝑖,_𝐶(1)Δ=1⊗𝑋+𝑋⊗1−ℎ1⊗1,_𝐶(𝑋)Δ=𝑋⊗𝑋+𝑎1⊗1,_𝑊(1)(),Δ=𝑖1⊗𝑋+𝑋⊗1−ℎ1⊗1_𝑊(𝑋)=𝑖(𝑋⊗𝑋+𝑎1⊗1). (114)ℱ_𝑊 is a twisting of ℱ_𝐶; that is, the comultiplication Δ_𝑊 and counit 𝜖_𝑊 are obtained from Δ_𝐶 and 𝜖_𝐶 by “twisting” them with invertible element −𝑖∈𝒜: 𝜖_𝑊(𝑥)=𝜖_𝐶(),Δ−𝑖𝑥_𝑊(𝑥)=Δ_𝐶(−𝑖)⁻¹𝑥=Δ_𝐶(𝑖𝑥),∀𝑥∈𝒜. (115) The fact that the above Frobenius systems 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 [17, Theorem 1.6]).

We obtain two commutative Frobenius structures on 𝒜: 𝒜_𝐶=𝒜,𝑚_𝐶,𝜄_𝐶,Δ_𝐶,𝜖_𝐶,𝒜_𝑊=𝒜,𝑚_𝑊,𝜄_𝑊,Δ_𝑊,𝜖_𝑊, (116) where 𝜄_𝐶=𝜄_𝑊=𝜄. Multiplication maps 𝑚_𝐶∶𝒜⊗𝒜→𝒜 and 𝑚_𝑊∶𝒜⊗𝒜→𝒜 are defined by the same rules: 𝑚_𝐶(1⊗𝑋)=𝑚_𝐶(𝑋⊗1)=𝑚_𝑊(1⊗𝑋)=𝑚_𝑊()𝑚𝑋⊗1=𝑋,_𝐶(1⊗1)=𝑚_𝑊()𝑚1⊗1=1,_𝐶(𝑋⊗𝑋)=𝑚_𝑊(𝑋⊗𝑋)=ℎ𝑋+𝑎. (117) We define the following homomorphisms: 𝑧∶𝒜_𝐶⟶𝒜_𝑊,𝑧(1)𝑧(𝑋)𝑧=1,=𝑋,_∗∶𝒜_𝑊⟶𝒜_𝐶,𝑧^∗(1)𝑧=−𝑖,_∗(𝑋)=−𝑖𝑋. (118) A straightforward computation shows that (𝒜_𝐶,𝒜_𝑊,𝑧,𝑧^∗) is “almost” twin Frobenius, in the sense that all properties of a twin Frobenius algebra are satisfied except for the “genus-one condition" which holds up to a minus sign.

Let R-Mod be the category of 𝑅-modules and module homomorphisms. We denote by 𝒯∶𝐒𝐢𝐧𝐠-𝟐𝐂𝐨𝐛→𝐑-𝐌𝐨𝐝 the TQFT corresponding to (𝒜_𝐶,𝒜_𝑊,𝑧,𝑧^∗) (c.f. Section 4), which assigns the ground ring 𝑅 to the empty 1-manifold and assigns 𝒜^⊗𝑘 to a generic object 𝐧=(𝑛₁,𝑛₂,…,𝑛_𝑘) in Sing-2Cob with |𝐧|=𝑘. The 𝑗th factor of 𝒜^⊗𝑘 is endowed with the structure 𝒜_𝐶 if 𝑛_𝑗=0=, or with the structure 𝒜_𝑊 if 𝑛_𝑗=1=.

On the generating morphisms of the category Sing-2Cob, the functor 𝒯 is defined as follows: (119)  It is worth noting that the TQFT defined in this example satisfies the local relations for the “dot free” version of the universal 𝔰𝔩(2) foam cohomology for links (see [15, Section 4]). To be precise, the following identities hold: (120)The last two identities are the “UFO” local relations used in [15] and depicted below: (121) Notice that we also have 𝑧∘𝑧^∗=−𝑖id() and 𝑧^∗∘𝑧=−𝑖id(), which are equivalent to (122) The latter are versions of the “curtain identities” used in [6, 9, 15].

Even though we have here only an “almost” twin Frobenius algebra, motivated by the above remarks, we believe that this example will play the key role in describing the universal 𝔰𝔩(2) foam link cohomology using twin TQFTs. We will consider this problem in a subsequent paper.

Example 2. Let 𝑘 be a field. Consider the commutative Frobenius algebra structure (𝐶,𝑚_𝐶,𝜄_𝐶,Δ_𝐶,𝜖_𝐶) on the truncated polynomial algebra 𝐶=𝑘[𝑥]/(𝑥²), where 𝑚_𝐶()𝑚1⊗1=1,_𝐶(1⊗𝑥)=𝑚_𝐶()𝑚𝑥⊗1=𝑥,_𝐶()Δ𝑥⊗𝑥=0,_𝐶(1)Δ=1⊗𝑥+𝑥⊗1,_𝐶(𝑥)𝜖=𝑥⊗𝑥,_𝐶(1)=0,𝜖_𝐶(𝑥)=1. (123) Consider also the truncated polynomial algebra 𝑊=𝑘[𝑦]/(𝑦^𝑛),𝑛≥2, which admits a commutative and therefore symmetric Frobenius algebra structure (𝑊,𝑚_𝑊,𝜄_𝑊,Δ_𝑊,𝜖_𝑊) with 𝜖_𝑊(𝑦^𝑛−1)=1 and 𝜖_𝑊(𝑦^𝑎)=0 for all 0≤𝑎≤𝑛−2. The rules for multiplication 𝑚_𝑊 are obvious from the definition of 𝑊. The comultiplication Δ_𝑊∶𝑊→𝑊⊗_𝑘𝑊 is dual to the multiplication via the counit map 𝜖_𝑊, and it is defined by Δ_𝑊𝑦^𝑎=^{𝑛−1−𝑎}_𝑖=0𝑦^𝑖+𝑎⊗𝑦^{𝑛−1−𝑖}∀0≤𝑎≤𝑛−1. (124) If char 𝑘=𝑛, then (𝐶,𝑊,𝑧,𝑧^∗) forms a twin Frobenius algebra with 𝑧(1)=1, 𝑧(𝑥)=0 and 𝑧^∗(𝑦^𝑛−1)=𝑥,𝑧^∗(𝑦^𝑎)=0, for all 0≤𝑎≤𝑛−2.

For a field 𝑘 with char 𝑘≠𝑛, the above example does not satisfy the genus-one condition, since 𝑧∘𝑚_𝐶∘Δ_𝐶∘𝑧^∗(𝑦^𝑎)=0 for all 0≤𝑎≤𝑛−1, while 𝑚_𝑊∘𝜏_𝑊,𝑊∘Δ_𝑊(1)=𝑛𝑦^𝑛−1 and 𝑚_𝑊∘𝜏_𝑊,𝑊∘Δ_𝑊(𝑦^𝑎)=0, for all 1≤𝑎≤𝑛−1.

Example 3. Consider the polynomial ring 𝑅=ℤ[𝑎,ℎ], and the truncated polynomial algebras 𝐶=𝑅[𝑥]/(𝑥²−ℎ𝑥−𝑎) and 𝑊=𝑅[𝑦]/(𝑦²−ℎ𝑦−𝑎). Both 𝐶 and 𝑊 are commutative Frobenius with structure maps Δ_𝐶(1)Δ=1⊗𝑥+𝑥⊗1−ℎ1⊗1,_𝐶(𝑥)𝜖=𝑥⊗𝑥+𝑎1⊗1,_𝐶(1)=0,𝜖_𝐶(𝑥)Δ=1,_𝑊(1)Δ=1⊗𝑦+𝑦⊗1−ℎ1⊗1,_𝑊𝑦𝜖=𝑦⊗𝑦+𝑎1⊗1,_𝑊(1)=0,𝜖_𝑊𝑦=1. (125) Then (𝐶,𝑊,𝑧,𝑧^∗) is twin Frobenius, with 𝑧(1)=1, 𝑧(𝑥)=𝑦 and 𝑧^∗(1)=1,𝑧^∗(𝑦)=𝑥.

Example 4. Let 𝑅 be a commutative ring, and let 𝛼∈𝑅 such that 𝛼²=1. Consider the algebras 𝐶=𝑅[𝑥]/(𝑥²−ℎ𝑥−𝑎)and𝑊=𝑅[𝑦]/(𝑦²−ℎ𝑦−𝑎), where 𝑎,ℎ∈𝑅, with Frobenius algebra structures given by Δ_𝐶(1)Δ=1⊗𝑥+𝑥⊗1−ℎ1⊗1,_𝐶(𝑥)𝜖=𝑥⊗𝑥+𝑎1⊗1,_𝐶(1)=0,𝜖_𝐶(𝑥)Δ=1,_𝑊(1),Δ=𝛼1⊗𝑦+𝑦⊗1−ℎ1⊗1_𝑊𝑦,𝜖=𝛼𝑦⊗𝑦+𝑎1⊗1_𝑊(1)=0,𝜖_𝑊𝑦=𝛼. (126) We have that (𝐶,𝑊,𝑧,𝑧^∗) is twin Frobenius, with 𝑧(1)=1,𝑧(𝑥)=𝑦 and 𝑧^∗(1)=𝛼,𝑧^∗(𝑦)=𝛼𝑥.

Other Examples. Examples of knowledgeable Frobenius algebras given by Lauda and Pfeiffer in [18, Examples 3.6–3.8] are also twin Frobenius algebras, with the same restrictions on char 𝑘. Example 3.13 from [18] is a twin Frobenius algebra as well, but we need no restriction on char 𝑘; actually, this is a particular case of our Example 3 over an arbitrary field 𝑘 and for ℎ=1.

In all of the above examples, the algebra 𝑊 is commutative. We provide now a couple of examples where 𝑊 is non commutative, but symmetric (we remark that these are particular cases of the algebras given in Examples 3.1 and  3.2 in [18]).

Example 5. Let 𝑘 be a field, 𝑛∈ℕ, and let 𝑊 be the matrix algebra of 𝑛×𝑛 matrices over 𝑘; that is 𝑊=𝑀_𝑛(𝑘). Denote by {𝑒_𝑖𝑗} the standard basis for 𝑊. There is a symmetric Frobenius algebra structure (𝑊,𝑚_𝑊,𝜄_𝑊,Δ_𝐶,𝜖_𝑊), with Δ_𝑊𝑒_𝑖𝑗=𝛼^𝑛_𝑘=1𝑒_𝑖𝑘⊗𝑒_𝑘𝑗,𝜖_𝑊𝑒_𝑖𝑗=𝛼⁻¹𝛿_𝑖𝑗,𝑚_𝑊𝑒_𝑖𝑗⊗𝑒_𝑘𝑙=𝛿_𝑗𝑘𝑒_𝑖𝑙,𝜄_𝑊(1)=^𝑛_𝑖=1𝑒_𝑖𝑖, (127) where 𝛼=±1∈𝑘.

Consider the (trivial) Frobenius algebra structure (𝐶,𝑚_𝐶,𝜄_𝐶,Δ_𝐶,𝜖_𝐶) on 𝐶∶=𝑘, with Δ_𝐶(1)=1⊗1, 𝜖_𝐶(1)=1, 𝑚_𝐶(1⊗1)=1, and 𝜄(1)=1. Then (𝐶,𝑊,𝑧,𝑧^∗) is a twin Frobenius algebra, with 𝑧(1)=∑^𝑛_𝑖=1𝑒_𝑖𝑖 and 𝑧^∗(𝑒_𝑖𝑗)=𝛼𝛿_𝑖𝑗.

Example 6. Let 𝑘 be a field containing 2. Consider the free associative unital 𝑘-algebra 𝑊∶=ℍ_𝑘 of quaternions, generated by 𝐼,𝐽,𝐾 subject to the relations 𝐼²=𝐽²=𝐾²=−1,𝐼𝐽=−𝐽𝐼=𝐾,𝐽𝐾=−𝐾𝐽=𝐼,𝐾𝐼=−𝐼𝐾=𝐽. (128) Then (𝑊,𝑚_𝑊,𝜄_𝑊,Δ_𝐶,𝜖_𝑊) is symmetric Frobenius, with Δ_𝑊(1)(),Δ=𝛼1⊗1−𝐼⊗𝐼−𝐽⊗𝐽−𝐾⊗𝐾_𝑊(𝐼)(),Δ=𝛼1⊗𝐼+𝐼⊗1+𝐽⊗𝐾−𝐾⊗𝐽_𝑊(𝐽)(),Δ=𝛼1⊗𝐽+𝐽⊗1+𝐾⊗𝐼−𝐼⊗𝐾_𝑊(𝐾)(),𝜖=𝛼1⊗𝐾+𝐾⊗1+𝐼⊗𝐽−𝐽⊗𝐼_𝑊(1)=𝛼⁻¹,𝜖_𝑊(𝐼)=𝜖_𝑊(𝐽)=𝜖_𝑊(𝐾)=0, (129) where 𝛼=±1/2.

There is a twin Frobenius algebra (𝐶,𝑊,𝑧,𝑧^∗), where 𝐶 is the underlying field 𝑘 equipped with the trivial Frobenius structure (as in the previous example), and where 𝑧(1)=1,𝑧^∗(1)𝑧=4𝛼,_∗(𝐼)=𝑧^∗(𝐽)=𝑧^∗(𝐾)=0. (130)

Remark 32. Considering any of these (last two) examples (𝐶₁,𝑊₁,𝑧₁,𝑧^∗₁), where 𝐶₁=𝑘 with the trivial Frobenius algebra structure, we can tensor it with one of the twin Frobenius algebras (𝐶₂,𝑊₂,𝑧₂,𝑧^∗₂) given in the first set of examples (with 𝐶₂ commutative). We obtain a twin Frobenius algebra (𝐶₂,𝑊₁⊗𝑊₂,𝑧₁⊗𝑧₂,𝑧^∗₁⊗𝑧^∗₂) with a bigger and possible noncommutative 𝑊₁⊗𝑊₂.

Acknowledgments

The author is grateful to the NSF for supporting her through Grant DMS-0906401. She would also like to thank Aaron Lauda for his helpful discussions that motivated her to return to this project.