Two-Dimensional Extended Homotopy Field Theories

We define $2$-dimensional extended homotopy field theories (E-HFTs) with aspherical targets and classify them. When target is a $K(G,1)$-space, oriented E-HFTs taking values in the symmetric monoidal bicategory of algebras, bimodules, and bimodule maps are classified by certain Frobenius $G$-algebras called quasi-biangular $G$-algebras. As an application, for any discrete group $G$ we verify a special case of the $(G \times SO(2))$-structured cobordism hypothesis due to Lurie.


Introduction
Extended topological field theories (E-TFTs) are generalizations of topological field theories (usually called TQFTs or TFTs) to manifolds with corners and higher categories ( [Fr], [La2], [BD]). A different generalization of TFTs is obtained by considering manifolds equipped with principal Gbundles. When G is a discrete group, such a generalization was introduced by V. Turaev [Tu2] who called them homotopy (quantum) field theories (HFTs). These theories are defined by applying axioms of TFTs to manifolds and cobordisms endowed with maps to a fixed target space. In this paper, we combine E-TFTs and HFTs in dimension 2. More precisely, we define 2-dimensional extended homotopy field theories (E-HFTs) with aspherical targets and classify them.
To define a 2-dimensional E-HFT with target X K(G, 1) we introduce a G-equivariant cobordism bicategory XBord 2 . The objects of XBord 2 are compact oriented 0-dimensional manifolds and the 1-morphisms are oriented cobordisms between such manifolds equipped with homotopy classes of maps to X. The 2-morphisms of XBord 2 are certain oriented surfaces with corners equipped with homotopy classes of maps to X and they are considered up to a diffeomorphism whose restriction to boundary is identity. The disjoint union operation turns XBord 2 into a symmetric monoidal bicategory and an oriented 2-dimensional E-HFT with target X is defined as a symmetric monoidal 2-functor from XBord 2 to any other symmetric monoidal bicategory.
For a commutative ring k, the symmetric monoidal bicategory Alg 2 k has k-algebras as objects, bimodules as 1-morphisms, and bimodule maps as 2-morphisms. Below we state a classification of Alg 2 k -valued oriented 2-dimensional E-HFTs. The following notions are the main ingredients of our result. For a discrete group G with identity element e, a strongly graded G-algebra is a G-graded associative k-algebra A = ⊕ g∈G A g with unity such that A g A g = A gg for all g, g ∈ G. The opposite G-algebra of A is A op = ⊕ g∈G A g −1 where the order of multiplication is reversed.
A Frobenius G-algebra is a pair (A, η) where A = ⊕ g∈G A g is a G-algebra such that each A g is a finitely generated projective k-module and η : A⊗A → k is a nondegenerate bilinear form satisfying η(ab, c) = η(a, bc) for any a, b, c ∈ A. A quasi-biangular G-algebra is a strongly graded Frobenius G-algebra (A, η) whose identity component A e is separable and η satisfies certain conditions (see Section 3.3). We also need G-graded Morita contexts between G-algebras which were introduced by P. Boisen [Bo]. We recall their definition and introduce a notion of compatibility with Frobenius structures in Section 3.3.
Theorem 3.5. Let k be a commutative ring and X be a pointed CW-complex which is an Eilenberg-MacLane space K(G, 1) for a group G. Then, any Alg 2 k -valued oriented 2-dimensional E-HFT with target X determines a triple (A, B, ζ) where A and B are quasi-biangular G-algebras and ζ is a compatible G-graded Morita context between A and B op . Moreover, any such triple (A, B, ζ) is realized by an oriented 2-dimensional E-HFT.
Theorem 3.5 generalizes Schommer-Pries' classification of Alg 2 k -valued oriented 2-dimensional E-TFTs ( [Sc]) which corresponds to the case G = {e}. Next, we upgrade Theorem 3.5 to an equivalence of two bicategories. The first bicategory SymMon(XBord 2 , Alg 2 k ) is the bicategory of symmetric monoidal 2-functors, transformations, and modifications (see [Sc]). We denote this bicategory by E-HFT(X, Alg 2 k ). Secondly, Frob G is the bicategory of quasi-biangular G-algebras, compatible G-graded Morita contexts, and equivalences of such Morita contexts (see Section 3.3). Finally, we define a forgetting 2-functor F : E-HFT(X, Alg 2 k ) → Frob G which assigns the quasi-biangular G-algebra A to each oriented E-HFT with target X K(G, 1) determining a triple of the form (A, B, ζ). On 1-and 2-morphisms the functor F similarly forgets the data coming from the last two components of the triples. Now we state our main theorem.
Definition 2.1. A linear G-data ξ is a pair (ξ 1 , ξ 2 ) where ξ 1 consists of finitely many nested finite subsets of [0, 1] and ξ 2 is a finite set of oriented open intervals contained in [0,1] such that each interval is labeled with an element from G. For each interval x ∈ ξ 2 we denote the left endpoint of its closure x by ∂ − x and the right endpoint of x by ∂ + x. We denote the largest set in ξ 1 by ξ 1 .
For a 1-dimensional compact X-manifold (M, T, g) equipped with a Morse function to [0, 1], ξ 1 is formed by the images of elements in T and ξ 2 is formed by the images of submanifolds of M obtained by removing critical points. Nested intervals are used to distinguish elements of T mapping to the same point.
Definition 2.2. A 1-dimensional graphic µ ( [Sc]) is a finite subset of (0, 1) where each point is labeled with either cup or cap. A linear G-data ξ = (ξ 1 , ξ 2 ) is said to be compatible with a 1-dimensional graphic µ if the following conditions hold: (i) The intersection µ ∩ ξ 1 is empty. (ii) For every a ∈ µ there exist two intervals x 1 , x 2 ∈ ξ 2 having the same G-labels but different orientations such that the intersection x 1 ∩ x 2 is nonempty and a ∈ ∂ + x i for i = 1, 2 if a is labeled with cap and a ∈ ∂ − x i for i = 1, 2 otherwise.
Critical values of f with their labels (see Figure 1) form µ while black points form ξ 1 = f (T) and ξ 1 = {ξ 1 } since f | T is an injection. Removing critical points divides M into five connected components whose images under f with their labels form ξ 2 . An open cover U = {U i } 4 i=1 of [0, 1] is a Ψ-compatible open cover and turquoise points form a chambering set subordinate to U.
is nonempty and identification with empty set otherwise. In this case, each {i} × V is called a sheet.
Trivializations of two neighboring chambers have the same number of sheets if chambers are separated by a point in Γ. If a point in µ separates chambers then by the Morse lemma (see [Mi]) the number of sheets differ by two (see Figure 1). A sheet data S for a pair (Ψ G , Γ) consists of a trivialization of each chamber and an injection or a permutation between trivializations of neighboring chambers describing how sheets are glued.
A linear G-data ξ supplements sheet data by encoding the characteristic map of X-manifold as follows. Each element of each set in ξ 1 is lifted to a sheet. Similarly, sheets are directed and labeled with elements of G according to those intervals in ξ 2 intersecting with a chamber. A G-sheet data S G associated to a pair (Ψ G , Γ) is a sheet data S with these additional assignments to sheets. We also require injections and permutations to preserve G-labelings. Note that a G-sheet data associated to a pair (Ψ G , Γ) produces a 1-dimensional compact X-manifold equipped with generic map to [0, 1].

Remark.
Oriented G-linear diagrams can be defined by assigning orientations to sheets and requiring permutations and injections to preserve these orientations. In this way, we obtain the oriented version of Proposition 2.1.
2.3. G-planar diagrams. C. Schommer-Pries [Sc] generalized the Morse theory of surfaces to a 2-dimensional Morse theory by stratifying jet spaces {J k (Σ, R 2 )} k≥0 for a surface Σ and introduced planar diagrams. We define G-planar diagrams by adding planar G-data so that each G-planar diagram produces an X-cobordism between 1-dimensional compact X-manifolds.
In this section and the following section we consider generic maps for certain stratifications of jet spaces. Let M and N be smooth manifolds and let jet spaces {J k (M, N)} k∈I be equipped with a stratification for k ∈ I ⊆ N. By a generic map we mean a smooth map f : M → N whose jet sections { j k f : M → J k (M, N)} k∈I are transverse to each stratum. In the following, for a closed surface Σ we call x ∈ Σ a singularity of f and f (x) ∈ R 2 its graphic if j 2 f (x) lies in codimension one or two stratum.
A generic map for Schommer-Pries' multijet stratification ( [Sc]) can have fold, Morse (cup, cap and saddle's) and cusp singularities. By the multijet transvesality theorem ( [GG]) generic maps are dense in C ∞ (Σ, R 2 ). Figure 3 shows some of the singularities of generic maps and their graphics in normal coordinates. Observe that singularities and graphics have symmetries such as changing the folding direction or reflecting the cup graphic. Each such symmetry is called an index. We use numbers to indicate different indices of fold and cusp singularities.

Cap
Cup Saddle-1 Saddle-2 Cusp-2 Cusp-1 Fold-2 Graphic of any generic map has the following properties (see Section 1.4 in [Sc]). Projection of each fold graphic to the last coordinate is a local diffeomorphism. Intersections of fold graphics are transversal and at most two fold graphics intersect at a point. Lastly, fold graphics do not intersect with Morse and cusp graphics. Now we describe how to add the data of the homotopy class of an X-cobordism on a graphic of a generic map.
Let Σ be a closed connected surface and let (Σ, P) be an X-cobordism. We know that [Σ, X] = Hom(π 1 (Σ), G)/G where G acts by conjugation. We fix a point σ ∈ Σ and finitely many G-labeled σ-based loops which are representatives of π 1 (Σ, σ)-generators and their G-labels are given by a representative P ∈ [(Σ, σ), (X, x)] of P. In other words, we choose an X-surface representative ((Σ, σ), P) of the X-cobordism (Σ, P). By choosing such a representative and considering images of points and loops under a generic map we have the following definition.
Definition 2.5. A planar G-data ξ is a pair (ξ 1 , ξ 2 ) where ξ 1 = {σ i } N i=1 consists of a finite subset of R 2 and ξ 2 = N i=1 {α consists of a finite number of immersed labeled oriented loops in R 2 such that for a fixed i, each element of {α j=1 is based at σ i , loops are transverse to each other, and each loop α g i,j i, j is labeled by g i,j ∈ G.
Definition 2.6. A 2-dimensional graphic ( [Sc]) Φ = (η, µ) is a diagram in R 2 consisting of a finite number of embedded labeled curves (η) and a finite number of labeled points (µ) satisfying the following conditions: (i) Elements of η can only have transversal intersections and no three or more elements intersect at a point. Each element of η is labeled with either Fold-1 or Fold-2. (ii) Projections of elements of η to the last coordinate of R 2 are local diffeomorphisms. (iii) Elements of µ are isolated and each element is labeled with one of the Cup, Cap, Saddle-1, Saddle-2, Cusp-i for i = 1, 2, 3, 4. (iv) Each element in µ has a neighborhood in which two elements of η form one of the Cup, Cap, Saddle-1, Saddle-2, Cusp-i graphic for i = 1, 2, 3, 4.
A planar G-data ξ = (ξ 1 , ξ 2 ) is said to be compatible with a 2-dimensional graphic Φ = (η, µ) if elements of ξ 2 are disjoint from µ and Proposition 2.2. Let Σ be a closed surface and f : Σ → R 2 be a generic map. Then for any X-cobordism (Σ, P) there exist a point σ i and σ i -based loops on each connected component of Σ representing generators of π 1 (Σ, σ i ) such that the graphic of f and the images of these loops under f form a 2-dimensional G-graphic.
Proof. Assume that Σ is connected. By the properties of the stratification, the graphic of f gives η and µ forming a 2-dimensional graphic Φ = (η, µ). First pick a point σ such that f (σ) is away from Morse and cusp graphics. Then pick σ-based loops which are in generic position representing π 1 (Σ, σ)-generators and consider their images under f . Since f is a generic map 1 each nontransversal intersection can be changed into a transversal one by modifying loops in their homotopy classes. Representative P of the homotopy class P determines the G-labelings of loops giving ξ 2 . If Σ is not connected, apply this process on each connected component.  Figure 4 shows an example of a 2-dimensional G-graphic induced from an X-torus (T 2 , P) equipped with a projection to the page map. In this example, the planar G-data ξ = (ξ 1 , ξ 2 ) is given by the point σ 1 and based loops labeled with α g i 1,1 and α g j 1,2 . The set η consists of four labeled arcs and the set µ consists of four labeled points. 1 In particular, f is a local diffeomorphism on connected open sets containing no singularity.
Loops in a planar G-data cannot be assumed to have transversal intersection with each element of η. Figure 4 shows an example of nontransversal intersection of an element of ξ 2 with a fold graphic.
Remark. Different finite presentations of π 1 (Σ, σ) lead to different G-graphics in R 2 . We restrict ourselves to 2-dimensional G-graphics whose planar G-data come from a fixed finite presentation of π 1 (Σ, σ) where Σ is a closed connected surface. We do not rename this subcollection and continue to say 2-dimensional G-graphics.
Let Φ G = (η, µ, ξ) be a 2-dimensional G-graphic, an open cover U = {U α } α∈J of R 2 with at most triple intersections is said to be Φ-compatible ( [Sc]) if each triple intersection is disjoint from µ and each double intersection is disjoint from η ∪ µ or contains a single element from η. Knowing the fact that R 2 has covering dimension two and sets η and µ are finite it is not hard to find Φ-compatible open covers for a given graphic Φ.
Definition 2.7. Let Φ G = (η, µ, ξ) be a 2-dimensional G-graphic. A chambering graph Γ for Φ G is a smoothly embedded graph in R 2 whose vertices are disjoint from Φ G and have degree either one or three. Edges of Γ are transverse to Φ G . Furthermore, projection of each edge to the last coordinate is a local diffeomorphism and around each trivalent vertex one of the edges projects to the opposite side of the projection of other two edges with respect to the image of the vertex.
Definition 2.8. Let Γ be a chambering graph for Φ G = (η, µ, ξ). Chambers of (Φ G , Γ) are the connected components of R 2 \(Γ ∪ η ∪ µ). A chambering graph Γ is said to be subordinate to an open cover U = {U α } α∈J of R 2 if each chamber is a subset of at least one U α with α ∈ J.   Figure 6 Example 2.3. Figure 5 shows an example of a chambering graph Γ for the 2-dimensional G-graphic given in Figure 4. Each colored region is a chamber.
Then there exists a chambering graph Γ for Φ G subordinate to U.
Proof. The 2-dimensional graphic version of this proposition was proven in [Sc] (see Proposition 1.46). In case of nontransversal intersections with planar G-data, edges and vertices of Γ can be slightly modified to make all intersections transversal while being compatible with U.
Let Φ G = (η, µ, ξ) be a 2-dimensional G-graphic induced from a generic map f on an X-cobordism (Σ, P) equipped with G-labeled based loops. Let Γ be a chambering graph for Φ G subordinate to a Φ-compatible open cover. Since f is generic the preimage f −1 (U β ) of a chamber consists of disjoint union of open sets (possibly empty) each mapping diffeomorphically onto U β . A trivialization of a chamber is the identification of f −1 (U β ) with N ≤N × U β for some N ∈ N if f −1 (U β ) is nonempty and identification with the empty set otherwise. In this case, each {i} × U β is called a sheet.
Trivializations of two neighboring chambers have the same number of sheets if chambers are separated by an edge of Γ. If an element in η separates chambers then the number of sheets differ by two (see Figure 3). A sheet data S for a pair (Φ G , Γ) consists of a trivialization of each chamber and an injection or a permutation (see Figure 6) between trivializations of neighboring chambers describing how sheets are glued (see [Sc] for details).
Gluing description of sheets requires the following conditions on permutations and injections. If three chambers are separated by edges of a trivalent vertex of Γ then the circular composition of permutations must be identity. We describe the sheet data of cusp graphic briefly on an example. Consider the Cusp-2 labeled point in Figure 7. Let N ≤N+3 ×U β 1 and N ≤N+1 ×U β 2 be the trivializations such that sheets N+3 i=N+1 i × U β 1 and (N + 1) × U β 2 belong to cusp singularity as shown in Figure 7. In this case, restriction of injections to cusp singularity gives σ 1 (N + 1) = N + 1 and σ 2 (N + 1) = N + 3.
. Sheet data for cusp graphic A G-sheet data S G is a sheet data S with additional lifts of ξ-elements, i.e. directed G-labeled arcs and points, to sheets. Permutations and injections are also required to preserve sheets with directed labeled arcs (see Figure 6). Definition 2.9. A G-planar diagram is a triple (Φ G , Γ, S G ) consisting of a 2-dimensional G-graphic Φ G , a chambering graph Γ for Φ G subordinate to a Φ-compatible open cover U = {U α } α∈J of R 2 , and a G-sheet data S G associated to the pair (Φ G , Γ).
Any G-planar diagram (Φ G , Γ, S G ) produces an X-cobordism (Σ, P) with a generic map f : Σ → R 2 where P is determined by the G-labeled based loops on Σ. Let (Σ, P) and (Σ , P ) be X-surfaces endowed with generic maps f : Σ → R 2 , f : Σ → R 2 . An X-homeomorphism F : Σ → Σ is said to be over R 2 if it commutes with generic maps, i.e. f • F = f . Proposition 2.4. For a closed surface Σ, let (Σ, P) be an X-cobordism with G-labeled based loops representing π 1 (Σ) and let f : Σ → R 2 be a generic map inducing Φ G . Let Γ be a chambering graph for Φ subordinate to a Φ-compatible open cover giving a G-planar diagram Proof. The diffeomorphism F : Σ → Σ maps inverse images of chambers to corresponding trivializations. Since G-labels of the fixed fundamental group generators coincide and both f and f • F restrict to the same map on f −1 (U β ) for any chamber U β , F is an X-homeomorphism over R 2 .
The definition of a 2-dimensional G-graphic is modified to include 1-dimensional G-graphics on R × {0, 1} and elements of η and ξ 2 are required to have transversal intersection with boundary components. A planar G-data ξ has additional G-labeled arcs between basepoints at boundary components. Edges of a chambering diagram are required to end on R×{0, 1} forming a chambering set subordinate to the induced open cover. All vertices of Γ must lie in the interior of R×I. A G-sheet data has additional trivializations of chambers with boundaries where injections and permutations form a 1-dimensional G-sheet data on the boundary.
To classify 2-dimensional E-HFTs we need to consider cobordisms between 1-dimensional compact X-manifolds. These cobordisms are surfaces with corners, more precisely 2 -surfaces ( [La1]) endowed with characteristic maps. A 2 -surface is a 2-dimensional compact manifold with faces S equipped with two submanifolds with faces ∂ h S and ∂ v S called horizontal and vertical faces respec- , (X, x)] to each connected component is the constant homotopy class (see Figure 8). Saddle-1 Fold-2 Fold-1 Figure 8. An example of a cobordism type 2 -X-surface and a chambering graph for the induced G-graphic Remark. We can glue two cobordism type 2 -X-surfaces along their common 3 horizontal or vertical faces and obtain a new one by forgetting the points lying in the complement of horizontal face.
Let (S, R, P) be a cobordism type 2 -X-surface and let I mn = [m, n] be an interval for m, n ∈ Z. A generic map is a smooth map of the form for the same stratifications as the compact case. Again by the relative Thom transversality theorem ( [GG]) such maps are dense in the space of smooth functions of this form.
In this case a 2-dimensional G-graphic lies in I mn × I and in addition to compact case elements of η are required to be disjoint from ∂I mn × I and transverse to I mn × ∂I. Edges of a chambering graph are additionally required to be disjoint from ∂I mn × I. Since there is no singularity on the vertical boundary, G-sheet data is similar to compact case producing a cobordism type 2 -X-manifold. An example of a 2-dimensional G-graphic under a projection to the page map and a chambering graph for the cobordism type 2 -X-surface is given in Figure 8.

Remark.
Oriented G-planar diagrams can be defined by assigning orientations to sheets and requiring permutations and injections to preserve these orientations. In this way, we obtain the oriented version of Proposition 2.4.
2.4. G-spatial diagrams. C. Schommer-Pries [Sc] introduced spatial diagrams to identify planar diagrams which produce homeomorphic surfaces. We define G-spatial diagrams to identify Gplanar diagrams giving X-homeomorphic X-cobordisms. Using G-spatial diagrams we define an equivalence relation among G-planar diagrams and prove the G-planar decomposition theorem.
Schommer-Pries [Sc] stratified jet spaces {J k ((Σ × I, Σ × ∂I), (R 2 × I, R 2 × ∂I))} k≥0 by applying ideas from Cerf theory to generic maps for his stratification of jet spaces {J k (Σ, R 2 )} k≥0 . By the relative transversality theorem ( [GG]) generic maps are dense in C ∞ ((Σ × I, Σ × ∂I), (R 2 × I, R 2 × ∂I)). In this case generic maps can have eight singularities whose graphics in normal coordinates are shown in Figure 9. Just as in the previous section indices of a singularity are symmetries such that different indices either give the same or symmetric graphics.  Figure 9. Graphics of the singularities in R 2 × I Graphic of any generic map for Schommer-Pries' multijet stratification has the following properties. There are only transversal intersections and at most three fold graphics can intersect at a point. Moreover, when two surfaces intersect along an arc the projection of the arc to the last coordinate is a local diffeomorphism except for finitely many points. Thus, the restriction of a graphic to R 2 × {t} is a 2-dimensional graphic except for finitely many t ∈ (0, 1). Now we describe how two planar G-data coming from different X-surface representatives of an X-cobordism are related.
Proof. The bijection is determined by X-surface representatives. Since F is generic there exists a straight arc on each connected component of Σ × I transverse to the graphic of F. The bijection determines the G-labels of straight arcs.
Let ∆ G = (δ, η, µ, τ) be a 3-dimensional G-graphic. An open cover of R 2 × I with at most 4-fold intersections is said to be ∆-compatible if each 4-fold intersection is disjoint from δ ∪ η ∪ µ, each 3-fold intersection is disjoint from µ ∪ η and contains at most a single component of surfaces in δ and each double intersection is disjoint from points in µ. Since R 2 × I has covering dimension 3 and there are only finitely many elements in δ, η, and µ there exist ∆-compatible open covers. Figure 11. Local models for a chambering foam Definition 2.13. ( [Sc]) Let ∆ G = (δ, η, µ, τ) be a 3-dimensional G-graphic. A chambering foam Γ for ∆ G is a smooth embedding of 2-dimensional locally conical stratified space Γ of compact type (see [Sc]) into R 2 × I with the following properties. Γ is locally conical with respect to the system of local models I 2 , I × C 1 , I × C 3 , CP, and CK 4 shown in Figure 11. Vertices are disjoint from ∆ G . Edges can only intersect with a surface from δ and with an arc from ξ 1 2 and ξ 2 2 . Faces can only intersect with surfaces from δ and arcs from η and ζ. All intersections are transversal and Γ additionally satisfies the following conditions: (I) Projection p : Γ → R × I to the last two coordinates has no singularity and projection of faces to the last coordinate has no singularity. (II) For every t ∈ I satisfying; R 2 × {t} ∩ µ = ∅, t is not a critical value of projection p : Γ → I, and R 2 × {t} ∩ Γ does not include a vertex of Γ, the graph R 2 × {t} ∩ Γ forms a chambering graph for the 2-dimensional graphic ∆ ∩ R 2 × {t}. (III) Projection of each one of four edges in CK 4 -model connecting at the cone point to the last coordinate is a local diffeomorphism. Additionally, at least one of them must map to downward of the cone point and at least one of them must map to upward of the cone point. (IV) Projection of the two edges in CP-model connecting at the cone point to the last coordinate maps both edges to the same direction with respect to the image of cone point.
Definition 2.14. Let ∆ G = (δ, η, µ, τ) be a 3-dimensional G-graphic and let Γ be a chambering foam for ∆ G . Chambers of Γ are the connected components of R 2 × I\(Γ ∪ δ ∪ η ∪ µ). A chambering foam Γ is said to be subordinate to an open cover O = {O α } α∈J of R 2 × I if each chamber is a subset of at least one O α with α ∈ J.
Proof. In [Sc] the corresponding statement for a 3-dimensional graphic was proven (see Corollary 1.47 in [Sc]). In case of nontransversal intersections with spatial G-data elements, Γ can be slightly modified to make all intersections transversal while being compatible with O.
Let ∆ G = (δ, η, µ, τ) be a 3-dimensional G-graphic induced from a generic map F defined on an X-cylinder (Σ × I, P) equipped with based loops on ∂(Σ × I) and straight arcs. Let Γ be a chambering foam subordinate to a ∆-compatible cover. Since F is generic the preimage F −1 (O β ) of a chamber consists of disjoint union of open sets each mapping diffeomorphically onto O β . A trivialization of a chamber is the identification of is nonempty and identification with the empty set otherwise. Each {i} × O β is called a sheet.
Trivializations of two neighboring chambers have the same number of sheets if chambers are separated by a 2-dimensional strata of Γ. If an element in δ separates chambers then the number of sheets differ by two. A sheet data S for a pair (∆, Γ) consists of trivialization of each chamber and an injection or a permutation between trivializations of neighboring chambers describing how sheets are glued (see [Sc] for details).
Gluing description of sheets requires the following conditions on permutations and injections. In the local models I × C 3 , CP, and CK 4 circular compositions of three or four permutations must be identity. Since cusp graphic is a path of cusp graphic in previous section sheet data is the same. According to properties of multijet stratification transversal double and triple fold intersections are possible. There are four chambers for the double and eight for the triple fold intersection. In both cases different compositions of injections starting from the chamber with the least number of sheets and ending at the chamber with the maximum number of sheets must be the same. Figure 12. The Swallowtail-1 sheet data We shortly describe the sheet data of Swallowtail-1 graphic shown in Figure 12 where two (blue and green) out of three fold singularities form a double fold crossing. Let N ≤N × U β 1 , N ≤N+2 × U β 2 , and N ≤N+4 × U β 3 be trivializations of chambers such that sheets N+2 i=N+1 i × U β 2 and N+4 j=N+1 j × U β 3 belong to swallowtail singularity as shown in Figure 12. Using the sheet data for cusp singularities restrictions of injections to these sheets give A G-sheet data S G is a sheet data S with additional lifts of τ-elements i.e. G-labeled arcs and points to sheets. Permutations and injections are also required to preserve sheets with directed labeled arcs.
be G-planar diagrams and let (Σ 1 , P 1 ) and (Σ 2 , P 2 ) be the constructed closed X-cobordisms respectively. Then, Proof. Assume that F : (Σ 1 , P 1 ) → (Σ 2 , P 2 ) is an X-homeomorphism and f i : Σ i → R 2 are generic maps for i = 1, 2. Consider the mapping cylinder M F = (Σ 1 × I Σ 2 )/(x, 1) ∼ F(x). Then there exists a generic map F on X-mapping cylinder (M F , P) which restricts to f 1 and f 2 on boundary components. Choosing straight arcs leads to a 3-dimensional G-graphic ∆ G which restricts to Φ G 1 and Φ G 2 on R 2 × {0, 1}. Lemma 2.1 states that there exists a chambering foam Γ restricting to Γ 1 and Γ 2 . Lastly, the generic map F induces a G-sheet data S G producing a G-spatial diagram (∆ G , Γ, S G ).
on boundary components. Then the boundary components of the constructed manifold are clearly diffeomorphic and an X-homeomorphism is defined using the lifts of the spatial G-data.
We define a relation among G-planar diagrams by restricting to the given G-planar diagrams on its boundary components. It is not hard to see that ∼ is an equivalence relation. Proposition 2.6 implies the following theorem which is the main result of the first part of the paper.
Theorem 2.1 (G-planar decomposition theorem). The X-homeomorphism classes of 2-dimensional closed X-cobordisms are in bijection with the equivalence classes of G-planar diagrams.
The notions and results of this section can be generalized to compact X-cobordisms and cobordism type 2 -X-surfaces. We briefly describe changes for the cobordism type 2 -X-surfaces. Let F : (S 1 , R 1 , P 1 ) → (S 2 , R 2 , P 2 ) be an X-homeomorphism of cobordism type 2 -X-surfaces relative to their boundary. Then we form the mapping cylinder M F = (S 1 × I S 2 )/(x, 1) ∼ F(x) and the relative Thom transversality theorem ( [GG]) implies that generic maps are dense in , and elements of δ, η, and µ are disjoint from ∂I mn × I 2 .
A chambering foam Γ is generalized as follows; restriction of Γ to I mn × I × {0, 1} gives chambering graphs for 2 -X-surfaces, Γ has transversal intersections with I mn × {0, 1} × I, Γ is disjoint from ∂I mn × I 2 , and vertices of Γ is disjoint from I mn × I 2 . A G-sheet data has additional trivializations of chambers with boundary and additional injections and permutations coming from G-sheet data of a cobordism type 2 -X-surface.
These additional modifications and conditions allow us to define G-spatial diagrams for cobordism type 2 -X-surfaces and results in this section extend to such X-surfaces. In particular, there is a bijection between the set of X-homeomorphism classes (relative to boundary) of cobordism type 2 -X-surfaces and equivalence classes of G-planar diagrams for cobordism type 2 -X-surfaces.

Remark.
Oriented G-spatial diagrams can be defined by assigning orientations to sheets and requiring injections and permutations to preserve these orientations. In this way, we obtain a version of G-planar decomposition theorem for oriented surfaces and oriented diagrams.

Extended Homotopy Field Theories
3.1. G-equivariant cobordism bicategories. In this section we define oriented and unoriented G-equivariant cobordism bicategories using halations introduced by Schommer-Pries in [Sc].
Definition 3.1. Let Man X be the category of smooth X-manifolds and smooth pointed maps commuting with characteristic maps and let I 1 , I 2 be small cofiltered categories. Then objects of the category pro-Man X are functors F 1 : I 1 → Man X , F 2 : I 2 → Man X called pro-X-manifolds and morphisms are given by where limit and colimit are taken in sets.
Let (M, g) and (N, h) be X-manifolds possibly with boundary or corners, and let ι : Consider the following cofiltered directed set consisting of codimension zero X-submanifolds of N and let (M,ĝ) = (M ⊂ N,ĥ| M ) be the corresponding pro-X-manifold for this directed set. An Xmanifold (M, g) is a pro-X-manifold by the constant directed set. The natural inclusion map between pro-X-manifolds (M, g) → (M,ĝ) is called an X-halation and denoted by a triple (M,M,ĝ). An Xmanifold with an X-halation is called an X-haloed manifold. A map between X-haloed manifolds (A,Â,ĝ) and (B,B,ĥ) is a commutative square of pro-X-manifold morphisms. Since X-halations are defined for X-manifolds we omit X and use manifold for brevity.
Let (S, R, P) be a 2 -surface and p : (E,P) → (S, P) be a vector bundle withP| s 0 (S) = P where s 0 is the zero section. A choice of a collar neighborhood (see [La1] for existence) and the directed set I S with the embedding ι = s 0 gives an X-halation. Different choices of collars give (noncanonically) isomorphic X-halations and all X-halations are isomorphic to a one having this form (see [Sc]). Codimension of an X-halation is the codimension of embedding.
A codimension one X-halation on ∂ h S and ∂ v S or a codimension two X-halation on ∂ h S ∩ ∂ v S are restrictions of a codimension zero X-halation on S if the corresponding vector bundles are trivial. An isotopy class of ordered nonzero sections trivializing the vector bundle is called a co-orientation. An X-halation is called co-oriented if it is equipped with a co-orientation. Figure 13. Co-oriented X-halations and an X-haloed 1-cobordism Let (M, g) be a compact 0-manifold with a pair of co-oriented X-halations with inclusions (M, g) → (M 1 ,ĝ 1 ) → (M 2 ,ĝ 2 ) where (M,M 1 ,ĝ 1 ) is a codimension one X-halation and (M,M 2 ,ĝ 2 ) is a codimension two X-halation. We denote such a pair of co-oriented X-halations with inclusions by a quadruple (M,M 1 ,M 2 ,ĝ 2 ). Similarly, let (N,N 1 ,N 2 ,ĥ 2 ) be another such quadruple for a compact 0-manifold (N, h). A pointed 1-cobordism between (M, g) and (N, h) is a 1-dimensional compact manifold (A, T, p) with ∂A = M N such that T contains at least two points from each connected component of A. Then an X-haloed 1-cobordism from (M,M 1 ,M 2 ,ĝ 2 ) to (N,N 1 ,N 2 ,ĥ 2 ) is a pointed 1-cobordism (A, T, p) with a codimension zero X-halation (A,Â 0 ,p 0 ) and a co-oriented codimension one X-halation (A,Â 1 ,p 1 ) along with a decomposition of the boundary of (A, T, p) as ∂A = ∂ in A ∂ out A with isomorphisms of X-halations preserving co-orientations (see Figure 13) HereÂ 0 | ∂ in is co-oriented by an inward pointing normal vector andÂ 0 | ∂ out is co-oriented by an outward pointing normal vector. We denote this X-haloed 1-cobordism by quintuple (A,Â 0 ,Â 1 , T,p 1 ).
Composition of 1-morphisms in XBord un 2 is defined as the pushout 7 of X-haloed manifolds. To compose 2-morphisms one needs to choose collar neighborhoods and glue cobordisms. While composing 2-morphisms vertically we require restrictions of characteristic maps to boundaries to match and we forget points lying on the identified horizontal faces (see Definition 2.10).
The proof repeats verbatim the proof for Bord un 2 (see [Sc]) using M. Shulman's method [Sh]. Additionally, we need to use constant characteristic maps to form companions of vertical 1-morphisms. Definition 3.3. Let C be a symmetric monoidal bicategory. A C-valued unoriented 2-dimensional extended homotopy field theory (E-HFT) with target X is a symmetric monoidal 2-functor from XBord un 2 to C.
Remark. The symmetric monoidal G-equivariant oriented cobordism bicategory XBord 2 is defined using oriented manifolds equipped with oriented X-halations 8 . Correspondingly, oriented 2dimensional E-HFT with target X is a symmetric monoidal 2-functor from XBord 2 .
In our classification of E-HFTs, the cofibrancy theorem (see Theorem 3.3) of Schommer-Pries [Sc] plays a key role. This theorem states that symmetric monoidal 2-functors out of a freely generated symmetric monoidal bicategory F(P) are determined by their values on generators subject to relations. Here P is a presentation from which F(P) is constructed (see Appendix for definitions).
The symmetric monoidal G-equivariant cobordism bicategories are not freely generated. However, Schommer-Pries [Sc] proved that every symmetric monoidal bicategory is equivalent to a freely generated one. Accordingly, in order to classify 2-dimensional E-HFTs, we need to find freely generated symmetric monoidal bicategories which are symmetric monoidally equivalent to G-equivariant cobordism bicategories. This is the content of the next section.

Presentations of the G-equivariant cobordism bicategories.
In this section we use the Gplanar decomposition theorem for cobordism type 2 -X-surfaces to introduce symmetric monoidal G-equivariant cobordism bicategories with diagrams. A 1-morphism is a quadruple ((A,Â 0 ,Â 1 , T,p 1 ), θ, L, ν) where (A,Â 0 ,Â 1 , T,p 1 ) is an X-haloed 1-cobordism, θ : A → I mn is a Morse function with distinct critical values, L = (Ψ G , Γ, S G ) is a G-linear diagram whose graphic and sheet data are induced by θ, and ν : is an isomorphism class of an X-haloed 2-cobordism, : S → I mn × I is a generic map for a representative (S,Ŝ, R,F), P = [(Φ G , Γ, S G )] is the corresponding equivalence class of cobordism type G-planar diagram, and κ : (S, R, F) → (S, R, F) is an X-homeomorphism over I mn × I where (S, R, F) is a cobordism type 2 -X-manifold constructed from the representative (Φ G , Γ, S G ) whose graphic and sheet data are induced by .
The second bicategory XB PD,un is defined by forgetting X-haloed manifolds and cobordisms in XBord PD,un 2 and taking isotopy classes of G-linear diagrams. In order to define isotopic G-linear diagrams we first need to explain compositions and tensor products of diagrams.
Horizontal compositions of G-linear and G-planar diagrams are given by the horizontal concatenation of diagrams where both G-sheet data agree and form a new G-sheet data. Vertical composition of equivalence classes of G-planar diagrams is vertical concatenation of diagrams followed by an isomorphism I ∪ pt I I and forgetting the G-linear diagram on the face along which two G-planar diagrams are concatenated. Figure 15 shows an example of horizontal and vertical compositions of 2-morphisms in XB PD,un whose labels are omitted.
2-morphisms in XB PD,un Horizantal and vertical compositions Symmetric monoidal product P 1 ⊗ P 2 Figure 15. Compositions and symmetric monoidal product of 2-morphisms in XB PD,un Symmetric monoidal structure on XB PD,un is defined as follows. Let P 1 = (Φ G 1 , Γ 1 , S G 1 ) and P 2 = (Φ G 1 , Γ 1 , S G 2 ) be two G-planar diagrams on I mn × I and on I ab × I respectively. Let V left be the leftmost chamber of P 1 and V right be the rightmost chamber of P 2 . Then, P 1 ⊗ P 2 is defined by stretching V left to the left by b − a units and stretching V right to the right by n − m units and joining the stretched diagrams (see Figure 15). Symmetric monoidal structure on G-linear diagrams can be deduced from this description. It is not hard to see that symmetric monoidal structures of diagrams is compatible with the disjoint union of X-haloed manifolds.
Recall that objects of XB PD,un are finite set of ordered points, 1-morphisms are isotopy classes of G-linear diagrams, and 2-morphisms are equivalence classes of G-planar diagrams. The notion of isotopy between G-linear diagrams is generated by the following identifications. Let L = (Ψ G , Γ, S G ) be any G-linear diagram, ∅ be the empty G-linear diagram for the empty 1-manifold, and id a be the identity G-linear diagram of the ordered set a.
In this case, it is not hard to see that XB PD,un is a strict 2-category. The proof for XB PD,un is very similar to the proof of Theorem 3.1. The case of XBord PD,un 2 follows from the compatibility of symmetric monoidal structures.
Remark. The symmetric monoidal bicategories XBord PD 2 and XB PD are defined similarly using oriented X-halations and the oriented G-planar decomposition theorem. These bicategories are generalizations 9 of the bicategories Bord PD 2 and B PD defined by Schommer-Pries ( [Sc]) to X-manifolds. Similarly, unoriented versions generalize Bord PD,un 2 and B PD,un . 9 Symmetric monodial bicategories Bord PD 2 and B PD are respectively equivalent to XBord PD Considering the results in G-planar decompositions section a natural question is whether symmetric monoidal bicategories defined by using diagrams are symmetric monoidally equivalent to G-equivariant cobordism bicategories. We give a positive answer using the following theorem.
Theorem 3.1 (Whitehead theorem for symmetric monoidal bicategories, [Sc]). Let B and C be symmetric monoidal bicategories. A symmetric monoidal 2-functor F : B → C is a symmetric monoidal equivalence if and only if it is an equivalance of underlying bicategories. That is, F is essentially surjective on objects, essentially full on 1-morphisms, and fully-faithful on 2-morphisms.
Proposition 3.1. The forgetful 2-functors F un and G un (F and G) given by forgetting (oriented) X-haloed cobordisms and (oriented) diagrams respectively Proof. For any given finite set W of ordered points or a compact 0-manifold with co-oriented codimension two X-halation (Y,Ŷ 0 ,Ŷ 1 ,ĝ) there exist objects in XBord PD,un 2 whose images under F un and G un are isomorphic to W and (Y,Ŷ 0 ,Ŷ 1 ,ĝ) respectively. For any given X-haloed 1-cobordism there exists a Morse function with distinct critical values leading to a G-linear diagram and any G-linear diagram produces an X-haloed 10 1-cobordism. Thus, by Proposition 2.1 each 2-functor is (essentially) full on 1-morphisms. Lastly, by the G-planar decomposition theorem for cobordism type 2 -X-surfaces 2-functors are fully-faithfull on 2-morphisms. Oriented case follows in the same way.
Proposition 3.1 implies that G-equivariant cobordism bicategories are symmetric monoidally equivalent to XB PD,un and XB PD . Furthermore, as we shall see below bicategories XB PD,un and XB PD are freely generated unbiased semistrict symmetric monoidal 2-categories (see Appendix).
In the oriented case, these arguments give XBord 2 which means that there exist four sets namely generating objects XG 0 , generating 1-morphisms XG 1 , generating 2-morphisms XG 2 , and generating relations XR among 2-morphisms forming the presentation XP such that there exists an (canonical) isomorphism of unbiased semistrict symmetric monoidal 2-categories where F uss (XP) is constructed from XP freely (see Appendix).
Notational remark. In Figures 16, 17, 22, and 23 each element labeled with group element g or g is indexed over G.
Proof of the following theorem as well as the definitions of a (unbiased semistrict) presentation and a freely generated (unbiased semistrict) symmetric monoidal 2-category are given in Appendix.
Theorem 3.2. The symmetric monoidal bicategories XB PD and XB PD,un are freely generated unbiased semistrict symmetric monoidal 2-categories with the following presentations: • The presentation XP = (XG 0 , XG 1 , XG 2 , XR) for XB PD is given by the diagram versions of elements in Figures 16 and 17. 10 Halation can be encoded into a G-sheet data by equipping trivializations of chambers with halations.

Classification of oriented 2-dimensional E-HFTs.
In this section we classify oriented 2dimensional E-HFTs using the cofibrancy theorem of Schommer-Pries [Sc].
Let F(P) be a freely generated symmetric monoidal bicategory for a given presentation P = (G 0 , G 1 , G 2 , R). The cofibrancy theorem is a coherence theorem for symmetric monoidal 2-functors from F(P) to a symmetric monoidal bicategory C. To state this theorem we recall the bicategory of P-data P(C) in C (see [Sc], [Ps] for details). In the following paragraphs, we denote the collection of objects of C by C 0 , 1-morphisms by C 1 , and 2-morphisms by C 2 .
An object A of P(C) is a collection of objects A 0 (G 0 ), 1-morphisms A 1 (G 1 ), and 2-morphisms A 2 (G 2 ) in C given by assignments A i : G i → C i for i = 0, 1, 2 such that A 1 and A 2 are invariant under source and target maps (globular) and the assignment A 2 is subject to relations in R.  Figure 17. Generating relations (XR) among 2-morphisms A 1-morphism α : A → B of P(C) is a collection of 1-morphisms α 0 (G 0 ) and 2-morphisms α 1 (G 1 ) given by assignments α i : These assignments are also required to be natural with respect to generating 2-morphisms i.e. for every ξ : A 2-morphism θ : α 1 → α 2 of P(C) is a collection of 2-morphisms θ 0 (G 0 ) in C given by an assignment θ 0 : G 0 → C 2 such that θ 0 (a) : α 1 0 (a) → α 2 0 (a) for every a ∈ G 0 and θ 0 is natural with respect to generating 1-morphisms i.e. for every f : a → b in G 1 horizontal compositions α 1 1 ( f ) * θ 0 (a) and θ 0 (b) * α 2 1 ( f ) are equal. The bicategory SymMon(F(P), C) has symmetric monoidal 2-functors as objects, symmetric monoidal transformations as 1-morphisms, and symmetric monoidal modifications as 2morphisms (see [Sc] and references therein for definitions). Theorem 3.3 (Cofibrancy Theorem, [Sc]). Let C be a symmetric monoidal bicategory and let F(P) be a freely generated symmetric monoidal bicategory for a presentation P. Then, there is an equivalence of bicategories SymMon(F(P), C) P(C).
We denote the bicategory SymMon(XBord 2 , C) by E-HFT(X, C) and state the classification of oriented 2-dimensional E-HFTs with target X as follows.
Theorem 3.4. Let XP be the presentation of XB PD given in Theorem 3.2. Then for any symmetric monoidal bicategory C there is an equivalence of bicategories E-HFT(X, C) XP(C).
Proof. Theorem 3.2 gives a presentation XP of XB PD as a freely generated symmetric monoidal bicategory. By the cofibrancy theorem we have SymMon(XB PD , C) XP(C). Using the symmetric monoidal equivalence between XB PD and XBord 2 in Proposition 3.1 we obtain the result.
3.3.1. Classification of Alg 2 k -valued oriented 2-dimensional E-HFTs. Every oriented 2-dimensional E-HFT with target X gives a nonextended one by restricting to oriented X-circles and X-cobordisms between them. A natural question is how the classification of oriented 2-dimensional E-HFTs is related to Turaev's classification ( [Tu2]) of oriented 2-dimensional HFTs by crossed Frobenius G-algebras. To understand this relation we study oriented E-HFTs taking values in Alg 2 k which has k-algebras as objects, bimodules as 1-morphisms, and bimodule maps as 2-morphisms for a commutative ring k with unity.
The symmetric monoidal structure of Alg 2 k is given by tensoring over k. We denote (E, C)bimodule D by E D C and omit the symbol k when either C or E is k. We regard E D C as a 1-morphism from C to E which is in line with the composition in XBord 2 (see Figure 14). Composition of 1morphisms E D C and C B A is the bimodule E (D ⊗ C B) A .
Before studying Alg 2 k -valued oriented 2-dimensional E-HFTs we recall necessary algebraic notions and introduce quasi-biangular G-algebras. Recall that a G-algebra over a commutative ring k is an associative k-algebra K equipped with a decomposition K = ⊕ g∈G K g such that K g K h ⊆ K gh for any g, h ∈ G. In this case, K e is the principal component and K is called strongly graded if K g K h = K gh for all g, h ∈ G. The opposite G-algebra of K is defined as K op = ⊕ g∈G K g −1 where the order of multiplication is reversed.
Definition 3.5. ([Tu1]) Let K = ⊕ g∈G K g be a G-algebra over a commutative ring k. An inner product on K is a symmetric bilinear form η : K ⊗ K → k satisfying η(ab, c) = η(a, bc) for any a, b, c ∈ K such that η| K g ⊗K h is nondegenerate when gh = e and zero otherwise. A Frobenius G-algebra is a G-algebra K with an inner product η and components of K are finitely generated projective k-modules.
Let (K = ⊕ g∈G K g , η) be a Frobenius G-algebra over k. Each nondegenerate form η| K g ⊗K g −1 yields an element η − g = i∈I g p g i ⊗ q g i ∈ K g ⊗ K g −1 , called an inner product element, where I g is finite and η − g is characterized by a = i∈I g η a, q g i p g i for any a ∈ K g . Since η is symmetric we have i p Recall that an associative k-algebra A is separable if there exists an element a = n i=1 p i ⊗q i ∈ A⊗ k A op called separability idempotent such that n i=1 p i q i = 1 and ab = ba for all b ∈ A. A separable algebra A is called strongly separable if the separability idempotent is symmetric i.e. a = n i=1 p i ⊗ q i = n i=1 q i ⊗ p i . and a central element z ∈ K e i.e. az = za for all a ∈ K. Then, for any g, h ∈ G and b ∈ K g −1 we have In particular, for any b ∈ K and c ∈ K h −1 we have j p g j bzq g j c = k cp hg k bzq hg k .
Proof. Since both sides belong to K h ⊗K h −1 g −1 it is enough to check that they give the same functionals on the dual k-module K h −1 ⊗ K gh . For any x ∈ K h −1 and y ∈ K gh applying x ⊗ y to the left hand side of equation (2) and using cyclic symmetry property of η we obtain Similarly applying x ⊗ y to the right hand side of the equation (2)  We generalize biangular G-algebras which were introduced by Turaev [Tu2] as follows.
Definition 3.6. A strongly graded Frobenius G-algebra (K, η) is called quasi-biangular if there exists a central element z ∈ K e such that for some collection of inner product elements i p g i ⊗ q g i g∈G equations i p g i zq g i = 1 hold for all g ∈ G. Remark. By the Lemma 3.3 the principal component of a quasi-biangular G-algebra is a separable algebra with separability idempotent i p e i ⊗ zq e i . A biangular G-algebra is a quasi-biangular G-algebra with z = 1. Similarly, the principal component of a biangular G-algebra is strongly separable.
One way of studying an algebra is to study the category of modules over it. Recall that Morita equivalence of algebras is the equivalence of categories of modules. In the case of a graded algebra one studies the category of graded modules. An equivalence of such categories is called a graded Morita equivalence which was introduced by P. Boisen [Bo] as follows.
Definition 3.7. ( [Bo]) A G-graded Morita equivalence ζ between G-algebras K = ⊕ g∈G K g and L = ⊕ g∈G L g is a quadruple ( L U K , K V L , τ, µ) where L U K = ⊕ g∈G U g is a graded (L, K)-bimodule that is L g U h K g ⊂ U ghg , K V L = ⊕ g∈G V g is a graded (K, L)-bimodule, and τ : K K K → K V ⊗ L U K and µ : L U ⊗ K V L → L L L are graded (K, K) and (L, L) bimodule maps respectively such that the following compositions are id U and id V respectively. When τ and ε are invertible as G-graded bimodule maps it is called a G-graded Morita context.
Lemma 3.4. ( [Ha]) Assume that G-algebras K = ⊕ g∈G K g and L = ⊕ g∈G L g are G-graded Morita equivalent. Then, if K is strongly graded then L is also strongly graded.
Now we transfer the inner product of one Frobenius G-algebra to another using a graded Morita context between them.
The inner product η of a Frobenius G-algebra (K, η) is determined at its principal component by η(a, b · 1) = η(ab, 1). This allows us to denote (K, η) by (K, Λ) where Λ : K e → k is a nondegenerate trace. Since η is symmetric Λ factors through K e /[K e , K e ]. Lemma 3.3 implies that for a symmetric Frobenius algebra (K e , Λ e ) an inner product element i p e i ⊗q e i can be considered as the image of 1⊗1 under a bimodule map ξ : where numbers indicate module actions i.e. K i e = K e for i = 1, 2, 3, 4. In the case of a quasi-biangular G-algebra (K = ⊕ g∈G K g , Λ) inner product elements i p g i ⊗ q g i g∈G\{e} are the image of 1 ⊗ 1 under the following composition where the second homomorphism is identity on K g −1 and K g , and ξ on K e ⊗ K e . In the following we consider inner product elements as the images of 1 ⊗ 1 under the above bimodule maps.
Definition 3.9. Let (K, Λ K ) and (L, Λ L ) be quasi-biangular G-algebras over k with collections of inner product elements {η K g } g∈G and {η L g } g∈G respectively. A G-graded Morita context ζ = ( L U K , K V L , τ, µ) between K and L is said to be compatible if Remaining inner product elements are obtained from ξ as described above.
Theorem 3.5. Any Alg 2 k -valued oriented 2-dimensional E-HFT with target X K(G, 1) determines a triple (A, B, ζ) where A and B are quasi-biangular G-algebras, and ζ is a compatible G-graded Morita context between A and B op . Moreover, any such triple (A, B, ζ) is realized by an oriented 2-dimensional E-HFT with target X.
Proof. Let Z : XBord 2 → Alg 2 k be an oriented 2-dimensional E-HFT and Z be an object of XP(Alg 2 k ) corresponding to Z under the equivalence of bicategories. Corresponding to two generating objects of XP we have k-algebras Z ( + ) = A e and Z ( -) = B e in Z 0 (XG 0 ). There are four types of generating 1-morphisms and each is indexed by the elements of G. For every g ∈ G they give the following bimodules in Z 1 (XG 1 )  Figure 18. Part of generators and relations giving G-algebra and G-graded module The first 2-morphism in Figure 18 defines a G-graded product on (A e , A e )-bimodules {A g } g∈G .
Associativity of this product is the obvious relation in Figure 17. Denote the corresponding Galgebra by A = ⊕ g∈G A g . The first relation in Figure 17 shows that the bimodule map is invertible for all g, g ∈ G. Since multiplication of G-algebra A is defined using (4) we have A g A g = A gg for all g, g ∈ G i.e. A is strongly graded. Similar arguments for (B e , B e )-bimodules {B g } g∈G yield another strongly graded G-algebra B = ⊕ g∈G B g .
Using the opposite algebra we can turn algebra actions on bimodules around. More precisely, a left B e action on A e ⊗B e M g can be turned into a right B op e action and the right B e module action on (N g ) A e ⊗B e can be turned into a left B op e action. The second 2-morphism in Figure 18 gives There are four types of cusp generators and each is indexed by two elements of G. For every g, g ∈ G they give the following bimodule maps in Z 2 (XG 2 ) f gg 1 Remaining generators are Morse generators consisting of saddles, cup, and cap 2-morphisms. The collection of bimodule maps in Z 2 (XG 2 ) for the first saddle morphism in Figure 16 Figure 19. Morphism giving inner product elements and cusp flip relation actions can be replaced by A-module actions and we get a graded (A 1 ⊗ A 3 , A 2 ⊗ A 4 ) bimodule map of the form where numbers indicate module actions i.e. A i = A for i = 1, 2, 3, 4. The graded bimodule map ξ is determined at 1 ⊗ 1 ∈ A e ⊗ A e which we denote by a finite sum i p e i ⊗ q e i and it satisfies i ap e i ⊗ q e i = i p e i ⊗ q e i a for all a ∈ A. Similarly, we denote the image of 1 ⊗ 1 under the first 2-morphism in Figure 19 by η A g = i p g i ⊗ q g i for all g ∈ G (compare with equation (3)). In the same way, the collection of bimodule maps in Z 2 (XG 2 ) for the second saddle morphism gives a graded (A 1 ⊗A 3 , A 2 ⊗A 4 ) bimodule map of the form η : The cusp flip relation shown in Figure 19 implies that ξ = η. Before considering cup and cap generators note that using ζ we can assign the collection of all g, g −1 labeled circles to ⊕ g∈G A g ⊗ (A e ⊗A op e ) A g −1 . The collections of 2-morphisms in Figure 20 give the following bimodule maps  Figure 20 shows that on nonprincipal components cup morphism is given by multiplication followed by Λ leading to a symmetric k-bilinear map η g : A g ⊗ A g −1 → k. Morse relations involving cup morphism indicates the nondegeneracy of η g as follows. Assuming β g ⊗ 1 ⊗ β g −1 as the image of 1 under A e ∼ − → A g ⊗ A e A e ⊗ A e A g −1 the first (left) 2-morphism in Figure 21 corresponds to following compositions and Morse relation implies that it is equivalent to id A g . Similarly, reflection of this morphism with g −1 label gives b = i η g (b, p g i )q g i for any b ∈ A g −1 , which shows that η g is nondegenerate. Thus, (A, η A ) is a Frobenius G-algebra where (η A )| A g ⊗A h is η g when h = g −1 and zero otherwise.
Remaining Morse relations contain cap morphisms which are determined on the principal component. For any c ∈ A e , assuming u(1)| A e ⊗A e = j a j ⊗ b j the second 2-morphism in Figure 21 corresponds to the following compositions where z = j b j a j ∈ A e . Morse relation implies that i p e i zq e i = 1 and consequently i p e i ⊗ zq e i is a separability idempotent of the algebra A e . Thus, (A e , η e ) is a separable symmetric Frobenius algebra as shown in [Sc]. Similarly, we have i p g i zq g i = 1 using the saddle whose image gives η A g . Until now we used ζ to replace B op actions by A actions. By changing the roles of A and B we obtain a quasi-biangular G-algebra B and ζ is a compatible graded Morita context between B and A op .
Thus, any oriented 2-dimensional E-HFT with target X determines a triple (A, B, ζ). For any such triple there exists an oriented 2-dimensional E-HFT by forming a strict 2-functor Z : XB PD → Alg 2 k using the given triple and precomposing with the equivalence XBord 2 ∼ − → XB PD .
We recall the 2-dimensional G-equivariant cobordism category XCob 2 which was introduced by Turaev [Tu2] to define oriented 2-dimensional (nonextended) HFT with target X. The objects of XCob 2 are oriented X-circles 11 and the morphisms are X-cobordisms considered up to X-homeomorphisms relative to boundary. Then, an oriented 2-dimensional HFT with target X is a symmetric monoidal functor from XCob 2 to any symmetric monoidal category. In the study of such theories crossed Frobenius G-algebras plays a significant role and they are defined as follows.

Definition 3.10. ([Tu2]) A Frobenius
is crossed if K is endowed with a group homomorphism ϕ : G → Aut(K) satisfying the following conditions (i) ϕ is conjugation type i.e. ϕ h (K g ) = K hgh −1 and ϕ h | K h = id K h for every g, h ∈ G, (ii) ba = ϕ h (a)b for any a ∈ K and b ∈ K h , (iii) Tr(µ c ϕ h : K g → K g ) = Tr(ϕ g −1 µ c : K h → K h ) for all g, h ∈ G and c ∈ K ghg −1 h −1 where µ c : K → K is a left multiplication by c and Tr is the trace of a map, (iv) η is invariant under ϕ.
Turaev [Tu1] defined the G-center of a biangular G-algebra. We extend this notion to a G-center of a quasi-biangular G-algebra (K, η) as Z G (K) = ⊕ g∈G Ψ(K g ) where Ψ(a) = i p e i aq e i for inner product 11 Closed oriented 1-dimensional X-manifolds. In general, G-center is not commutative and it differs from the usual center of the algebra. However, it has a crossed Frobenius G-algebra structure.
Lemma 3.6. Let (K, η) be a quasi-biangular G-algebra with a central element z ∈ K e and a collection of inner product elements i p g i ⊗ q g i g∈G . Then, Z G (K) is a G-subalgebra and the triple i for all a ∈ K and all g ∈ G.
Proof. Since i p e i ⊗ zq e i is a separability idempotent we have Ψ(z) = 1 ∈ Z G (K) e . By Lemma 3.3 we have the equality which implies that Z G (K) is a G-subalgebra of K. Restriction of η to Z G (K) is an inner product and hence (Z G (K), η| Z G (K) ) is a Frobenius G-algebra. For any b ∈ K and for all h ∈ G we have showing ϕ e | Z G (K) = id Z G (K) . Note that for any g ∈ G and a ∈ K we have showing ϕ g is an algebra homomorphism. Using equation (7)  Taking a = 1 gives ϕ g • ϕ h = ϕ gh for all g, h ∈ G, which also implies that ϕ g −1 is the inverse of ϕ g for all g ∈ G. For all a, b ∈ K and g ∈ G using the cyclic symmetry of η we have showing the inner product η is invariant under ϕ : G → Aut(Z G (K)). Lemma 3.3 implies that for any b ∈ K h −1 and g, h ∈ G and by taking g −1 = e we obtain ϕ h −1 (b) = ϕ e (b) = b. This implies that ϕ g acts by identity on Z G (K) g for all g ∈ G. Equation (6) gives ϕ g (a)b = bϕ h −1 g (a) for a ∈ K, b ∈ K h and g, h ∈ G. In this case by taking g = h we have ϕ h (a)b = ba. Let µ c : K → K be a multiplication by c ∈ K then for any g, h ∈ G and c ∈ K ghg −1 h −1 we have 12 Lastly, we show that the G-center of a quasi-biangular G-algebra is well-defined. By definition of inner product elements Z G (K) = ⊕ g∈G Ψ(K g ) is independent of the choice of inner product elements. for any g ∈ G. For any ∈ K e , Lemma 1.4 in [Tu1] gives the following equalities η , We have the same equality using r g i ⊗ s g i g∈G for all g ∈ G, which implies that z = z . Theorem 3.6. (Turaev,[Tu2]) There is a bijection between the isomorphism classes of oriented 2-dimensional HFTs with target X K(G, 1) and the isomorphism classes of crossed Frobenius G-algebras.
Any oriented 2-dimensional E-HFT produces a nonextended one by restricting it to a symmetric monoidal full subcategory XCob 2 of XBord 2 defined as follows. The objects of XCob 2 are { g e } g∈G , the empty 1-morphism in XBord 2 , and disjoint union of these 1-morphisms. The morphisms of XCob 2 are the 2-morphisms of XBord 2 among these 1-morphisms. We define a symmetric monoidal functor D : XCob 2 → XCob 2 by g e → g for any g ∈ G. On morphisms D forgets a point on each boundary component and takes the corresponding relative homotopy class. Using definitions it is not hard to see that D is an equivalence of categories. Then, by restriction of Z : XBord 2 → Alg 2 k to XCob 2 above we mean precomposing Z with D −1 : XCob 2 → XCob 2 .
Corollary 3.6.1. Let Z : XBord 2 → Alg 2 k be an oriented E-HFT giving (A, B, ζ). Then, the nonextended oriented HFT obtained from Z by restricting to XCob 2 is the nonextended oriented HFT associated to the G-center of quasi-biangular G-algebra (A, η A ).
Proof. Proceeding with the notation used in the proof of Theorem 3.5 the image of a g-labeled circle under Z is given by The G-center of (A, η A ) is given by Z G (A) = ⊕ g∈G Ψ(A g ). For any a ∈ A e ⊗ A e ⊗A op The third 2-morphism in Figure 21 gives the crossed structure on the restricted HFT and it corresponds to following sequence of compositions which coincides with the crossed structure of Z G (A).
Example 3.1. Let k be an algebraically closed field. Then separable k-algebras are the same as semisimple k-algebras. By Artin-Wedderburn structure theorem any separable algebra is isomorphic to a product of finitely many matrix algebras over k. Consider the G-algebra A = ⊕ g∈G A g whose principal component is a product A e = n i=1 M k i (k) of (k i × k i )-matrix algebras over k such that each k i is invertible in k and each component is given by A g = g A e where g is a basis. Define an inner product η on A as where r ∈ k is invertible and Tr(L ab ) is the trace of left multiplication by ab map. We can express inner product concretely as η( g For each g ∈ G an inner product element can be chosen as In this case, the central element z ∈ A e is given by (rI k 1 , . . . , rI k n ) where I k i denote (k i × k i ) identity matrix. Note that n i=1 k −1 i k i α,β=1 E α,β ⊗ E β,α is a separability idempotent of A e . Thus, the map Ψ : A g → A g is given by which is a projection onto its center g k n .
Until know we have studied the objects of XP(Alg 2 k ). Theorem 3.4 implies that studying 1-and 2-morphisms of XP(Alg 2 k ) leads us to a bicategory equivalent to E-HFT(X, Alg 2 k ). Let Z 0 and Z 1 be oriented E-HFTs with target X giving triples (A, B, ζ) and (A , B , ζ ) respectively. A 1-morphism α : Z 0 → Z 1 in XP(Alg 2 k ) gives 1-morphisms α 0 ( + ) = A e R A e and α 0 ( -) = B e S B e , and 2-morphisms α 1 ( which are isomorphisms for all g ∈ G and G-graded bimodules M, M , N, and N are the components of ζ and ζ . These morphisms are natural with respect to generating 2-morphisms. Naturality with respect to graded multiplication, g + g' gg' + + + + , leads to the commutativity of the diagram for all g, g ∈ G. We denote bimodules A e A g ⊗ A e R A e and A e R ⊗ A e (A g ) A e by A e (R g ) A e and A e (R g ) A e respectively. Commutativity of the above diagram implies that they are naturally isomorphic. Thus, we can use one of them and denote it by R g . Similarly, S g denotes (B e , B e )-bimodule. These assignments and naturality with respect to Using α 0 ( -) we define a 1-morphism α 0 ( + ) = A e R A e as follows where σ is the symmetric braiding of Alg 2 k . Using α 0 ( + ) we define a 2-morphism α 1 ( Using the naturality R is turned into a G-graded (A, A )-bimodule R = ⊕ g∈G R g . The 1-morphism and similarly α 1 (--g ) can be obtained from α 1 ( + + g ). Likewise, using α 0 ( + ) in the images of cusps generators under Z 0 , the 2-morphisms α 1 ( g + -) and α 1 ( g -+) are defined and both α 1 ( g + -) and α 1 ( g -+) can be obtained from these 2-morphisms. As in the proof of Theorem 3.5 using G-graded Morita contexts ζ and ζ graded bimodules M and M can be replaced by A⊗A op A and A ⊗(A ) op A . We can also replace the graded bimodule S by R using α 0 ( + ). Thus, naturality with respect to G-module generators turn the collection {α 1 (

Naturality with respect to cusp generators indicate that compositions
are id R and id R respectively. In other words, α gives a G-graded Morita context between A and A . Similarly, one can define α 0 (--g ) and obtain a G-graded Morita context between B and B . Naturality with respect to Morse generators indicates that G-graded Morita contexts are compatible. Hence, α leads to two compatible G-graded Morita contexts. In the theory of bicategories this means that both α 0 ( + ) and α 0 ( -) are parts of two adjoint equivalences. Since an adjoint equivalence is the same as an equivalence (see Proposition A.27 in [Sc]) Z 0 and Z 1 are equivalent E-HFTs.
Let α 1 , α 2 : Z 0 → Z 1 be 1-morphisms in XP(Alg 2 k ) and θ : α 1 → α 2 be a 2-morphism in XP(Alg 2 k ). Assume that Z 0 and Z 1 give triples (A, B, ζ) and (A , B , ζ ) as before and 1-morphisms give α 1 0 ( + ) = A e R A e and α 2 0 ( + ) = A e P A e . Then, θ 0 ( + ) = A e R A e → A e P A e and the naturality of θ 0 ( + ) with respect to + + g is the commutativity of the following diagram A e A g ⊗ A e R A e α 1 1 ( which shows that θ 0 ( + ) is a G-graded bimodule map. Assuming (α 0 ) 1 ( + ) = A e R A e and (α 0 ) 2 ( + ) = A e P A e we similarly have a graded bimodule map θ 0 ( + ) : A e R A e → A e P A e using θ 0 ( -) and (α 0 ) i ( + ) for i = 1, 2. Naturality with respect to g + -and g -+ corresponds to the commutativity of these bimodule maps with the unit and counit of the adjunctions. In other words, θ leads to an equivalence of graded Morita contexts. In the same way, using B and B one gets another equivalence of graded Morita contexts.
Motivated by these observations we define a bicategory Frob G and a forgetting 2-functor F : XP(Alg 2 k ) → Frob G as follows. The bicategory Frob G has quasi-biangular G-algebras as objects, compatible G-graded Morita contexts as 1-morphisms, and equivalences of G-graded Morita contexts as 2-morphisms. The forgetting 2-functor F maps an object of XP(Alg 2 k ) giving (A, B, ζ) to A. On 1-morphisms F maps α : Z 0 → Z 1 to a compatible G-graded Morita context between quasi-biangular G-algebras whose principal components are Z 0 ( + ) and Z 1 ( + ). On 2-morphisms F maps θ : α 1 → α 2 to an equivalence of the compatible G-graded Morita contexts. Composing F with the equivalence E-HFT(X, Alg 2 k ) XP(Alg 2 k ) we define F. Theorem 3.7. The 2-functor F is an equivalence of bicategories E-HFT(X, Alg 2 k ) Frob G .
Proof. It is enough to show that F is an equivalence and we use Whitehead theorem (Theorem 3.1). For a given quasi-biangular G-algebra A, the triple (A, A op , id) gives an object Z of XP(Alg 2 k ) such that F (Z) = A. Let α be a compatible G-graded Morita context between quasi-biangular G-algebras A and A . Then triples (A, (A ) op , α) and (A , A op , α) give objects Z 0 and Z 1 in XP(Alg 2 k ) such that F (α ) = α where α : Z 0 → Z 1 .
3.3.2. The G × SO(2)-structured cobordism hypothesis. A different approach to categorical classification of (fully-)extended oriented HFTs is given by the structured cobordism hypothesis due to J. Lurie [Lu]. Cobordism hypothesis ( [AF], [Lu], [BD]) is conjectured by J. Baez and J. Dolan in their seminal paper [BD]. Lurie [Lu] reformulated the cobordism hypothesis using (∞, n)-categories and generalized it to a structured cobordism hypothesis using homotopy fixed points as follows.
Structured Cobordism Hypothesis. (Lurie, [Lu]) Let C be a symmetric monoidal (∞, n)-category (see [CS]) and Bord Γ n be the Γ-equivariant extended cobordism (∞, n)-category 13 for a group Γ. Then, there is a canonical equivalence of (∞, n)-categories where Fun ⊗ is the (∞, n)-category of symmetric monoidal functors between symmetric monoidal (∞, n)categories, C f d is the sub-(∞, n)-category of fully dualizable objects with duality data, (C f d ) ∼ is the underlying ∞-groupoid and ((C f d ) ∼ ) hΓ is the ∞-groupoid of homotopy Γ-fixed points given by where EΓ is a weakly contractible ∞-groupoid equipped with a free Γ-action.
Remark. An oriented 2-dimensional E-HFT with target X K(G, 1) i.e. a classifying space BG, is a (G × SO(2))-structured 2-dimensional fully-extended TFT by pulling back universal bundle along characteristic maps of oriented X-manifolds. The fact that characteristic maps are relative homotopy classes instead of pointed maps does not lead to a problem because in the context of structured E-TFTs we would like to glue cobordisms along submanifolds equipped with isomorphic bundles not necessarily the same bundles.
When k is an algebraically closed field of characteristic zero O. Davidovich [Da] showed that for a finite group G homotopy (G×SO(2))-fixed points in Alg 2 k are given by G-equivariant algebras. A Gequivariant algebra is a strongly graded Frobenius G-algebra with semisimple principal component. Her methods do not particularly require group G to be finite and can be extended to discrete groups directly. Thus, the objects of Frob G and the objects of Alg Assume that k is an algebraically closed field of characteristic zero. Artin-Wedderburn theorem implies that any separable k-algebra is isomorphic to a product of matrix algebras over k. Let A e = End(V 1 )×End(V 2 )×· · ·×End(V n ) be a such algebra where V 1 , V 2 , . . . , V n are finite dimensional k-vector spaces. Recall that A = ⊕ g∈G A g is strongly graded by the generators leading to bimodule isomorphisms {τ g,g : A g ⊗ A e A g − → A gg } g,g∈G , that is each A g is an invertible (A e , A e )-bimodule. Under the above assumption on A e , these isomorphisms form a function τ : G × G → (k * ) n . 13 Manifolds equipped with principal Γ-bundle.
Moreover, the relations involving these generators give the following commutative diagram for all g, g , g ∈ G id⊗τ(g,g ) / / A g ⊗ A e A gg τ(gg ,g ) / / A gg g and isotopy classes of G-linear diagrams generate the relations which can be expressed as the following commutative diagram for all g ∈ G A g ⊗ A e A e τ(e,g) which imply that τ is a normalized 2-cocycle. Davidovich [Da] showed that any invertible (A e , A e )bimodule is isomorphic to one of the form for some permutation σ ∈ S n and denote this bimodule by A σ . Since the direct sum A = ⊕ g∈G A g forms a G-algebra permutations indeed form a homomorphism σ : G → S n .
It is known that all traces on a matrix algebra are given as some (nonzero) constant multiple of the matrix trace. Thus, in the case of A e = End(V 1 )×End(V 2 )×· · ·×End(V n ) there are constants r i ∈ k * for i = 1, . . . , n and the inner product of quasi-biangular G-algebra A = ⊕ g∈G A σ(g) is given by η( f, g) = Tr(r • (g • σ f )) for any f ∈ A g and g ∈ A g −1 where r = (r 1 id V 1 , . . . , r n id V n ) and • σ is the composition of morphisms under σ such as f i • g σ(i) for f i ∈ Hom k (V σ(i) , V i ), g σ(i) ∈ Hom k (V σ(σ(i)) , V σ(i) ). Since the inner product is invariant under cyclic order i.e. η( f, g · h) = η(h · f, g) = η(h, f · g), the vector r ∈ (k * ) n must satisfy Im(σ) ⊆ Stab S n (r) where S n acts on r by permuting the entries. More explicitly, as an example consider the products (h • σ g) • σ f and g • σ ( f • σ h) for f ∈ A g , g ∈ A g , and h ∈ A (gg ) −1 . Then, the corresponding traces of these morphisms in A e = End(V 1 ) × End(V 2 ) × · · · × End(V n ) are related by the permutation σ(gg ) ∈ S n .
Using the above arguments when k is algebraically closed field of characteristic zero we can conclude that up to an isomorphism a quasi-biangular G-algebra (A = ⊕ g∈G A g , η) is determined by a Morita class of the principal component (n ≥ 1), a normalized 2-cocycle τ : G × G → (k * ) n , a homomorphism σ : G → S n , and an element r ∈ (k * ) n with Im(σ) ⊆ Stab S n (r).
Let ( A 2 M A 1 , A 1 N A 2 , κ, µ) be a graded Morita context between two quasi-biangular G-algebras (A 1 , η 1 ) and (A 2 , η 2 ) which are determined by the normalized 2-cocycles τ i , homomorphisms σ i : G → S n , and elements r i ∈ (k * ) n with Im(σ i ) ⊆ Stab S n (r i ) for i = 1, 2. Then M and N are invertible (A 2 , A 1 ) and (A 1 , A 2 ) bimodules respectively, which means there exists σ ∈ S n such that M e is isomorphic to , V n ) for all g ∈ G and σ g i = σ i (g) ∈ S n for i = 1, 2. Being a graded (A 2 , A 1 )-bimodule forces σ to satisfy σ g 2 = σσ g 1 σ −1 for all g ∈ G. In this case, nonprincipal components are given as M g = Hom(V σ g (1) , W 1 ) × · · · × Hom(V σ g (n) , W n ) where σ g = σσ g 1 = σ g 2 σ for all g ∈ G. Using the similar arguments for the invertible (A 1 , A 2 )-bimodule N, we obtain N g = Hom(W σ g (1) , V 1 ) × · · · × Hom(W σ g (n) , V n ) for σ g = σ Transferring the Frobenius form via the graded Morita context amounts to finding a central element in (A 2 ) e corresponding to r 1 ∈ (k * ) n . Using the identity component M e this element is given by σ(r 1 )(id W 1 , id W 2 , . . . , id W n ). Thus, we have the equality σ(r 1 ) = r 2 ∈ (k * ) n . The bimodule isomorphisms κ : A 1 (A 1 ) A 1 → A 1 N ⊗ A 2 M A 1 and µ : A 2 M ⊗ A 1 N A 2 → A 2 (A 2 ) A 2 lead to a map φ : G → (k * ) n and graded Morita context equations produce a map φ : (A 1 ) g → (A 2 ) g for all g ∈ G so that the diagram commutes for all g, h ∈ G. This means that normalized 2-cocycles φ 1 , φ 2 : G × G → (k * ) n differ by a coboundary ∂φ. Thus, we conclude that quasi-biangular G-algebras up to compatible G-graded Morita contexts are in bijection with ∞ r=1 [r]∈(k * ) n /S n H 2 (G; (k * ) n ) × Hom(G, Stab S n (r))/ ∼ where the equivalence ∼ is given by conjugation. Using Theorem 3.7 we derive the following proposition which was previously proven by Davidovich [Da].
where the equivalence ∼ is given by conjugation.
The fact that Proposition 3.2 is proven in two different ways, namely using the structured cobordism hypothesis and without using it, implies the conjecture in this special case.
Corollary 3.7.1. For any algebraically closed field k of characteristic zero, the (G × SO(2))-structured cobordism hypothesis for Alg 2 k -valued oriented E-HFTs with target X K(G, 1) holds true.

Classification of unoriented 2-dimensional E-HFTs.
In this section we define and classify unoriented 2-dimensional E-HFTs with target X K(G, 1) where every element of G has order two. From now on we assume that G is such a group and X is a pointed K(G, 1)-space.
For a symmetric monoidal bicategory C, a C-valued unoriented 2-dimensional E-HFT with target X is a symmetric monoidal 2-functor from XBord un 2 to C. The bicategory E-HFT un (X, C) has C-valued unoriented E-HFTs as objects, symmetric monoidal transformations as 1-morphisms, and symmetric monoidal modifications as 2-morphisms.
Remark. There is a symmetric monoidal 2-functor Forget or : XBord 2 → XBord un 2 given by forgetting the orientation. In the same way, any oriented or unoriented 2-dimensional E-TFT leads to an oriented or unoriented E-HFT respectively by forgetting the X-manifold data. The following diagram indicates the universality of unoriented 2-dimensional E-TFTs in this context where E-TFT un (C) and E-TFT(C) are defined similarly using Bord un 2 and Bord 2 respectively. For any given symmetric monoidal bicategory C using the cofibrancy theorem, we state the classification of C-valued unoriented 2-dimensional E-HFTs as the equivalence of bicategories E-HFT un (X, C) XP un (C).

Classification of Alg 2
k -valued unoriented 2-dimensional E-HFTs. K. Tagami [Ta] classified nonextended unoriented 2-dimensional HFTs by extended crossed Frobenius G-algebras. Similar to oriented case our goal is to understand the relation between his classification and the restriction of Alg 2 k -valued unoriented 2-dimensional E-HFTs to circles and cobordisms between them. Firstly, we introduce necessary algebraic notions. Let K be a G-algebra and V be a (K, K op ) bimodule. Conjugate of V is the (K, K op ) bimodule V obtained by turning actions around. Similarly, the conjugate of a graded Morita context ζ = ( K op U K , K V K op , τ, µ) is given by ζ = ( K op U K , K V K op , τ, µ). We generalize stellar algebras introduced in [Sc] to stellar G-algebras as follows.  Figure 23. Additional generating relations of X un R Definition 3.11. A stellar G-algebra is a G-algebra K = ⊕ g∈G K g , equipped with a G-graded Morita context ζ = ( K op U K , K V K op , τ, µ) together with an isomorphism of G-graded Morita contexts σ : ζ ζ such that σ • σ is the identity isomorphism where σ is the induced isomorphism between ζ and ζ.
Stellar structure on a G-algebra can be transferred along a graded Morita context as follows. Let ρ = ( K U L , L V K , κ, ν) be a G-graded Morita context between G-algebras K and L and let (K, ζ, σ) be a stellar structure on K with ζ = ( K op U K , K V K op , τ, µ). Then, (L, ρ * ζ, ρ * σ) is a stellar algebra where Definition 3.12. Let (K, ζ, σ) be a stellar G-algebra with ζ = ( K op U K , K V K op , τ, µ) and let (K, η) be a quasi-biangular G-algebra. Stellar structure is said to be compatible with quasi-biangular G-algebra if there exists an element j a j ⊗ b j ∈ K e ⊗ K e giving the central element z = j b j a j such that the following diagrams 14 commute Following the proof of Theorem 3.5 we have a strongly graded G-algebra A = ⊕ g∈G A g where Z ( ) = A e . We also have G-graded (A ⊗ A, k) and (k, A ⊗ A)-bimodules M = ⊕ g∈G M g , N = ⊕ g∈G N g respectively. By turning actions around we obtain (A, A op )-bimodule M and (A op , A)-bimodule N.
Bimodule maps in Z 2 (X un G 2 ) corresponding to cusp generators (subject to relations) yield a G- A op are invertible G-graded bimodule maps. Bimodule maps in Z 2 (X un G 2 ) for the Morse generators satisfying relations imply that (A, η) is a quasi-biangular Galgebra. The generators in Figure 22 give the following graded bimodule maps in Z 2 (X un G 2 ) These graded bimodule maps are subject to the relations in Figure 23. Thereby, we have σ 1 •σ 1 = id M , σ 1 • σ 1 = id M , σ 2 • σ 2 = id N , and σ 2 • σ 2 = id N . These isomorphisms of bimodules lead to an isomorphism σ : ζ ζ. Applying σ to ζ gives another isomorphism σ : ζ → ζ whose composition with σ gives σ • σ : ζ ζ. Third relation on the first row of Figure 23 and its reflection indicate that compositions of bimodule maps M → M → M and N → N → N are identity maps.
Thus, additional generators and relations among them lead to a stellar structure (ζ, σ) on the quasi-biangular G-algebra A. Remaining relations imply the compatibility giving the quasibiangular stellar G-algebra (A, η, ζ, σ). For any quasi-biangular stellar G-algebra there exists an unoriented 2-dimensional E-HFT by forming a strict 2-functor Z : XB PD,un → Alg 2 k using the quadruple and precomposing with the equivalence XBord un 2 ∼ − → XB PD,un . Similar to oriented case every unoriented 2-dimensional E-HFT with target X produces a nonextended one by precomposition XCob un 2 → XCob un 2 → Alg 2 k where XCob un 2 and XCob un 2 are defined just as XCob 2 and XCob 2 using unoriented X-manifolds. In the unoriented case extended crossed Frobenius G-algebras plays an important role in the study of unoriented 2-dimensional nonextended HTFs and they are defined as follows.
Definition 3.14. ( [Ta]) Let (K, η, ϕ) be a crossed Frobenius G-algebra over k. An extended structure on K consists of a k-module homomorphism Φ : K → K and a family of elements {θ g ∈ K e } g∈G satisfying the following conditions (1) Φ(K g ) ⊂ K g and Φ(θ g ) = θ g for all g ∈ G, for any g, h, l ∈ G and v ∈ K gh , we have where ∆ g,h : K gh → K g ⊗ K h is defined by the equation (id g ⊗ η) • (∆ g,h ⊗ id h ) = m. Such a map ∆ g,h is uniquely determined since η is nondegenerate and each K g is finitely generated, (7) Φ(θ h v) = ϕ hg (θ hg v) for any g, h ∈ G and v ∈ K g , (8) ϕ h (θ g ) = θ g for any g, h ∈ G, (9) for any g, h, l ∈ G, we have θ g θ h θ l = q(1)θ ghl where q : k → K e is defined as follows; let {a i ∈ K gh } n i=1 and {b i ∈ K gh } n i=1 be families of elements of K gh satisfying the equation i η(b i ⊗ v)a i = ϕ hl (v) for any v ∈ K gh . As in (3), such a i and b i are uniquely determined and q(1) = i a i b i . Theorem 3.9. (Tagami, [Ta]) Let G be a group with each nonidentity element having order 2. There is a bijection between the isomorphism classes of unoriented 2-dimensional HFTs with target X K(G, 1) and the isomorphism classes of extended crossed Frobenius G-algebras. Corollary 3.9.1. Assume that Z : XBord un 2 → Alg 2 k determines a quasi-biangular stellar G-algebra (A, η, ζ, σ). The stellar structure (ζ, σ) gives an extended structure on the crossed Frobenius G-algebra Z G (A). Moreover, the corresponding 2-dimensional HFT is the unoriented HFT obtained by restricting Z to Cob un 2 .
Proof. We have a crossed Frobenius G-algebra (Z G (A), η| Z G (A) , {ϕ| Z G (A) } g∈G ). By Tagami's classification, the unoriented 2-dimensional HFT given by the restriction of Z to circles and cobordism between them induces an extended structure on Z G (A). We claim that homomorphism Φ and elements {θ g ∈ Z G (A) e } g∈G come from the stellar structure (ζ, σ) on A.
In [Ta], for each g ∈ G the restriction Φ| Z G (A) g : Z G (A) g → Z G (A) g is the involution induced by an orientation reversing homeomorphism of a g-labeled circle. In the extended case this morphism is given by additional 2-morphisms ( Figure 22). More precisely, commutes. It is not hard to see that Φ reverses the orientation of the oriented (input) circle. In [Ta], for every g ∈ G the element θ g is the image of HFT under the Möbius strip whose boundary is labeled by g 2 = e where the Möbius strip is considered as the cobordism from the empty 1-manifold to the boundary circle. In the extended case, θ g ∈ A e ⊗ A e ⊗A op e A e is the image of 1 ∈ k under the following composition; {g, g}-labeled cap morphism followed by new generators (see Figure 23) which is composed with module actions turning boundary labels into {e, e} (see Figure 20).
We see that the involution Φ and elements {θ g } g∈G are defined according to their topological description given in [Ta]. Hence, (Z G (A), η| Z G (A) , {ϕ g | Z G (A) } g∈G , Φ, {θ g } g∈G ) is an extended crossed Frobenius G-algebra which by definition corresponds to the restriction of Z : XBord un 2 → Alg 2 k to X-circles and unoriented X-cobordisms between them.
In order to upgrade Theorem 3.8 to an equivalence of bicategories we study morphisms in the bicategory XP un (Alg 2 k ). Let α be a 1-morphism from Z 0 to Z 1 giving quasi-biangular stellar Galgebras (A, η, ζ, σ) and (A , η , ζ , σ ) respectively. We know from the oriented case that α gives a compatible G-graded Morita context ξ between G-algebras A and A . Assuming α 0 ( ) = A e R A e and ξ = ( A R A , A R A , τ, µ) naturality with respect to the first generator in Figure 22 is the commutativity of the following diagram where M and M are components of the graded Morita contexts ζ and ζ respectively. There are similar commutative diagrams for the remaining three generators. These diagrams indicate that the G-graded Morita context ξ gives an equivalence of G-graded Morita contexts ζ and ξ * ζ with α • σ = ξ * σ • α. In other words, α leads to a morphism of stellar G-algebras (see Definition 3.13).
Let θ : α 1 → α 2 be a 2-morphism in XP(Alg 2 k ) with θ 0 ( ) = A e R A e → A e P A e . In the oriented case we observed that θ induces an equivalence of G-graded Morita contexts ξ = ( A R A , A R A , τ, µ) and ρ = ( A P A , A P A , κ, ν). Naturality of θ 0 ( ) with respect to g is the commutativity of the following diagram and there is a similar diagram for the naturality with respect to g . Naturality for { g } g∈G and { g } g∈G gives α 2 = θ • α 1 and naturality for { g } g∈G and { g } g∈G gives α 2 = θ • α 1 . In other words, θ gives an isomorphism of stellar G-algebra morphisms (see Definition 3.13).
The stellar structure is compatible with the Frobenius form as follows. A Frobenius form on a k-algebra A is determined by a central element which is the image of 1 under a bimodule map z : A A A → A A A . Geometrically, z(1) is an element of π 2 (Map(BSO(2), χ r ), f ) (k × ) r where the algebra A ∈ χ r ⊂ χ = ∞ r=1 χ r is isomorphic to End(V 1 ) × End(V 2 ) × · · · × End(V r ) under Artin-Wedderburn isomorphism for finite dimensional k-vector spaces V 1 , . . . , V r .
Compatibility means that (horizontal) composition of z with ζ yields z again. Geometrically, this corresponds to conjugating the representing sphere based at f with loops in χ r given by bimodules of ζ. Since this loop is contractible conjugation does not change z(1) in the second homotopy group. Thus, we have a compatible stellar structure and following Davidovich's methods [Da] we obtain that for a discrete group G homotopy (G × O(2))-fixed points are quasi-biangular stellar G-algebras.
In the light of above lemmas and Theorem 3.10 we verify the special case of (G × O(2))-structured cobordism hypothesis.
Corollary 3.10.1. For any algebraically closed field k of characteristic zero, the (G × O(2))-structured cobordism hypothesis for Alg 2 k -valued unoriented E-HFTs with target X holds true.
Lastly, we comment on the relation between E-HFTs whose targets are related by pointed coverings. Let G be any discrete group, Y K(H, 1) be a pointed CW-complex for a subgroup H ≤ G and p : (Y, y) → (X, x) be a covering. Then, any Y-manifold (Y-cobordism) can be turned into an X-manifold (X-cobordism) by postcomposing a representative of characteristic map with p. This gives a symmetric monoidal 2-functor ι H : YBord 2 → XBord 2 and precomposing any oriented E-HFT with target X with ι H yields an oriented E-HFT with target Y. Moreover, for any symmetric monoidal bicategory C precomposition of C-valued E-HFT with ι lifts to a 2-functor E-HFT(X, C) → E-HFT(Y, C) by forgetting the naturality of transformations with respect to G\H labeled 1-morphisms. Correspondingly, there is a 2-functor XP(C) → YP(C) where XP and YP are the presentations of XBord 2 and YBord 2 respectively. When C is Alg 2 k the functor Frob G → Frob H is given by forgetting the G\H components of quasibiangular G-algebras, compatible G-graded Morita contexts, and equivalences of G-graded Morita contexts. In other words, a G-graded Morita context can be considered as a collection of Morita contexts indexed by the subgroups of G (see [Bo]). There are similar 2-functors in the unoriented case.

Appendix A. Freely Generated Symmetric Monoidal Bicategories
In this section we recall the freely generated symmetric monoidal bicategories. Our main reference is [Sc] in which these bicategories are called computadic symmetric monoidal bicategories. A similar exposition for freely generated monoidal bicategories is given in the appendix of [Ps].
A freely generated symmetric monoidal bicategory F(P) is constructed from a presentation 15 . A presentation P consists of four sets G 0 , G 1 , G 2 , R together with source and target maps defined on G 1 and G 2 . These sets are generating sets; namely, generating objects G 0 , generating 1-morhisms G 1 , generating 2-morphisms G 2 , and generating relations R among 2-morphisms. We describe the conditions on these sets and the codomains of source and target maps below.
For a given presentation P = (G 0 , G 1 , G 2 , R) the symmetric monoidal bicategory F(P) is constructed in two steps. Ignoring relations R first the symmetric monoidal bicategory F(P G ) is constructed for P G = (G 0 , G 1 , G 2 ). Then, F(P) is defined from F(P G ) by considering the equivalence classes of 2-morphisms defined by R. The first step can be described as follows.
Definition A.1. For a given presentation P = (G 0 , G 1 , G 2 , R) the objects of F(P G ) are binary words in G 0 , the 1-morphisms are binary sentences in G 1 , and the 2-morphisms are equivalence classes of paragraphs in G 2 .
Definition A.2. Let G 0 be a set. The set BW(G 0 ) of binary words in G 0 contains symbol (ı), the parenthesized elements of G 0 i.e. (a) ∈ BW(G 0 ) for all a ∈ G 0 , and parenthesized ⊗-products i.e. (a ⊗ b) ∈ BW(G 0 ) for all a, b ∈ BW(G 0 ).
Since binary words in G 0 form the objects of F(P G ) the set G 1 of generating 1-morphisms are required to be equipped with source and target maps s, t : G 1 → BW(G 0 ). Definition A.3. Let G 1 be a set equipped with maps s, t : G 1 → BW(G 0 ). The set BW(G 1 ) of binary words in G 1 contains parenthesized elements of G 1 , parenthesized symbols in Table 1 for any a, b, c ∈ BW(G 0 ) i.e. ( ) ∈ BW(G 1 ) for any symbol in Table 1, and ( * ) for every symbol in Table 1. Table 1 extends the source and target maps to parenthesized symbols. If a symbol in BW(G 1 ) is of Table 1. Binary words in G 1 the form ( * ) for some symbol in Table 1 then s(( * )) = t(( )) and t(( * )) = s(( )). Definition A.4. Let BW(G 1 ) be a set of binary words in G 1 with s, t : BW(G 1 ) → BW(G 0 ). The set BS(G 1 ) of binary sentences in G 1 contains parenthesized binary words in G 1 , g• f for any f, g ∈ BW(G 1 ) with s(g) = t( f ), and f ⊗ g for any f, g ∈ BW(G 1 ). Source and target maps extend naturally to BS(G 1 ) by and t( f ⊗ g) = t( f ) ⊗ t(g) for any f, g ∈ BW(G 1 ).
Since binary sentences in G 1 form the 1-morphisms of F(P G ) the set G 2 of generating 2-morphisms are required to be equipped with source and target maps s, t : G 2 → BS(G 1 ) satisfying s • s = s • t and t • s = t • t. This leads to a definition of presentation without a relation (R = ∅). A free presentation 16 P G consists of generating sets G 0 ,G 1 ,G 2 together with maps s, t : G 1 → BW(G 0 ), and s, t : G 2 → BS(G 1 ) satisfying s • s = s • t and t • s = t • t.
Definition A.5. Let G 2 be a set equipped with s, t : G 2 → BS(G 1 ) satisfying s • s = s • t and t • s = t • t. The set BW(G 2 ) of binary words in G 2 contains parenthesized elements of G 2 and parenthesized symbols in Table 2 for any x, x , y, z, u ∈ BW(G 0 ) and f, f , g, g , h ∈ BS(G 1 ) with x = s( f ) = t( f ), t( f ) = y, s( f ) = t( f ), and s(g) = t(g ). In addition, BW(G 2 ) contains ( −1 ) for any symbol in Table 2. This table extends the source and target maps to parenthesized symbols. If an element in BW(G 1 ) is of the form ( −1 ) for some symbol in Table 2 then s(( −1 )) = t(( )) and t(( −1 )) = s(( )).
• If p, p −1 ∈ PG(G 2 ) with s(p) = f and t(p) = g then p • p −1 ∼ id g and p −1 • p ∼ id f . • The binary words a c , r c , l c , φ ⊗ ( f,g),( f ,g ) , φ ⊗ x,x , α f,g,h , f , r f , and β f,g are natural with respect to binary sentences.
In the symmetric monoidal bicategory F(P G ) the composition of 1-morphisms (binary sentences in G 1 ) is given by •. Horizontal composition of 2-morphisms (equivalence classes of paragraphs) is given by * while vertical composition is given by concatenation.
Definition A.7. The set R of generating relations among 2-morphisms for a free presentation P G consists of pairs (F, G) of 2-morphisms in F(P G ) with s(F) = s(G) and t(F) = t(G). A presentation P consists of a free presentation P G and a set R of generating relations among 2-morphisms for P G .
The symmetric monoidal bicategory F(P) is obtained from F(P G ) by considering the -equivalence classes of 2-morphisms where is the smallest equivalence relation on 2-morphisms of F(P G ) such that is generated by R and closed under compositions and tensor product. A symmetric monoidal bicategory C is called freely generated if there exists a strict symmetric monoidal equivalence F : F(P) → C for some presentation P.
Schommer-Pries [Sc] proved that every symmetric monoidal bicategory is equivalent to a freely generated symmetric monoidal bicategory. In this paper, we are interested in certain stricter version of symmetric monoidal bicategories called unbiased semistrict symmetric monoidal 2-categories ( [Sc]). More precisely, our goal here is to recall freely generated unbiased semistrict symmetric monoidal 2-categories ( [Sc]). To do this, we review string diagrams for bicategories. Figure 25. Pasting diagram with the corresponding string diagram and a string diagram for a certain symmetric monoidal bicategory Alternative to pasting diagrams, string diagrams are tools describing morphisms in a bicategory. Instead of arrows between objects and 1-morphisms, a string diagram consists of regions, arcs, and vertices. Each region represents an object and each arc represents a 1-morphism between objects whose corresponding regions share this arc as a common boundary. Each vertex represents a 2-morphism between 1-morphisms whose corresponding arcs are connected with each other via this vertex. On the left hand side of Figure 25, a pasting diagram and the corresponding string diagram is shown. Note that we read string diagrams from right to left and from top to bottom.
Unbiased semistrict symmetric monoidal 2-categories are strict enough to admit a version of string diagram ( [Sc]). An example of such a string diagram is shown in Figure 25 in which regions are labeled with objects {ω i } 7 i=1 , red arcs are labeled with 1-morphisms { f j } 3 j=1 , and a red vertex is labeled with a 2-morphism {α}. However, there are additional strings and vertices of different colors coming from the structure morphisms of an unbiased semistrict symmetric monoidal 2-category.
In order to prove Theorem 3.2 we first need to show that the symmetric monoidal bicategories XB PD and XB PD,un are unbiased semistrict symmetric monoidal 2-categories. In the following we only consider the oriented case. Unoriented case is similar except for minor changes which we left to the reader. Recall that objects of XB PD are finite set of ordered oriented points, 1-morphisms are isotopy classes of oriented G-linear diagrams, and 2-morphisms are equivalence classes of oriented G-planar diagrams.
As the first step of showing XB PD is an unbiased semistrict symmetric monoidal 2-category we express 1-and 2-morphisms in terms of string diagrams. We start with modifying oriented G-linear diagrams. Let (M, T, g) be a 1-dimensional compact oriented X-manifold equipped with a generic map f : (M, ∂M) → ([0, 1], {0, 1}). Let Γ be a chambering set subordinate to a Ψ-compatible open cover where Ψ is the induced 1-dimensional graphic.
The set of chambers is modified as the connected components of [0, 1]\(Γ ∪ Ψ ∪ f (T)). Let Ψ G be the 1-dimensional G-graphic Ψ G formed by a Ψ-compatible linear G-data ξ = (ξ 1 , ξ 2 ). We first equip the boundaries of sheets except critical points of f with oriented points using the orientation of M. Then, using ξ we label each sheet with a group element as follows. For any element (a, b) g of ξ 2 , i.e. an open interval (a, b) labeled with g ∈ G, if the open submanifold of M labeled with g by g ∈ [(M, T), (X, x)] mapping to (a, b) under f contains critical points of f then the sheets containing the rightmost 17 critical point as boundary are labeled with g . If there is no critical point on this submanifold then any sheet can be labeled with g . After repeating this process for every element of ξ 2 we label rest of the sheets by identity element e.
We subsequently add labeled points to [0, 1] as follows. If the preimage of a chamber does not have any singularity then the midpoint of that chamber is added to Ψ G . Then we think of a label P g 1 for a sheet in the preimage of this chamber having positive boundary points and labeled by g 1 , similarly we consider the label N g 2 for a sheet having negative boundary points and labeled by g 2 . The label of the added point is then given by the tensor product of these assignments according to trivialization of the chamber (see Figure 27).  Figure 27. String diagram version of Figure 2 Labels of elements of Ψ are also modified. Note that a cap singularity appears in the preimage of a chamber whose right endpoint is labeled with cap. Let V be such a chamber whose trivialization has k 1 -sheets with plus endpoints and k 2 -sheets with minus endpoints. Then, the cap label of the right endpoint of V is replaced with (k 1 + k 2 + 1)-fold tensor product of labels F g 2 , P g 1 , . . . , P g k 1 , N g k 1 +1 , . . . , N g k 1 +k 2 where g ∈ G is the label of cap singularity and each g k i ∈ G is a label of one of the other sheets. Here F 2 stands for Fold-2 18 and the order of tensor product is 17 The corresponding critical value is closest to 1. 18 Note that Fold-2 singularity is the path of cap singularities. determined by the trivialization of the chamber. In the same way, a cup label is replaced with tensor product of labels using F 1 instead of F 2 and using the trivialization of a chamber whose left endpoints is the cup labeled point we started with.
Lastly, we label each element of the chambering set by β σ where σ ∈ S n is the permutation coming from the G-sheet data. It is not hard to see that 1-dimensional compact oriented X-manifold (M, T, g) is still recovered up to an X-homeomorphism over [0, 1] from this more detailed G-linear diagram. To see the changes on an example, we provided the modified version of Example 19 2.1 in Figure 27.
As the next step we express oriented G-planar diagrams as string diagrams. Let (Φ G , Γ, S G ) be an oriented G-planar diagram whose horizontal boundaries are string diagrams for the corresponding oriented G-linear diagrams. As the continuation of G-linear diagrams Fold-1 and Fold-2 labels are replaced with F 1 and F 2 along with superscripts coming from the boundary G-labels. Similarly, labels of Saddle-1, Saddle-2, and Cusp-i are abbreviated as S 1 , S 2 , C i for i = 1, 2, 3, 4 with superscripts coming from their boundary G-labels.
In addition to arcs coming from fold singularities and edges of chambering graph, there are arcs whose boundary on G-linear diagram is labeled with P g 1 , N g 2 for some g 1 , g 2 ∈ G or finite tensor product of these. Moreover, intersection of these arcs produces new 2-morphisms such as τ g 1 ,g 2 + : P g 2 • P g 1 → P g 1 g 2 or its inverse (see Figure 18 for the corresponding oriented 2 -Xsurface). Similarly, there are 2-morphisms given by the intersection of these arcs with fold graphics such as τ N g 1 ⊗P e ,F g 1 2 F g 1 g 2 2 : (N g 1 ⊗ P e ) • F g 1 2 → F g 1 g 2 2 or its inverse. Such intersection points (trivalent vertices) are labeled with τ which has the labels of incoming (upper) arcs as superscript and the labels of outgoing (lower) arcs as subscript. As a continuation of chambering sets we label the edges of chambering graph by β σ where σ ∈ S n is the permutation given by the G-sheet data. The vertices of chambering graph are labeled depending on the position of edges and their labels. Recall that all of the three edges of a trivalent vertex cannot be directed upward or downward. When two of the edges are directed upward, the vertex is labeled with X σ,σ where σ, σ ∈ S n are the labels of these edges. When two of the edges are directed downward then the vertex is labeled with (X σ,σ ) −1 . Similarly, a univalent vertex is labeled with X e if the edge is directed downward and it is labeled with (X e ) −1 if the edge is directed upward.
We place a point to each intersection of an arc with an edge of the chambering graph. This point is labeled with β σ f where σ ∈ S n is the label of the upper edge and f is the label of the arc. Lastly, we label the regions (chambers) of modified G-planar diagrams with words in + andusing the trivializations of oriented sheets. We do not label the regions whose trivializations are empty set. Here note that new arcs can split chambers into smaller regions and in this case these smaller regions still have the same label as the main chamber. That is, regions sharing a piece of new arc as the common boundary have the same label.
An example of a cobordism type 2 -X-surface and its modified G-planar diagram is shown in Figure 28 where the generic map is projection to the page map and numbers on the left hand side of the figure indicate the trivializations of the chambers. The modified oriented G-planar diagram encodes more data to the diagram. It is not hard to see that results in Section 2.3 still hold for modified oriented G-planar diagrams.
Lemma A.1. Chambering sets, graphs, and foams equip XB PD with the structure of an unbiased semistrict symmetric monoidal 2-category.
Proof. Recall that compositions of morphisms in XB PD are given by the concatenation of diagrams.
Since 1-morphisms are isotopy classes of oriented G-linear diagrams and 2-morphisms are equivalence classes of oriented G-planar diagrams the underlying bicategory is a strict 2-category. We know that the symmetric monoidal structure of XB PD is cubical. The transformations α, , r, π, λ, and ρ are identity since 2-morphisms are equivalence classes of G-planar diagrams.
The local models CP and CK 4 of chambering foam shown in Figure 11 give two conditions on the left hand side of Figure 26. For the remaining two conditions recall that a chambering graph can only have univalent and trivalent vertices. Even if we assume the existence of four-valent vertices labeled as in Figure 26 it is not hard to see that these conditions are satisfied by the G-sheet data.
Let (C, β, X) be an unbiased semistrict symmetric monoidal 2-category. The invertibility of modifications {X σ,σ , X e } σ,σ ,e∈S n ,n≥0 and axioms of unbiased semistrict symmetric monoidal 2-category generate relations between structure morphisms. These relations are given on the left hand side of Figure 20 29 in terms of string diagrams. Since chambering foams are responsible for the relations between boundary chambering graphs, on the right hand side of Figure 29 chambering foams corresponding to these relations are shown.
To finish the proof of Theorem 3.2 we need to show that the unbiased semistrict symmetric monoidal 2-category XB PD is freely generated. As the next step we define freely generated G-equivariant unbiased semistrict symmetric monoidal 2-categories and show that XB PD is an example. Such a 2-category is constructed from a certain presentation which we call G-equivariant unbiased semistrict presentation. This type of presentation P consists of four sets (G 0 , G 1 , G 2 , R) together with source and target maps s, t : G 1 → BW uss (G 0 ), and s, t : G 2 → BS uss G (G 1 ) as before. The main difference between G-equivariant unbiased semistrict presentation and the presentation given in Definition A.7 is the constructions of binary words and binary sentences from the generating sets.
Elements of BW uss (G 0 ) have no parentheses around them compared to elements of BW(G 0 ) given in Definition A.2. The main reason for this is the fact that the underlying bicategory of an unbiased semistrict symmetric monoidal 2-category is a strict 2-category and the tensor product is strictly associative. For the same reason in the following there will be no parentheses around the so that nonidentity 1-morphisms of F G uss (P) are given by compositions (horizontal concatenation) of basic 1-morphisms. The 2-morphisms of F G uss (P) are the equivalence classes of string diagrams where two string diagrams are equivalent if they can be related by finitely many (local) moves coming from generating relations R. Part of these moves are shown in Figures 26 and 29. Compositions of morphisms are given by horizontal and vertical concatenations of string diagrams while (cubical) monoidal product is given by stretching out diagrams from different horizontal directions and merging them (see Figure 15).

Remark.
Unbiased semistrict presentation can be defined by taking G = {e}, removing G-labels, and the 2-morphisms involving τ. In this case, the bicategory F uss (P) is called the freely generated unbiased semistrict symmetric monoidal 2-category (see [Sc]). From this point of view a freely generated G-equivariant unbiased semistrict symmetric monoidal 2-category is a freely generated unbiased semistrict symmetric monoidal 2-category where the generating sets and relations include additional elements.
Example A.1. Consider a G-equivariant unbiased semistrict presentation XP = (XG 0 , XG 1 , XG 2 , XR) whose generating sets are given as; XG 0 = { + , -}, XG 1 consists of oriented linear diagrams of { + -, -+} without chambering sets, and XG 2 consists of oriented planar diagrams without chambering graphs of the first two rows of generating 2-morphisms in Figure 16 with only e ∈ G labels. The set of relations XR consists of pairs of oriented G-planar diagrams corresponding to equalities given in Figure 17 and string diagrams given in Figures 26 and 29. The objects of F G uss (XP) are words in + and -. Each 1-morphism is a composition of the following two types of basic 1-morphisms; β σ a,σ(a) where σ ∈ S n and a is a word of length n and an oriented G-linear diagram whose 1-morphism is labeled with I g 1 a 1 ⊗ · · · ⊗ I g k−1 a k−1 ⊗ f g ⊗ I g k+1 a k+1 ⊗ · · · ⊗ I g n a n for some 0 < k ≤ n where a i is either + orand g i ∈ G for all i = 1, . . . , k − 1, k + 1, . . . , n, and f g ∈ {F g 1 , F g 2 } for some g ∈ G. Comparing the latter basic 1-morphism with string diagram in Figure 27 we can identify I g + with P g and I g with N g for any g ∈ G. In this case, note that oriented G-linear diagrams corresponding to and I e -= --N e . The 2-morphisms of F G uss (XP) are equivalence classes of paragraphs PG uss G (XG 2 ) where equivalence classes are generated by the set of generating relations XR. Note that the string diagram interpretation of elements of PG uss G (XG 2 ) coincides with the string diagram interpretation of 2morphisms of XB PD (see Figure 28). More precisely, the sets of labels for regions coincide, in both string diagrams there are two types of basic 1-morphisms whose sets of labels and possible intersecting patterns coincide, and in both string diagrams there are three types of vertices whose sets of labels coincide for each type of vertex. Lastly, equivalence relations on both string diagrams are generated by the same local moves.
Above observation suggests an isomorphism between F G uss (XP) and XB PD namely a symmetric monoidal equivalance preserving the unbiased semistrict symmetric monoidal structures. The following lemma shows that this is indeed the case and finishes the proof of Theorem 3.2.
Proof. Comparing the descriptions of unbiased semistrict symmetric monoidal 2-categories F G uss (XP) and XB PD given above it is not hard to define the 2-functor Θ. On the level of objecs Θ maps nontrivial words in set { + , -} to the finite ordered oriented points given by the words. The empty word is sent to ı.
On 1-morphisms it is enough to specify the images of basic 1-morphisms {β σ , I g 1 a 1 ⊗· · ·⊗ f g ⊗· · ·⊗I g n a n } where σ ∈ S n , n ≥ 0, f ∈ XG 1 , a i ∈ { + , -}, and g, g 1 , . . . , g n ∈ G. For σ ∈ S n and word a the 1-morphism Θ(β σ a,σ(a) ) is an oriented G-linear diagram whose chambering set has only one element labeled by β σ . The latter 1-morphism is mapped to an oriented G-linear diagram described in Example A.1. Recall that 1-morphisms of F G uss (XP) are equivalence classes determined by certain identifications and 1-morphisms of XB PD are isotopy classes of G-linear diagrams. It is not hard to see that above assignments are well-defined on 1-morphisms.
The 2-functor Θ maps an equivalence class [P] of paragraph P ∈ PG uss G (XG 2 )/ ∼ to the equivalence class of string diagram corresponding to P. This assignment makes sense because as mentioned in Example A.1 any representative string diagram can be interpreted in both 2-categories. Since in both 2-categories F G uss (XP) and XB PD string diagrams are considered up to local moves described in Figures 17 21 , 26 and 29 this assignment is well defined.
We use Whitehead theorem for symmetric monoidal bicategories (Theorem 3.1) to show that Θ is a symmetric monoidal equivalence. It is clear that Θ is essentially surjective on objects. We claim that Θ is essentially full on 1-morphisms. To prove this it is enough to show that every oriented Glinear diagram is a composition of 1-morphisms Θ(β σ a,σ(a) ), Θ(I g 1 a 1 ⊗· · ·⊗I g n a n ), Θ(I g 1 a 1 ⊗· · ·⊗ f g ⊗· · ·⊗I g n a n ) for some n ≥ 0, σ ∈ S n , f ∈ XG 1 , and g, g 1 , . . . , g n ∈ G. The compatibility of open cover with 1dimensional graphic and the condition that chambering sets subordinate to such covers imply that elements of chambering graph separate the elements of 1-dimensional graphic. This implies that every G-linear diagram can be written as compositions of above 1-morphisms.
Recall that oriented G-planar diagrams are formed using generic maps and the characteristic maps of 2 -X-surfaces. Thus, any oriented G-planar diagram can be obtained from generating 2-morphisms in Figure 16 under horizontal and vertical concatenations with an addition of a chambering graph. It is not hard to see that each G-labeled generating 2-morphism in the first two rows of Figure 16 can be obtained from composition of e-labeled versions with the last two rows of generating 2-morphisms in Figure 16 (see Figures 19 and 20). Note that in F G uss (XP) the 2-morphisms given by τ correspond to the last two rows of generating 2-morphisms in Figure 16. Thus, for any G-planar diagram there exists a paragraph such that their equivalence classes are matched by Θ. Consequently, θ is fully-faithfull on 2-morphisms.
Thus, Whitehead theorem implies that Θ is an equivalence. By definition Θ preserves the unbiased semistrict symmetric monoidal structures. That is, XB PD is a freely generated G-equivariant unbiased semistrict symmetric monoidal 2-category. By the above remark XB PD is a freely generated unbiased semistrict symmetric monoidal 2-category.