A pair of homotopy-theoretic version of TQFT's induced by a Brown functor
AA FAMILY OF TQFT’S ASSOCIATED WITH HOMOLOGY THEORY
MINKYU KIMA bstract . The purpose of this paper is to give a categorical construction of a homotopy the-oretic analogue of C k -valued TQFT from a Hopf bc k -valued Brown functor. It is formally givenby a projective symmetric monoidal functor from a cospan category of spaces to C k . Here, C k is a category of finite-dimensional, semisimple, cosemisimple, bicommutative Hopf al-gebras over a field k , which is introduced in this paper. As an analogue of the obstructionclass of a projective representation, we construct an obstruction class in the second coho-mology of a cospan category of spaces. A Hopf bc k -valued homology theory consists of asequence of Hopf bc k -valued Brown functors. We show that the obstruction classes induced bysome Hopf bc k -valued homology theory vanish in two cases. One is related with the dimen-sion reduction in the literature of topological field theory ; and the other one is the case ofbounded-below (or bounded-above) homology theories. The latter case gives a generaliza-tion of higher abelian Dijkgraaf-Witten-Freed-Quinn TQFT and bicommutative Turaev-Viro-Barrett-Westbury TQFT. C ontents
1. Introduction 1Acknowledgements 32. Overview of previous study 32.1. Some extensions of Brown functors 42.2. Integrals along bialgebra homomorphisms 53. The category C k
64. Path-integral along (co)span diagrams 85. Applications of the Path-integral 105.1. Brown functor 105.2. Homology theory 116. Vanishing of the obstruction classes 156.1. Dimension reduction 156.2. Bounded-below homology theory 196.3. DWFQ TQFT and TVBW TQFT 20Appendix A. Projective symmetric monoidal functor 22References 241. I ntroduction
The purpose of this paper is to give a categorical construction of a homotopy theoreticanalogue of C k -valued TQFT from a Hopf bc k -valued Brown functor. Here, C k is a symmet-ric monoidal category of finite-dimensional semisimple cosemisimple bicommutative Hopfalgebras over a field k (see section 3 for definition). The symbol Hopf bc k represents the cat-egory of bicommutative Hopf algebras over k and Hopf homomorphisms. We note that thehomotopy theoretic analogue of C k -valued TQFT is formally given by a projective symmetricmonoidal functor from a cospan category of spaces [10] (instead of a cobordism category) to C k . It is projective in the sense that it preserves compositions up to a scalar in k ˚ “ k zt u . As a r X i v : . [ m a t h . A T ] J u l MINKYU KIM an analogue of the obstruction class of a projective representation, we construct an obstruc-tion class in the second cohomology of a cospan category of spaces with coe ffi cients in themultiplicative group k ˚ .A Hopf bc k -valued homology theory consists of a sequence of Hopf bc k -valued Brown functors.We show that the obstruction classes induced by some Hopf bc k -valued homology theory van-ish and obtain two main theorems. In the following statements, the symbol Cosp »ď d p CW fin ˚ q denotes the d -dimensional cospan category of pointed finite CW-spaces for d P N > t8u [10]. We have a faithful bijective functor from the homotopy category Ho p CW fin ˚ , ďp d ´ q q into Cosp »ď d p CW fin ˚ q . The symbol Hopf bc , vol k denotes the category of finite-dimensional semisim-ple cosemisimple bicommutative Hopf algebras and Hopf homomorphisms. The first maintheorem is given as follows : Theorem 1.1.
Let r E ‚ be a Hopf bc k -valued reduced homology theory. Suppose that for anyr P Z , the Hopf algebra r E r p S q is finite-dimensional, semisimple and cosemisimple. Letq P Z . For the induced functor r E q : Ho p CW fin ˚ q Ñ Hopf bc , vol k , there exists a symmetricmonoidal functor Z “ ˆ Z p r E ‚ ; q q : Cosp »ď8 p CW fin ˚ q Ñ C k satisfying the following conditions : (1) The following diagram strictly commutes where W T ` p K q “ K ^ T ` . In particular, wehave Z p K q “ r E q p K ^ T ` q for a space K. (1) Ho p CW fin ˚ q Ho p CW fin ˚ q Hopf bc , vol k Cosp »ď8 p CW fin ˚ q C kW T ` r E q Z (2) For a pointed finite CW-space L, denote by L the cospan diagram p pt Ñ L Ð pt q with some abuse of notations. The corresponding endomorphism Z p L q on Z p pt q – kdefines a homotopy invariant valued in k ˚ . The induced homotopy invariant is givenby Z p L q “ dim r E q p L q P k ˚ . Theorem 1.1 implies that the obstruction class induced by the dimension reduction van-ishes. In the literature of topological field theory, the cartesian product of manifolds witha circle T induces the dimension reduction (for example see [5]). The smash product with T ` “ T > t pt u gives a pointed version.On the one hand, the second main theorem is related with bounded homology theories : Theorem 1.2.
Let r E ‚ be a Hopf bc k -valued reduced homology theory which is bounded below.Suppose that for any r P Z , the Hopf algebra r E r p S q is finite-dimensional, semisimple andcosemisimple. Let q P Z . For the induced functor r E q : Ho p CW fin ˚ q Ñ Hopf bc , vol k , there existsa symmetric monoidal functor Z “ ¯ Z p r E ‚ ; q q : Cosp »ď8 p CW fin ˚ q Ñ C k satisfying the followingconditions : (1) The following diagram strictly commutes. In particular, we have Z p K q “ r E q p K q fora space K. (2) Ho p CW fin ˚ q Hopf bc , vol k Cosp »ď8 p CW fin ˚ q C k r E q Z (2) The induced homotopy invariant is given by Z p L q “ ś l ě dim r E q ´ l p L q p´ q l P k ˚ . FAMILY OF TQFT’S ASSOCIATED WITH HOMOLOGY THEORY 3
A refinement of Theorem 1.1 (Theorem 1.2, resp.) is given by Theorem 6.5, 6.6 (Theorem6.13, resp.). For an arbitrary
Hopf bc k -valued homology theory r E ‚ , we introduce a (possiblyempty) subset Γ p r E ‚ q Ă Z . Then an analogous statement is true for q P Γ p r E ‚ q . In fact, wehave Γ p r E ‚ q “ Z under the assumptions in the main theorems.There are various ways to obtain some nontrivial Hopf bc k -valued homology theories [11].We apply our main theorems to such examples. See Example 6.8, 6.14.The homotopy-theoretic analogue of TQFT in the main theorems induces an n -dimensional C k -valued TQFT for arbitrary n . In fact, we have a canonical functor from the cobordismcategory to the cospan category of pointed finite CW-spaces. See Definition 6.3.The application of Theorem 1.2 to the first singular homology theory gives a generaliza-tion of untwisted Dijkgraaf-Witten-Freed-Quinn TQFT [4] [19] [7] [17] [8] [6] [16] of abeliangroups and Turaev-Viro-Barrett-Westbury TQFT [18] [13] of bicommutative Hopf algebras.The TQFT’s obtained by DWFQ and TVBW factor through Z in Theorem 1.2 for an appro-priate k .The path-integral and state-sum in the literature are formulated in di ff erent ways althoughthey stem from the same idea. We give a new approach to the path-integral and state-sum byusing a notion of integral along bialgebra homomorphisms and our previous results in [9].The vector spaces assgined to surfaces in TVBW theory are naturally isomorphic to theground-state spaces in the Kitaev lattice Hamiltonian model (a.k.a. toric code ) [1]. Thereader is referred to [14] [3] for Kitaev lattice Hamiltonian model. In [12], we gave a gener-alization of the Kitaev lattice Hamiltonian model based on bicommutative Hopf algebras : thesingular (co)homology theory of any finite CW-complex is realized as the ground-state spaceof some lattice Hamiltonian model. As a consequence of this paper and the previous study,the relationship between TVBW theory and Kitaev lattice Hamiltonian model is generalizedto an arbitrary ground field k and pointed finite CW-complexes.The results in this paper hold for cohomology theories in a dual way.This paper is organized as follows. In section 2, we give some overviews of our previousstudies [9] [10]. In section 3, we introduce the symmetric monoidal category C k . In section 4,we introduce path-integral along cospan diagrams based on the results in [9]. In particular,we introduce the path-integral projective functors ˆ PI k and ˇ PI k . In section 5, we apply the path-integral to Hopf bc , vol k -valued Brown functors. In subsection 5.1, we introduce an obstructioncocycle and class for a Brown functor E to extend to a homotopy-theoretic analogue of C k -valued TQFT. In subsection 5.2, we study the obstruction cocycle for the case of homologytheory. Especially, in subsection 5.2.3, we prove some inversion formula of the obstructionclass. In section 6, we compute the obstruction cocycle and class for some special cases andgive concrete examples. In subsection 6.1, we consider the dimension reduction. In subsec-tion 6.2, we consider bounded-below homology theories. In subsection 6.3, we reconstructthe abelian DWFQ TQFT and TVBW TQFT from the results in subsection 6.2. In appendixA, we give an overview of projective symmetric monoidal functors and the obstruction class.A cknowledgements The author was supported by FMSP, a JSPS Program for Leading Graduate Schools inthe University of Tokyo, and JPSJ Grant-in-Aid for Scientific Research on Innovative AreasGrant Number JP17H06461. 2. O verview of previous study
In this section, we give an overview of our previous study. In subsection 2.1, we give maintheorems in [10]. In subsection 2.2, we give an overview of the results in [9].
MINKYU KIM
Some extensions of Brown functors.Definition 2.1.
Let A be an abelian category. A square diagram is a quadruple p g , f , g , f q ofmorphisms in A such that g , f and g , f are composable respectively. Consider a followingsquare diagram l in A .(3) B DA C gf f g The morphism f induces a morphism k l : Ker p f q Ñ Ker p g q . The morphism g induces amorphism c l : Cok p f q Ñ Cok p g q . The square diagram is exact if the morphism k l is anepimorphism and the morphism c l is a monomorphism.The exactness of a square diagram is represented in a familiar way as follows. See [10] forthe proof. Proposition 2.2.
Consider the square diagram (3). We define a chain complex C p l q byA u l Ñ B ‘ C v l Ñ D (4) where u l def . “ p f ‘ p´ f qq ˝ ∆ A and v l def . “ ∇ D ˝ p g ‘ g q . Then the following conditions areequivalent : (1) The square diagram l is exact. (2) The induced chain complex C p l q is exact. Definition 2.3.
For cospan diagrams Λ “ ´ A f Ñ B f Ð A ¯ and Λ “ ˆ A f Ñ B f Ð A ˙ ,we denote by Λ ĺ Λ if A “ A , A “ A and there exists a monomorphism g : B Ñ B in A such that g ˝ f “ f and g ˝ f “ f . For such a monomorphism g , we say that themonomorphism g gives Λ ĺ Λ . The relation ĺ gives a preorder of cospan diagrams in A . Definition 2.4.
We define an equivalence relation « of cospan diagrams in A . We define Λ « Λ if there exists an upper bound of t Λ , Λ u with respect to the preorder ĺ in Definition2.3. In fact, Λ « Λ is equivalent with the condition that there exists a lower bound of t Λ , Λ u . Definition 2.5.
The equivalence relation « is compatible with the direct sum of cospan di-agrams and the composition of cospan diagrams. The equivalence classes form a daggersymmetric monoidal category denoted by Cosp « p A q . It contains A as a subcategory. In thesame manner, one can define a dagger symmetric monoidal category Sp « p A q consisting of(equivalence classes of) span diagrams in A .We have an isomorphism T : Cosp « p A q – Sp « p A q . Under the isomorphism, we denoteby A one of them. We have a bijective and faithful functor ι A : A Ñ A .The following theorem is the main theorem in [10]. Theorem 2.6.
For d P N Y t8u , let E : Ho ` CW fin ˚ , ď d ˘ Ñ A be a d-dimensional A -valuedBrown functor. (1) There exists a unique dagger-preserving symmetric monoidal extension of ι A ˝ E ˝ Σ to Cosp »ď d p CW fin ˚ q . Here, Σ : Ho ´ CW fin ˚ , ďp d ´ q ¯ Ñ Ho ` CW fin ˚ , ď d ˘ is the suspension FAMILY OF TQFT’S ASSOCIATED WITH HOMOLOGY THEORY 5 functor. In other words, the following diagram commutes : (5) Ho p CW ˚ , ďp d ´ q q Ho p CW ˚ , ď d q A Cosp »ď d p CW fin ˚ q A Σ E D ! (2) There exists a unique dagger-preserving symmetric monoidal extension of ι A ˝ E ˝ i to Cosp »ď d p CW fin ˚ q . Here, i : Ho ´ CW fin ˚ , ďp d ´ q ¯ Ñ Ho ` CW fin ˚ , ď d ˘ is the inclusion functor.In other words, the following diagram commutes : (6) Ho p CW ˚ , ďp d ´ q q Ho p CW ˚ , ď d q A Cosp »ď d p CW fin ˚ q A i E D ! Integrals along bialgebra homomorphisms.
In [9], we consider a symmetric monoidalcategory C satisfying some assumptions. Here, we apply the results to the case C “ p Vec k , b k q ,the tensor category of vector spaces over a field k . In this paper, we freely use the results inthis subsection.For a bialgebra homomorphism ξ : A Ñ B , a normalized generator integral along ξ is amorphism µ : B Ñ A in C satisfying some axioms. For the application here, we consideronly bicommutative Hopf algebras. We describe a necessary and su ffi cient condition for theexistence of a normalized generator integral by the kernel of ξ and cokernel of ξ . Theorem 2.7.
Let A , B be bicommutative Hopf algebras and ξ : A Ñ B be a Hopf homomor-phism. There exists a normalized generator integral µ ξ along ξ if and only if the followingconditions hold : (1) the kernel Hopf algebra Ker p ξ q has a normalized integral. (2) the cokernel Hopf algebra Cok p ξ q has a normalized cointegral.Note that if a normalized integral along ξ exists, then it is unique. We introduce an invariant of bicommutative Hopf algebras A , called an inverse volume vol ´ p A q . It is defined as a composition σ A ˝ σ A P k where σ A is a normalized integral and σ A is a normalized cointegral. Definition 2.8.
A bicommutative Hopf algebra
A has a finite volume if(1) It has a normalized integral σ A : k Ñ A .(2) It has a normalized cointegral σ A : A Ñ k .(3) Its inverse volume vol ´ p A q “ p σ A ˝ σ A q P k is invertible.Denote by Hopf bc , vol k the category of bicommutative Hopf algebras with a finite volume andHopf homomorphisms.As a corollary of Theorem 2.7, we obtain the following statement. Corollary 2.9.
Let A , B be bicommutative Hopf algebras with a finite volume. For any bial-gebra homomorphism ξ : A Ñ B, there exists a unique normalized generator integral µ ξ along ξ . Proposition 2.10.
Let ξ : A Ñ B be a Hopf homomorphism between bicommutative Hopfalgebras with a finite volume. See Corollary 3.8 for an equivalent description.
MINKYU KIM (1) If ξ is an epimorphism in the category Hopf bc k , then we have ξ ˝ µ ξ “ id B . In otherwords, µ ξ is a section of ξ in the category Vec k . (2) If ξ is an monomorphism in the category Hopf bc k , then we have µ ξ ˝ ξ “ id A . In otherwords, µ ξ is a retract of ξ in the category Vec k .Proof. It is immediate from Lemma 7.3 [9]. (cid:3)
The inverse volume induces a volume on the abelian category
Hopf bc , bs k consisting of bi-commutative Hopf algebras with a normalized integral and a normalized cointegral. Here,the volume on the abelian category is a generalization of the dimension of vector spaces andthe order of abelian groups, which is also introduced in [9]. Theorem 2.11.
We regard the field k as the multiplicative monoid. Then the inverse volumevol ´ gives an k-valued volume on the abelian category Hopf bc , bs k , i.e. if A Ñ B Ñ C is ashort exact sequence in
Hopf bc , bs k , then we have vol ´ p B q “ vol ´ p A q ¨ vol ´ p C q . By Theorem 2.11,
Hopf bc , vol k Ă Hopf bc , bs k is closed under short exact sequences. In partic-ular, Hopf bc , vol k is also an abelian category. Then the following corollary is immediate fromTheorem 2.11. Corollary 2.12.
The inverse volume vol ´ gives an k ˚ -valued volume on the abelian category Hopf bc , vol k . Here, we regard k ˚ “ k zt u as the multiplicative group. Proposition 2.13.
Consider the exact square diagram (3) for A “ Hopf bc , vol k . Then we have µ g ˝ g “ f ˝ µ f . (7) Proof.
It follows from Theorem 3.4 in [9]. Note that an epimorphism π in the category Hopf bc , vol k has a section in Vec k . In fact, the normalized integral µ π along π is a section of π in Vec k by Lemma 9.3 in [9]. Similarly, any monomorphism in the category Hopf bc , vol k has aretract in in Vec k . (cid:3) The inverse volume of bicommutative Hopf algebras is generalized to Hopf homomor-phisms. For a Hopf homomorphism ξ : A Ñ B , we define x ξ y “ σ B ˝ ξ ˝ σ A P k . By usingthis notion, a composition rule of normalized integrals is represented as follows. Proposition 2.14.
Let ξ : A Ñ B , ξ : B Ñ C be morphisms in the category
Hopf bc , vol k . Thenfor some λ P k ˚ , we have µ ξ ˝ µ ξ “ λ ¨ µ ξ ˝ ξ . (8) Moreover, we have λ “ x cok p ξ q ˝ ker p ξ qy where ker p ξ q : Ker p ξ q Ñ B and cok p ξ q : B Ñ Cok p ξ q are the canonical morphisms.Proof. It follows from Theorem 3.6. or Theorem 12.1. in [9]. (cid:3)
3. T he category C k In this section, we introduce a symmetric monoidal category C k for a field k . Definition 3.1. (1) Consider a cospan diagram Λ “ ´ A ξ Ñ B ξ Ð A ¯ in the category Hopf bc , vol k . In other words, A , A , B are bicommutative Hopf algebras with a finitevolume and ξ , ξ are bialgebra homomorphisms. We define a linear homomorphism ş Λ : A Ñ A by ż Λ def . “ µ ξ ˝ ξ . (9) Here, µ ξ denotes the normalized integral along ξ . Analogously, for a span diagramV, we define a linear homomorphism ş V for a span diagram V in Hopf bc , vol k . FAMILY OF TQFT’S ASSOCIATED WITH HOMOLOGY THEORY 7 (2) Let A , A be bicommutative Hopf algbras with a finite volume. A linear homomor-phism (cid:37) : A Ñ A is realized as a nontrivial integral along a cospan diagram if thereexists a cospan diagram Λ in the category Hopf bc , vol k and λ P k ˚ such that (cid:37) “ λ ¨ ş Λ . Lemma 3.2.
Let A , A , A be bicommutative Hopf algebras with a finite volume. If (cid:37) : A Ñ A and (cid:37) : A Ñ A are realized as a nontrivial integral along a cospan diagram, then thecomposition (cid:37) ˝ (cid:37) is realized as a nontrivial integral along a cospan diagram.Proof. Suppose that (cid:37) “ λ ¨ ş Λ and (cid:37) “ λ ¨ ş Λ for some λ, λ P k ˚ and some cospan diagrams Λ , Λ in the category Hopf bc , vol k . Let Λ “ ´ A ξ Ñ B ξ Ð A ¯ and Λ “ ˆ A ξ Ñ B ξ Ð A ˙ .Recall that the composition Λ ˝ Λ is defined by ˆ A ϕ ˝ ξ Ñ B ϕ ˝ ξ Ð A ˙ where B is given by thepushout diagram (10). We obtain ş Λ ˝ ş Λ “ µ ξ ˝ µ ϕ ˝ ϕ ˝ ξ . Since we have µ ξ ˝ µ ϕ “ λ ¨ µ ϕ ˝ ξ for λ “ x cok p ξ q ˝ ker p ϕ qy P k ˚ by Proposition 2.14, we obtain ş Λ ˝ ş Λ “ λ ¨ ş Λ ˝ Λ , hence (cid:37) ˝ (cid:37) “ λ ¨ ş Λ ˝ Λ where λ “ λ ¨ λ ¨ λ . By definition, the composition (cid:37) ˝ (cid:37) is realized as anontrivial integral along a cospan diagram.(10) B B B A ϕ ϕ ξ ξ (cid:3) Definition 3.3.
We introduce a category C k of bicommutative Hopf algebras with a finitevolume. Its morphisms consist of morphisms realized as a nontrivial integral along a cospandiagram. The composition is well-defined due to Lemma 3.2. We have an obvious embeddingfunctor ι : Hopf bc , vol k ã Ñ C k . Remark 3.4.
Consider a category N of bicommutative Hopf algebras with a finite volumedefined as follows. For two objects A , B of N , the morphism set Mor N p A , B q consists of linearhomomorphisms. Then the category C k is the smallest subcategory of N which contains thefollowing three classes of morphisms : ‚ a Hopf homomorphism ξ : A Ñ B for objects A , B of N , ‚ a morphism µ : A Ñ B in N which is a normalized integral along some Hopf homo-morphism ξ : B Ñ A, ‚ an automorphism on the unit object k in N . Definition 3.5.
Let A be a bicommutative Hopf algebra with a finite volume. Let Λ be acospan diagram of Hopf algebras ´ k η Ñ A ∇ Ð A b A ¯ . We define morphisms i A : k Ñ A b A and e A : A b A Ñ k in C k by i A def . “ ż Λ , (11) e A def . “ a ´ ¨ ż Λ : . (12)Here, a “ vol ´ p A q P k ˚ denotes the inverse volume of A . Proposition 3.6.
The morphisms i A , e A give a symmetric self-duality of A in C k . MINKYU KIM
Proof.
Since A is bicommutative, we have τ ˝ i A “ i A , e A ˝ τ “ e A where τ : A b A Ñ A b A ; x b y ÞÑ y b x . All that remain is to prove that i A , e A form a duality. Let e A “ ş Λ : . Thena zigzag diagram is computed as Figure 1. The third equality holds since µ ∇ is an integralalong the multiplication ∇ . Note that the normalized cointegral σ A is a normalized integralalong the unit η : k Ñ A . The last morphism p id A b σ A q ˝ µ ∇ : A Ñ A is proportionalto a normalized integral along the identity on A . The proportional factor coincides with x id A y “ vol ´ p A q due to Proposition 2.14. It completes the proof since i A , e A are symmetric. (cid:3) F igure Corollary 3.7.
A bicommutative Hopf algebra with a finite volume is finite-dimensional.Moreover, we have vol ´ p A q ´ “ e A ˝ i A so that vol ´ p A q ´ coincides with the dimensionof A modulo the characteristic of k.Proof. Since a vector space with duality is finite-dimensional, we obtain the first claim byProposition 3.6. We prove the second claim. We use the fact that for a morphsim ξ : A Ñ B in Hopf bc , vol k , if ξ is an epimorphism, then we have ξ ˝ µ ξ “ id B . It follows from Lemma 9.3.in [9]. Then we obtain, vol ´ p A q ¨ p e A ˝ i A q “ σ A ˝ ∇ ˝ µ ∇ ˝ η, (13) “ σ A ˝ η, (14) “ . (15) (cid:3) Corollary 3.8.
A bicommutative Hopf algebra A has a finite volume if and only if it is finite-dimensional, semisimple and cosemisimple.Proof.
A finite-dimensional Hopf algebra is semisimple if and only if it has a normalizedintegral [15]. In the same manner, the cosemisimplicity is equivalent with an existence of anormalized cointegral. In Theorem 3.3 [2], it is proved that the composition of a left (right)
IntA -valued integral and a left (right)
IntA -based integral of finite-dimensional Hopf algebrais invertible. The other part follows from Corollary 3.7. (cid:3)
4. P ath - integral along ( co ) span diagrams In this section, we formulate the integral in Definition 3.1 as a projective symmetric monoidalfunctor valued in C k . For the definition of projective symmetric monoidal functor, see theappendix. We give some basic properties of the induced obstruction class. From the obser-vation, we also show some nontriviality of the second cohomology theory of the symmetricmonoidal category Hopf bc , vol k . FAMILY OF TQFT’S ASSOCIATED WITH HOMOLOGY THEORY 9
Lemma 4.1. If Λ ĺ Λ , then we have ş Λ “ ş Λ . In particular, the equivalence Λ « Λ implies ş Λ “ ş Λ .Proof. We use the notations in Definition 2.3. The inverse volume x cok p f q ˝ ker p g qy “ σ Cok p f q ˝ cok p f q ˝ ker p g q ˝ σ Ker p g q is 1 P k since Ker p g q – k and σ Cok p f q is a normalizedcointegral. By Proposition 2.14, we have µ f ˝ µ g “ µ g ˝ f . Hence, we obtain ş Λ “ µ g ˝ f ˝ g ˝ f “ µ f ˝ µ g ˝ g ˝ f . By the second part in Proposition 2.10, we have µ g ˝ g “ Id B so that weobtain ş Λ “ µ f ˝ f “ ş Λ . It completes the proof. (cid:3) Definition 4.2.
We define the path-integral projective functor ˆ PI k . It is a projective symmetricmonoidal functor ˆ PI k : Cosp « p Hopf bc , vol k q Ñ C k which is the identity on objects and assignsˆ PI k pr Λ sq def . “ ş Λ to morphisms r Λ s . The compositions are preserved up to a scalar due toProposition 2.14. It is a well-defined projective functor by Lemma 4.1. Analogously, wedefine a projective symmetric monoidal functor ˇ PI k : Sp « p Hopf bc , vol k q Ñ C k by using thepath-integral along spans in Definition 3.1.Recall Definition 2.5. Under the identification of Cosp « p Hopf bc , vol k q with Sp « p Hopf bc , vol k q ,the above projective functors ˆ PI k , ˇ PI k induce the same projective functor which we denote by PI k : Hopf bc , vol k Ñ C k . Proposition 4.3.
The following conditions are equivalent. (1)
The obstruction cocycle ω p PI k q vanishes. (2) The obstruction class O p PI k q vanishes. (3) The categorical dimension of any bicommutative Hopf algebra with a finite volume is P k.Proof. It su ffi ces to prove the statement for PI k “ ˆ PI k .(1) ñ (2) : It is obvious.(2) ñ (3) : Suppose that O p ˆ PI k q “
1. By Proposition A.7, there exists a symmetricmonoidal functor F : Cosp « p Hopf bc , vol k q Ñ C k such that F – proj ˆ PI k . Let A be a bicommu-tative Hopf algebra with a finite volume. Note that A is self-dualizable in Cosp « p Hopf bc , vol k q .Let d be the categorical dimension of A in Cosp « p Hopf bc , vol k q . Then d is the identity on theunit object k since the endomorphism set of k in Cosp « p Hopf bc , vol k q has only the identity.Since F is a symmetric monoidal functor, F p A q “ A has a trivial categorical dimension in C k .(3) ñ (1) : Suppose that every bicommutative Hopf algebra with a finite volume has atrivial categorical dimension. Let A be a bicommutative Hopf algebra with a finite volume.Then A is dualizable and its categorical dimension coincides with the inverse of the inversevolume vol ´ p A q by Corollary 3.7. By the assumption, the inverse volume vol ´ p A q is trivial.In other words, we have vol ´ p A q “ P k for any bicommutative Hopf algebra A with a finitevolume. By Proposition 11.9 in [9], we have x ξ y “ P k for any homomorphism ξ : A Ñ B between bicommutative Hopf algebras with a finite volume. Therefore, the cocycle ω p ˆ PI k q vanishes by definitions. (cid:3) Corollary 4.4.
Let p be the characteristic of the ground field k. (1)
The obstruction class O p PI k q vanishes if and only if p “ . (2) If p ‰ , then the second cohomology group H p Hopf bc , vol k ; k ˚ q is not trivial.Proof. Note that if p ‰
2, then there exists a bicommutative Hopf algebra with a finite volumewhose categorical dimension is not 1 P k . Such examples could be obtained from group Hopfalgebras. If p “
2, then the categorical dimension of any bicommutative Hopf algebra with afinite volume is 1 P k since the dimension should be invertible in k by Corollary 3.7. It provesthe first claim. By the first claim, the class O p PI k q ‰ p ‰
2. It proves the second claim. (cid:3)
Corollary 4.5.
Let p be the characteristic of the ground field k. (1)
If p ‰ , , then the second cohomology group H p Hopf bc , vol k ; F ˚ p q is not trivial. (2) If p “ , then the second cohomology group H p Hopf bc , vol k ; Q ą q is not trivial.Proof. Let G be the multiplicative group F ˚ p if p ‰ , Q ą if p “
0. Then the obstructioncocycle ω p PI k q has coe ffi cients in G due to Corollary 3.7. It induces a class r ω p PI k qs P H p Hopf bc , vol k ; G q . The induced map H p Hopf bc , vol k ; G q Ñ H p Hopf bc , vol k ; k ˚ q assigns O p PI k q to the class r ω p PI k qs ‰
1. By Corollary 4.4, r ω p PI k qs P H p Hopf bc , vol k ; G q is nontrivial. Itcompletes the proof. (cid:3)
5. A pplications of the P ath - integral In this section, we apply the path-integral projective functor to
Hopf bc , vol k -valued Brownfunctors. Roughly speaking, Brown functors induce a homotopy-theoretic analogue of C k -valued TQFT. In general, the obtained TQFT preserves compositions up to a scalar in k ˚ . Forhomology theories, we deduce some formulas to compute the induced obstruction classes.See subsection 5.2,5.1. Brown functor.
Let E : Ho p CW fin ˚ , ď d q Ñ Hopf bc , vol k be a d -dimensional Hopf bc , vol k -valuedBrown functor where d P N Y t8u . Definition 5.1.
Let ˆ E be the cospanical extension of E ˝ i by the second part of Theorem 2.6.We define a projective symmetric monoidal functor ˆ PI k p E q def . “ ˆ PI k ˝ ˆ E . Cosp »ď d p CW fin ˚ q ˆ E ÝÑ Cosp « p Hopf bc , vol k q ˆ PI k ÝÑ C k . (16)Analogously, we define a projective symmetric monoidal functor ˇ PI k p E q def . “ ˇ PI k ˝ ˇ E where ˇ E is the spanical extension of E ˝ Σ by the first part of Theorem 2.6. Cosp »ď d p CW fin ˚ q ˇ E ÝÑ Sp « p Hopf bc , vol k q ˇ PI k ÝÑ C k . (17) Remark 5.2.
The projective symmetric monoidal functor ˆ PI p E q : Cosp »ď d p CW ˚ q Ñ C k sat-isfies the following (strictly) commutative diagram by definitions. (18) Ho p CW ˚ , ďp d ´ q q Ho p CW ˚ , ď d q Hopf bc , vol k Cosp »ď d p CW ˚ q C ki E ˆ PI p E q In fact, the commutativity of the diagram follows from Theorem 2.6 and definitions. Theanalogous statement for ˇ PI p E q is true. In that case, consider the suspension functor insteadof the inclusion functor i. Definition 5.3.
We define two cohomology classes in H p Cosp »ď d p CW fin ˚ q ; k ˚ q by,ˇ O p E q def . “ O p ˇ PI k p E qq , (19) ˆ O p E q def . “ O p ˆ PI k p E qq . (20)In the following theorem, for Θ , see Definition A.5. FAMILY OF TQFT’S ASSOCIATED WITH HOMOLOGY THEORY 11
Theorem 5.4.
The cohomology class ˆ O p E q vanishes if and only if there exists θ P Θ p ˆ PI p E qq such that the symmetric monoidal functor θ ´ ¨ ˆ PI p E q : Cosp »ď d p CW fin ˚ q Ñ C k satisfies thefollowing (strictly) commutative diagram. (21) Ho p CW fin ˚ , ďp d ´ q q Ho p CW fin ˚ , ď d q Hopf bc , vol k Cosp »ď d p CW fin ˚ q C ki E θ ´ ¨ ˆ PI p E q The analogous statement for ˇ O p E q is true. In that case, consider the suspension functorinstead of the inclusion functor i. The proof follows from the following lemma.
Definition 5.5.
Let θ be a 1-cochain of Cosp »ď d p CW fin ˚ q with coe ffi cients in k ˚ . The cochain θ is good if ι ˚ θ is trivial where ι : Ho p CW fin ˚ , ďp d ´ q q Ñ Cosp »ď d p CW fin ˚ q is the embedding functor. Lemma 5.6.
If the obstruction class ˆ O p E q vanishes, then there exists θ P Θ p ˆ PI p E qq which isgood.Proof. By the assumption, we have Θ p ˆ PI p E qq ‰ H . We prove that there exists θ P Θ p ˆ PI p E qq such that θ p ι p f qq “ f : K Ñ L . Choose any θ P Θ p ˆ PI p E qq . By direct calculation, wehave ω p ι p f q , ι p g qq “
1. Hence, we obtain θ p ι p f qq θ p ι p g qq “ θ p ι p g ˝ f qq . (22)For Λ “ ´ K f Ñ L f Ð K ¯ , we define θ p Λ q “ θ p ι p f qq ´ θ p Λ q θ p ι p f qq . (23)By direct calculation, we obtain δ θ “ ω where we use (22). In other words, θ P Θ p ˆ PI p E qq .Then for f : K Ñ L , we have θ p ι p f qq “ θ p ι p f qq ´ θ p ι p f qq θ p ι p Id L qq , (24) “ . (25) (cid:3) Proof of Theorem 5.4.
The obstruction class of a projective symmetric monoidal func-tor vanishes if and only if it is naturally isomorphic to a symmetric monoidal functor. Forexample, see Proposition A.7. We call such a symmetric monoidal functor by a lift of theprojective symmetric monoidal functor. The nontrivial part for the proof of the theorem is toverify whether there exists a lift extending the functor E ˝ i . See Remark 5.2First suppose that the obstruction ˆ O p E q vanishes. Then there exists θ P Θ p ˆ PI p E qq such that θ p ι p f qq “ f : K Ñ L by Lemma 5.6. For such θ , the lift θ ´ ¨ ˆ PI p E q satisfies the claimby definition. The converse is obvious. It completes the proof of Theorem 5.4.5.2. Homology theory.
Definitions.
Let r E ‚ be a Hopf bc k -valued reduced homology theory. Proposition 5.7.
For ´8 ď q ď q ď 8 , the following conditions are equivalent : (1) Let q be an integer such that q ď q ď q . For any pointed finite CW-space K suchthat dim K ď p q ´ q q , the Hopf monoid r E q p K q has a finite volume. In other words,the restriction r E q : Ho p CW fin ˚ , ďp q ´ q q q Ñ Hopf bc k factors through Hopf bc , vol k . (2) The q-th coe ffi cient r E q p S q has a finite volume for any integer q such that q ď q ď q .Here, S denotes the pointed 0-dimensional sphere. (3) Let r be any integer such that ď r ď p q ´ q q and . If q is an integer such thatq ` r ď q ď q , then the Hopf algebra r E q p K q has a finite volume for any pointedr-dimensional finite CW-space K.Proof. (1) obviously implies (2).We prove (3) from (2). If r “
0, then K is 0-dimensional so that by (2), the Hopf algebra r E q p K q has a finite volume for q ď q ď q . Hence (3) holds for r “
0. Let r be an integersuch that 0 ď r ă p q ´ q q . Suppose that if q is an integer such that q ` r ă q ď q , then theHopf algebra r E q p K q has a finite volume for a pointed r -dimensional finite CW-space K . Let L be a p r ` q -dimensional finite CW-complex. Let q be an integer such that q ` r ` ď q ď q .Consider the long exact sequence associated with the pair p L , L p r q q where L p r q is the r -skeletonof L . r E q ` p L { L p r q q Ñ r E q p L p r q q Ñ r E q p L q Ñ r E q p L { L p r q q Ñ r E q ´ p L p r q q (26)By the assumption, the Hopf algebras r E q p L p r q q , r E q ´ p L p r q q have a finite volume. Moreover thequotient complex L { L p r q is homeomorphic to a finite bouquet Ž S r ` of the pointed p r ` q -dimensional spheres. From the isomorphism r E q ` p Ž S r ` q – r E q p Ž S r q and r E q p Ž S r ` q – r E q ´ p Ž S r q and the assumption, the Hopf algebras r E q ` p Ž S r ` q and r E q p Ž S r ` q have afinite volume. The Hopf algebra r E q p L q has a finite volume since Hopf bc , vol k Ă Hopf bc k is closedunder short exact sequences. It proves (3).We prove (1) from (3). Let q be an integer such that q ď q ď q . Let K be a pointedfinite CW-space with dim K ď p q ´ q q . Put r “ dim K . Since 0 ď r ď p q ´ q q and q ` r ď q ď q by definitions, the Hopf algebra r E q p K q has a finite volume by (3). Itcompletes the proof. (cid:3) Definition 5.8.
Denote by Γ p r E ‚ q the set of integer q P Z such that the q -th coe ffi cient r E q p S q has a finite volume. For q P Γ p r E ‚ q , we define d p r E ‚ ; q q def . “ p q ´ m p r E ‚ ; q qq ě m p r E ‚ ; q q def . “ inf t r P Γ p r E ‚ q ; r ď r ď q ñ r P Γ p r E ‚ qu ě ´8 . (27) Corollary 5.9.
For q P Γ p r E ‚ q , the restriction r E q : Ho p CW fin ˚ , ď d q Ñ Hopf bc k factors through Hopf bc , vol k where d “ d p r E ‚ ; q q . The induced symmetric monoidal functor r E q : Ho p CW fin ˚ , ď d q Ñ Hopf bc , vol k is a d-dimensional Hopf bc , vol k -valued Brown functor.Proof. It is immediate from Proposition 5.7. (cid:3)
Isomorphism between path-integrals along spans and cospans.
Lemma 5.10.
Let A be a small abelian category and B be an abelian subcategory of A .Let r E ‚ be an A -valued reduced homology theory. Suppose that r E q ` p K q lies in B for apointed finite CW-space K with dim K ď p d ` q . Denote by r E q ` : Ho p CW fin ˚ , ďp d ` q q Ñ B the induced functor. In particular, by the suspension isomorphism, r E q ` induces a functor r E q : Ho p CW fin ˚ , ď d q Ñ B such that r E q p K q “ r E q p K q . Denote by ˆ X the cospanical extensionof E ˝ i for E “ r E q in Theorem 2.6. Denote by ˇ Y the spanical extension of E ˝ Σ forE “ r E q ` in the first part of Theorem 2.6. Then the following diagram commutes up to a FAMILY OF TQFT’S ASSOCIATED WITH HOMOLOGY THEORY 13 natural isomorphism. (28)
Cosp »ď d p CW fin ˚ q Cosp « p B q Cosp »ďp d ` q p CW fin ˚ q Sp « p B q ˆ X ˇ Y T Proof.
We show that the dagger-preserving symmetric monoidal functor T ˝ ˇ Y is a cospanicalextension of the symmetric monoidal functor r E q : CW fin ˚ , ď d Ñ B . In fact, for a cospan Λ “ p K Ñ L Ð K q in the category CW fin ˚ , the square diagram (29) is exact by Proposition2.2. Hence, we obtain T p r E q ` p T Σ p Λ qqq ĺ r E q p Λ q where we identify r E q p K q with r E q ` p Σ K q bythe suspension isomorphism. Especially we obtain T p r E q ` p T Σ p Λ qqq « r E q p Λ q by definitions.(29) r E q p K q r E q p L q r E q ` p C p f _ f qq r E q p K q By the uniqueness in Theorem 2.6, we obtain a natural isomorphism ˆ X – T ˝ ˇ Y . (cid:3) Theorem 5.11.
Let r E ‚ be a Hopf bc k -valued reduced homology theory. Let q P Γ p r E ‚ q such that p q ` q P Γ p r E ‚ q . Denote by i : Cosp »ď d p CW fin ˚ q Ñ Cosp »ďp d ` q p CW fin ˚ q the inclusion functorwhere d “ d p r E ‚ ; q q . Then we have a natural isomorphism of projective symmetric monoidalfunctors in the strong sense, i ˚ p ˇ PI p r E q ` qq – ˆ PI p r E q q . (30) Proof.
Let ˆ F be the cospanical extension of r E q ˝ i and ˇ G be the spanical extension of r E q ` ˝ Σ .Consider the following diagram of functors. By considering A “ Hopf bc k , B “ Hopf bc , vol k , r E q ` “ r E q ` and r E q “ r E q in Lemma 5.10, the left diagram commutes up to the suspensionisomorphism. Furthermore, the right diagram commutes up to a natural isomorphism in thestrong sense. By composing the natural isomorphisms, we obtain the results.(31) Cosp »ď d p CW fin ˚ q Cosp « p Hopf bc , vol k q C k Cosp »ďp d ` q p CW fin ˚ q Sp « p Hopf bc , vol k q C ki ˆ F ˆ PI k Tˇ G ˇ PI k (cid:3) By definition of the obstruction class, we obtain the following formula.
Corollary 5.12.
Let q P Γ p r E ‚ q such that p q ` q P Γ p r E ‚ q . Then we havei ˚ p ˇ O p r E q ` qq “ ˆ O p r E q q . (32)5.2.3. Inversion formula of obstruction class.
Definition 5.13.
For q P Γ p r E ‚ q , we define a normalized 1-cochain θ q p r E ‚ q of the symmetricmonoidal category Cosp »ď d p CW fin ˚ q with coe ffi cients in k ˚ . Here, d “ d p r E ‚ ; q q . Let r Λ s be amorphism in Cosp »ď d p CW fin ˚ q where Λ “ ´ K f Ñ L f Ð K ¯ . Then, θ q p r E ‚ qpr Λ sq def . “ vol ´ p r E q p C p f qqq P k ˚ . (33) Here, C p f q denotes the mapping cone of the pointed map f . Since C p f q is a complex withthe dimension lower than d , the bicommutative Hopf monoid r E q p C p f qq has a finite volumeso that (33) is well-defined. Note that the 1-cochain θ q p r E ‚ q is good in the sense of Definition5.5. Remark 5.14.
By Corollary 3.7, the definition is equivalent with θ q p r E ‚ qpr Λ sq def . “ dim p r E q p C p f qqq ´ P k ˚ . (34) Lemma 5.15.
Suppose that we have an exact sequence in the abelian category
Hopf bc , vol k :C B Ñ A Ñ B Ñ C B Ñ A ´ . (35) Then we have xB y ¨ xB y “ vol ´ p A q ¨ vol ´ p B q ´ ¨ vol ´ p C q . (36) Proof.
Note that the exact sequence induces the following exact sequence : k Ñ Im pB q B Ñ A Ñ B Ñ C B Ñ Coim pB q Ñ k . (37)Since the inverse volume is a volume on the abelian category Hopf bc , vol k by Corollary 2.12, weobtain the following equation : vol ´ p Im pB q ¨ vol ´ p B q ¨ vol ´ p Coim pB qq “ vol ´ p A q ¨ vol ´ p C q . (38)Since vol ´ p Im pB q “ xB y and vol ´ p Coim pB qq “ xB y by Proposition 9.9 in [9], the claimis proved. (cid:3) Lemma 5.16.
Let r E ‚ be a Hopf bc k -valued homology theory. For q P Γ p r E ‚ q , we have ω p ˆ PI p r E q qq ¨ ω p ˇ PI p r E q qq “ δ p θ q p r E ‚ qq . (39) Proof.
Let d “ d p r E ‚ ; q q . Consider composable morphisms r Λ s , r Λ s in Cosp »ď d p CW fin ˚ q withthe representatives, Λ “ ´ K f Ñ L f Ð K ¯ , Λ “ ˆ K f Ñ L f Ð K ˙ . (40)We introduce notations of maps associated with the composition Λ ˝ Λ and T Σ p Λ q , T Σ p Λ q , T Σ p Λ ˝ Λ q following Figure 2, 3. For simplicity, denote by α “ ω p ˆ PI p r E q qq and β “ ω p ˇ PI p r E q qq .Recall the definition, α pr Λ s , r Λ sq “ x cok p r E q p f qq˝ ker p r E q p g qqy . Let B f , g q ` : r E q ` p C p g qq Ñ r E q p C p f qq be the connecting morphism in the long exact sequence associated with the map-ping cone sequence C p f q Ñ C p g ˝ f q Ñ C p g q . Note that the images of cok p r E q p f qq ˝ ker p r E q p g qq and B f , g q ` are canonically isomorphic with each other. In fact, each long ex-act sequence associated with g and f respectively implies that the image of cok p r E q p f qq ˝ ker p r E q p g qq coincides with the image of the composition r E q ` p C p g qq B g q ` Ñ r E q p L q r E q p f q Ñ r E q p L q .Then we have α pr Λ s , r Λ sq “ x cok p r E q p f qq ˝ ker p r E q p g qqy “ xB f , g q ` y by Proposition 9.9 [9].In the same manner, we calculate β pr Λ s , r Λ sq to obtain β pr Λ s , r Λ sq “ xB k , h q ` y . Note thatthe suspension of the mapping cone sequence C p f q Ñ C p g ˝ f q Ñ C p g q is homotopyequivalent with the mapping cone sequence C p k q Ñ C p h ˝ k q Ñ C p h q . It implies that xB f , g q y “ xB k , h q ` y “ β pr Λ s , r Λ sq .Above all, we obtain α pr Λ s , r Λ sq¨ β pr Λ s , r Λ sq “ xB f , g q ` y¨xB f , g q y . By applying Lemma 5.15,it is verified that α pr Λ s , r Λ sq ¨ β pr Λ s , r Λ sq coincides with δ p θ q p r E ‚ qqpr Λ s , r Λ sq . It completesthe proof. FAMILY OF TQFT’S ASSOCIATED WITH HOMOLOGY THEORY 15 F igure igure (cid:3) Theorem 5.17.
Let r E ‚ be a Hopf bc k -valued reduced homology theory. For q P Γ p r E ‚ q , we have ˆ O p r E q q “ ˇ O p r E q q ´ . (41) Proof.
It follows from Lemma 5.16. (cid:3)
Corollary 5.18.
If q , p q ` q P Γ p r E ‚ q , then we havei ˚ p ˇ O p r E q ` qq “ ˇ O p r E q q ´ . (42) Proof.
It is immediate from Corollary 5.12 and Theorem 5.17. (cid:3)
6. V anishing of the obstruction classes
In this section, we give our main results which imply the main theorems. We verify thatsome obstruction classes vanish mainly by using the formulas in the previous section. Oneclass of examples is obtained from a dimension reduction of homology theories, and theother class is from bounded homology theories. The latter examples contain a generalizationof DWFQ and TVBW TQFT’s.6.1.
Dimension reduction.Definition 6.1.
Let X be a pointed finite CW-space. For a pointed finite CW-space K , wedefine W X p K q def . “ K ^ X .By the functoriality of W X , an A -valued reduced homology theory r E ‚ induces a homologytheory. We denote it by W ˚ X r E ‚ . In particular, W ˚ X r E q p K q “ r E q p K ^ X q . Lemma 6.2.
Let r E ‚ be an A -valued homology theory. We have an isomorphism of homologytheories, W ˚ T ` r E ‚ – r E ‚ ‘ r E ‚´ .Proof. Denote a pointed n -sphere by S n . Denote T a 1-sphere without basepoint. By themapping cone sequence S Ñ T ` Ñ S , we obtain a split exact sequence for a pointed finiteCW-space K : 0 Ñ W ˚ S r E q p K q Ñ W ˚ T ` r E q p K q Ñ W ˚ S r E q p K q Ñ . (43)Furthermore, we obtain an isomorphism of homology theories, W ˚ T ` r E ‚ – W ˚ S r E ‚ ‘ W ˚ S r E ‚ .Here, ‘ denotes the biproduct in A . By the natural isomorphism X – X ^ S , we obtainan isomorphism of homology theories, W ˚ S r E ‚ – r E ‚ . With some careful treatment of signs,we also obtain an isomorphism, W ˚ S r E ‚ – r E ‚´ by using the suspension isomorphism. Itcompletes the proof. (cid:3) From now on, we consider A “ Hopf bc k the category of bicommutative Hopf algebras. Fixa Hopf bc k -valued reduced homology theory r E ‚ . We put r F ‚ “ W ˚ T ` r E ‚ . Proposition 6.3.
We have Γ p r F ‚ q “ Γ p r E ‚ q X ´ Γ p r E ‚ q ` ¯ . (44) Proof.
Note that the biproduct in the abelian category A “ Hopf bc k is the tensor product ofHopf algebras. By Lemma 6.2, it su ffi ces to prove that Γ p r E ‚ b r E ‚´ q “ Γ p r E ‚ q X Γ p r E ‚´ q . Γ p r E ‚ q X Γ p r E ‚´ q Ă Γ p r E ‚ b r E ‚´ q is clear. Let q P Γ p r E ‚ b r E ‚´ q , i.e. the Hopf algebra r E q p S q b r E q ´ p S q has a finite volume. We claim that r E q p S q and r E q ´ p S q have a finitevolume. More generally, for a bicommutative Hopf algebras A , B , if the tensor product A b B has a finite volume, then A has a finite volume. In fact, the composition of the inclusion i : A Ñ A b B and the normalized cointegral on A b B induces a normalized cointegral on A . In the same manner, A is proved to have a normalized integral by using the projection A b B Ñ A . Likewise, the Hopf algebra B has a normalized integral and a normalizedcointegral. In particular, the inverse volume of A , B are defined. By Theorem 2.11, we obtain vol ´ p A q ¨ vol ´ p B q “ vol ´ p A b B q . The inverse volume vol ´ p A b B q is invertible so that vol ´ p A q is invertible. It proves that A has a finite volume. (cid:3) Proposition 6.4.
Let q P Γ p r F ‚ q . (1) The obstruction classes ˇ O p r F q q , ˆ O p r F q q vanish. (2) ω p ˇ PI p r F q qq “ δ p θ q ´ p r E ‚ qq . (3) ω p ˆ PI p r F q qq “ δ p i ˚ p θ q p r E ‚ qqq . Here, i is the inclusion from the p d ´ q -dimensionalcospan category of spaces to the d-dimensional one.Proof. The first part follows from the last two statements, but here we give a way to computethe classes by using formulas of obstruction classes. We first prove that the class ˇ O p r F q q van-ishes. By Proposition 6.3, q P Γ p r F ‚ q implies q P Γ p r E ‚ q X ´ Γ p r E ‚ q ` ¯ so that the obstructionclasses ˇ O p r E q ´ q , ˇ O p r E q q are defined. By Lemma 6.2, we have ˇ O p r F q q “ i ˚ p ˇ O p r E q qq ¨ ˇ O p r E q ´ q .By Corollary 5.18, we obtain ˇ O p r F q q “
1. By Theorem 5.17, the obstruction class ˆ O p r F q q alsovanishes.Note that the Brown functor r F q is isomorphic to i ˚ r E q b r E q ´ by definitions where i : Ho p CW fin ˚ , ďp d ´ q q Ñ Ho p CW fin ˚ , ď d q is the inclusion. Hence, ˇ PI p r F q q – ˇ PI p i ˚ r E q q b ˇ PI p r E q ´ q – i ˚ ˇ PI p r E q q b ˇ PI p r E q ´ q where the last i denotes the inclusion from p d ´ q -dimensional cospan FAMILY OF TQFT’S ASSOCIATED WITH HOMOLOGY THEORY 17 category of spaces to d -dimensional one. By Theorem 5.11, we obtain an isomorphism ofprojective symmetric monoidal functors in the strong sense :ˇ PI p r F q q – ˆ PI p r E q ´ q b ˇ PI p r E q ´ q . (45)By Lemma 5.16, we obtain ω p ˇ PI p r F q qq “ δ p θ q ´ p r E ‚ qq .We compute ω p ˆ PI p r F q qq as follows. We have ω p ˆ PI p r F q qq “ δ p θ q p r F ‚ qq ¨ ω p ˇ PI p r F q qq ´ byLemma 5.16. By the previous result, we obtain ω p ˆ PI p r F q qq “ δ p θ q p r F ‚ q ¨ θ q ´ p r E ‚ q ´ q . ByLemma 6.2, we have θ q p r F ‚ q “ i ˚ p θ q p r E ‚ qq ¨ θ q ´ p r E ‚ q so that we obtain the claim. (cid:3) Theorem 6.5.
For q P Γ p r F ‚ q , let d “ d p r E ‚ ; q q “ d p r F ‚ ; q q ` . Then there exists a canon-ical symmetric monoidal functor Z “ ˇ Z p r E ‚ ; q q : Cosp »ďp d ´ q p CW fin ˚ q Ñ C k satisfying thefollowing conditions : (1) The diagram below commutes strictly. (46) Ho p CW fin ˚ , ďp d ´ q q Ho p CW fin ˚ , ďp d ´ q q Hopf bc , vol k Cosp »ďp d ´ q p CW fin ˚ q C k Σ r F q Z (2) The induced homotopy invariant is given byZ p L q “ dim p r E q ´ p L qq P k . (47) Proof.
The existence of such Z satisfying only the first condition is easily deduced from thefirst part of Proposition 6.4. To construct such Z also satisfying the second condition, we needto choose some concrete lift of the projective functor obtained from path-integral. In fact, bythe second part of Proposition 6.4, we could choose θ q ´ p r E ‚ q as a canonical complementary1-cochain. Then Z “ θ q ´ p r E ‚ q ´ ¨ ˇ PI p r F q q satisfies all the conditions by definitions. (cid:3) In a parallel way, one can use the path-integral along cospan diagrams. In that case, we usethe third part of Proposition 6.4 instead of the second part. We give the result without proofas follows.
Theorem 6.6.
For q P Γ p r F ‚ q , let d “ d p r E ‚ ; q q “ d p r F ‚ ; q q ` . Then there exists a canon-ical symmetric monoidal functor Z “ ˆ Z p r E ‚ ; q q : Cosp »ďp d ´ q p CW fin ˚ q Ñ C k satisfying thefollowing conditions : (1) The diagram below commutes strictly. (48) Ho p CW fin ˚ , ďp d ´ q q Ho p CW fin ˚ , ďp d ´ q q Hopf bc , vol k Cosp »ďp d ´ q p CW fin ˚ q C ki r F q Z (2) The induced homotopy invariant is given byZ p L q “ dim p r E q p L qq P k . (49) Proposition 6.7.
If q , p q ` q P Γ p r F ‚ q , then the restriction of ˇ Z p r E ‚ ; q ` q coincides with ˆ Z p r E ‚ ; q q . In other words, if d “ d p r E ‚ ; q q , then the diagram below commutes strictly : (50) Cosp »ďp d ´ q p CW fin ˚ q C k Cosp »ď d p CW fin ˚ q ˆ Z p r E ‚ ; q q ˇ Z p r E ‚ ; q ` q Proof.
It follows from Theorem 5.11 and definitions of ˆ Z p r E ‚ ; q q , ˇ Z p r E ‚ ; q ` q . (cid:3) Example 6.8.
In [11], we give three ways to construct some nontrivial class of
Hopf bc k -valuedhomology theories. For such homology theories r E ‚ , one can obtain homotopy-theoretic ana-logue of TQFT’s by Theorem 6.5, 6.6. In particular, they are parametrized by the set Γ p r F ‚ q .(1) Let r D ‚ be a generalized homology theory. The group Hopf algebra functor withcoe ffi cients in k induces a Hopf bc k -valued homology theory r E ‚ “ k r D ‚ . Then Γ p r E ‚ q consists of q P Z such that r D q p S q is finite and its order is coprime to the characteristicof k . For example, let k be a field with characteristic zero. Consider the generalizedreduced homology theory r D ‚ “ r π s ‚ induced by the sphere spectrum. It is well-knownthat the q -th coe ffi cient r π sq p S q is finite if q ‰
0. Hence, the obtained
Hopf bc k -valuedhomology theory r E ‚ “ k r D ‚ satisfies Γ p r E ‚ q “ Z zt u . By Proposition 6.3, we obtain Γ p r F ‚ q “ Γ p r E ‚ q X p Γ p r E ‚ q ` q “ Z zt , u . For an integer q ě
2, we have d p r E ‚ ; q q “p q ´ q by definitions. The symmetric monoidal functors ˇ Z p r E ‚ ; q q in Theorem 6.5 andˆ Z p r E ‚ ; q q in 6.6 are defined on p q ´ q -dimensional cospan category of spaces. For apointed finite CW-space K with dim K ď p q ´ q , we haveˇ Z p r E ‚ ; q qp K q “ r F q ´ p K q – k `r π sq ´ p K q ˆ r π sq ´ p K q ˘ , (51) ˆ Z p r E ‚ ; q qp K q “ r F q p K q – k `r π sq p K q ˆ r π sq ´ p K q ˘ . (52) Furthermore, for a pointed finite CW-space L with dim L ď p q ´ q , we have ˇ Z p r E ‚ ; q qp L q “| r π sq ´ p L q| and ˆ Z p r E ‚ ; q qp L q “ | r π sq p L q| .(2) An ordinary homology theory with coe ffi cients in a Hopf algebra provides a classof examples. Let A be a bicommutative Hopf algebra with a finite volume. Thenthere exists an ordinary homology theory with coe ffi cients in A which we denote by r E ‚ p´q “ r H ‚ p´ ; A q . By definitions, we have Γ p r E ‚ q “ Z so that Γ p r F ‚ q “ Z byProposition 6.3. For q P Γ p r F ‚ q , we have d p r E ‚ ; q q “ 8 . The symmetric monoidalfunctors ˇ Z p r E ‚ ; q q in Theorem 6.5 and ˆ Z p r E ‚ ; q q in 6.6 are defined on -dimensionalcospan category of spaces. For a pointed finite CW-space K , we haveˇ Z p r E ‚ ; q qp K q – r H q ´ p K ; A q b r H q ´ p K ; A q , (53) ˆ Z p r E ‚ ; q qp K q – r H q p K ; A q b r H q ´ p K ; A q . (54)(3) An exponential functor gives a class of examples. Let h be another field. Considera bicommutative Hopf algebra A over k with an h -action α . Then there exists anassignment of a Hopf bc k -valued homology theory r E ‚ “ p A , α q r D ‚ to a Vec fin h -valuedhomology theory r D ‚ . Note that a Vec fin h -valued homology theory is nothing but ageneralized homology theory such that r D r p K q are finite-dimensional vector spacesover h . If A is not a group Hopf algebra, then the obtained homology theory r E ‚ isnot induced by the group Hopf algebra functor as the first part above. By definitions,we have Γ p r E ‚ q “ Z and d p r E ‚ ; q q “ 8 for any q P Γ p r E ‚ q . The symmetric monoidal FAMILY OF TQFT’S ASSOCIATED WITH HOMOLOGY THEORY 19 functors ˇ Z p r E ‚ ; q q in Theorem 6.5 and ˆ Z p r E ‚ ; q q in 6.6 are defined on -dimensionalcospan category of spaces. For a pointed finite CW-space K , we have isomorphismsof Hopf algebras ˇ Z p r E ‚ ; q qp K q – â b q ´ p K q A b â b q ´ p K q A , (55) ˆ Z p r E ‚ ; q qp K q – â b q p K q A b â b q ´ p K q A . (56) Here, we fix a basis b q p K q of the vector space r D q p K q . Furthermore, for a pointedfinite CW-space L with dim L ď p q ´ q , we have ˇ Z p r E ‚ ; q qp L q “ p dim A q dim r D q ´ p L q and ˆ Z p r E ‚ ; q qp L q “ p dim A q dim r D q p L q . Remark 6.9.
Note that the set Γ p r F ‚ q could be empty in general even if Γ p r E ‚ q is not. Forexample, consider reduced K-theory r K ‚ . Then the group Hopf algebra functor over a fieldk induces a Hopf bc k -valued homology theory r E ‚ “ k r K ‚ . Then Γ p r E ‚ q “ Z ` since thecoe ffi cient Hopf algebra k r K q p S q has a finite volume if and only if q is odd by definitions. ByProposition 6.3, we obtain Γ p r F ‚ q “ H . Remark 6.10. In [6] [8] [16] , gauge fields in DWFQ theory are described by classifyingmaps. In this sense, DWFQ theory is a sigma-model whose target space is the classifyingspace. The first part of Example 6.8 or 6.14 gives more examples of possible sigma-modelswhich naturally have a quantization by path-integral. In fact, a spectrum in algebraic-topological sense plays a role of the target space. It is well-known that a spectrum induces ageneralized cohomology theory of CW-spaces. Such generalized cohomology theory is con-structed by a homotopy set of maps from (the suspension spectrum of) spaces to the spectrum. Bounded-below homology theory.
Let r E ‚ be a Hopf bc k -valued reduced homology the-ory which is bounded below. In other words, there exists q P Z such that q ă q implies r E q p K q – k for any pointed finite CW-space K . In this subsection, we prove that the obstruc-tion class ˆ O p r E q q vanishes for q P Γ p r E ‚ q such that m p r E ‚ ; q q “ ´8 (see Definition 5.8). Morestrongly, we give a canonical complementary 1-cochain for the projective functor ˆ PI p r E q q .There is an analogous result for bounded above homology theory and appropriate degree q . In that case, the obstruction class ˇ O p r E q q “ O p ˇ PI p r E q qq vanishes due to an analogue ofProposition 6.12 Definition 6.11.
We define a normalized 1-cochain θ ď q p r E ‚ q of the symmetric monoidal cat-egory Cosp »ď8 p CW fin ˚ q with coe ffi cients in the multiplicative group k ˚ by θ ď q p r E ‚ q def . “ ź l ě θ q ´ l p r E ‚ q p´ q l (57)Here, the normalized 1-cochains θ q p r E ‚ q are defined in Definition 5.13. Note that the 1-cochain θ ď q p r E ‚ q is good in the sense of Definition 5.5 since so does θ q p r E ‚ q . Proposition 6.12.
Let r E ‚ be a Hopf bc k -valued reduced homology theory which is boundedbelow. For q P Γ p r E ‚ q such that m p r E ‚ ; q q “ ´8 , we have ω p ˆ PI p r E q qq “ δ θ ď q p r E ‚ q . (58) In particular, the obstruction class ˆ O p r E q q vanishes.Proof. By Lemma 5.16, we obtain ω p ˆ PI p r E q qq “ ω p ˇ PI p r E q qq ´ ¨ δ p θ q p r E ‚ qq . By Theorem 5.11,we have a natural isomorphism ˇ PI p r E q q – ˆ PI p r E q ´ q in the strong sense so that we obtain ω p ˆ PI p r E q qq “ ω p ˆ PI p r E q ´ qq ´ ¨ δ p θ q p r E ‚ qq . We repeat this formula until the integer q . Sincethe homology theory r E ‚ is assumed to be bounded below, we obtain the claim. (cid:3) Theorem 6.13.
Let q P Γ p r E ‚ q such that m p r E ‚ ; q q “ ´8 . Then there exists a canonicalsymmetric monoidal functor Z “ ¯ Z p r E ‚ ; q q : Cosp »ď8 p CW fin ˚ q Ñ C k satisfying the followingconditions : (1) The diagram below commutes strictly. (59) Ho p CW fin ˚ q Hopf bc , vol k Cosp »ď8 p CW fin ˚ q C k r E q Z (2) The induced homotopy invariant is given byZ p L q “ ź l ě dim p r E q ´ l p L qq p´ q l P k . (60) Proof.
Note that d p r E ‚ ; q q “ q ´ m p r E ‚ ; q q “ 8 . We choose some concrete lift of the projectivefunctor obtained from path-integral. In fact, by Proposition 6.4, we could choose θ ď q p r E ‚ q asa canonical complementary 1-cochain. Then ¯ Z p r E ‚ ; q q “ θ ď q p r E ‚ q ´ ¨ ˇ PI p r E q q satisfies all theconditions by definitions. (cid:3) Example 6.14.
Recall Example 6.8.(1) Let r D ‚ be a generalized homology theory which is bounded below. Then the induced Hopf bc k -valued homology theory r E ‚ “ k r D ‚ is bounded below. Let q P Z such that q ă q implies that the q -th coe ffi cient r D q p S q is finite and its order is coprime to thecharacteristic of k . Then q P Γ p r E ‚ q satisfies m p r E ‚ ; q q “ ´8 by definitions. Hence,for such q P Γ p r E ‚ q we obtain a symmetric monoidal functor Z “ ¯ Z p r E ‚ ; q q satisfyingthe conditions in Theorem 6.13. In particular, the induced homotopy invariant is givenby Z p L q “ ś l ě | r D q ´ l p L q| p´ q l P k .(2) Let A be a bicommutative Hopf algebra over k . Let r E ‚ p´q “ r H ‚ p´ ; A q be the reducedordinary homology theory with coe ffi cients in A . The homology theory r E ‚ is boundedbelow. Suppose that the Hopf algebra A has a finite volume. Then we have Γ p r E ‚ q “ Z and m p r E ‚ ; q q “ ´8 for any q P Γ p r E ‚ q by definitions. The application of Theorem6.13 gives a generalization of Dijkgraaf-Witten-Freed-Quinn TQFT for an abeliangroups and Turaev-Viro-Barrett-Westbury TQFT for a bicommutative Hopf algebra.See subsection 6.3.(3) Let h be another field. Consider a bicommutative Hopf algebra A over k with an h -action α . Then there exists an assignment of a Hopf bc k -valued homology theory r E ‚ “p A , α q r D ‚ to a Vec fin h -valued homology theory r D ‚ . By definitions, we have Γ p r E ‚ q “ Z and m p r E ‚ ; q q “ ´8 for any q P Γ p r E ‚ q . If r D ‚ is bounded below, then so is r E ‚ .Theapplication of Theorem 6.13 gives a symmetric monoidal functor Z “ ¯ Z p r E ; q q for q P Γ p r E ‚ q . In particular, we have Z p L q “ p dim A q χ ď q p L ; r D ‚ q where χ ď q p L ; r D ‚ q “ ř l ě p´ q l ¨ dim r D q ´ l p L q .6.3. DWFQ TQFT and TVBW TQFT.Definition 6.15.
Let d P N > t8u . For n P N such that n ď d , let Cob n be the n -dimensional cobordism category of unoriented smooth compact manifolds. We define asymmetric monoidal functor Φ n , d : Cob n Ñ Cosp ď d p CW fin ˚ q . It assigns M ` “ M > t pt u FAMILY OF TQFT’S ASSOCIATED WITH HOMOLOGY THEORY 21 to a closed p n ´ q -manifold M . It assigns the homotopy equivalence of the cospan diagraminduced by embeddings to an n -cobordism N from M to M : Φ n , d pr N ; M , M sq def . “ r M ` ã Ñ N ` Ð â M ` s . (61)It is obviously well-defined. Especially, the composition is preserved since boundaries havea collar neighborhood. Definition 6.16.
Consider k “ C , the complex number field. Note that any bicommutativeHopf algebra A with a finite volume is a function Hopf algebra by Wedderburn’s theorem. Inother words, there naturally exists a group with an isomorphism A – C G . Via the isomor-phism, the Hopf algebra A has a unique Hilbert inner product whose orthonormal basis isgiven by δ g P C G , g P G , i.e. the delta functionals. It induces a symmetric monoidal functor U Hil : C C Ñ Hil fin where Hil fin is the category of finite-dimensional Hilbert spaces. Proposition 6.17.
Let n P N such that n ď d. Let k “ C . Every symmetric monoidal functorfrom Cosp »ď d p CW fin ˚ q to C k induces a Hil fin -valued n-dimensional TQFT by Φ n , d and U Hil . Remark 6.18.
Even if the ground field k is general, there is an assignment of nondegeneratepairing to bicommutative Hopf algebras with a finite volume. Denote by
Vec ssd k the categoryof vector spaces p V , i : k Ñ V b V , e : V b V Ñ k q equipped with a symmetric self-duality.Recall Definition 3.5. We define a symmetric monoidal functor U ssd : C k Ñ Vec ssd k byU ssd p A q def . “ p U p A q , i A , e A q where U p A q is the underlying vector space of A. By Proposition3.6, it is well-defined. Analogously to Proposition 6.17, every symmetric monoidal functorfrom Cosp »ď d p CW fin ˚ q to C k induces a Vec ssd k -valued n-dimensional TQFT by Φ n , d and U ssd . DWFQ TQFT for abelian groups.
For a finite abelian group G , let A “ C G be thefunction Hopf algebra. Note that the Hopf algebra A has a finite volume. Proposition 6.19.
Put Z “ ¯ Z p r E ‚ ; 1 q “ θ ´ ď p r E ‚ q ¨ ˇ PI p r E q in Theorem 6.13. Then the composi-tion U Hil ˝ Z ˝ Φ , n : Cob n Ñ Hil fin is the untwisted DWFQ TQFT associated with G. Here,U Hil , Φ , n are defined in Definition 6.15, 6.16.Proof. We sketch the proof. Note that the set of isomorphism classes of principal bundlesover a pointed space K is given by r H p K ; G q . For a cobordism B from M to M , we havethe following cospan diagram (62). Then the path-integral along this cospan is nothing butthe finite path-integral in the literature. Moreover, we have a natural isomorphism C r H p K ; G q – r H p K ; C G q as Hopf algebras.(62) C r H p B ; G q C r H p M ; G q C r H p M ; G q (cid:3) In the same manner, the untwisted higher abelian Dijkgraaf-Witten (unextended) TQFT[16] is reproduced by our result. The proof is similar with that of the above proposition. Notethat the TQFT in [16] is extended to codimension two.
Proposition 6.20.
Put Z “ ¯ Z p r E ; q q “ θ ´ ď q p r E ‚ q ¨ ˇ PI p r E q q . The composition U Hil ˝ Z ˝ Φ , n isthe untwisted higher abelian Dijkgraaf-Witten TQFT obtained by q-form gauge fields . TVBW TQFT for bicommutative Hopf algebras.
Let k be an algebraically closed fieldof characteristic zero. Let A be a bicommutative Hopf algebra over k with a finite volume.Then A is finite-dimensional by Corollary 3.7. Moreover, A is involutory since it is bicommu-tative. Hence, the category of finite-dimensional left A modules Rep p A q is spherical fusioncategory in the sense of [13]. Proposition 6.21.
For Z “ ¯ Z p r E ‚ ; 1 q “ θ ´ ď p r E ‚ q ¨ ˇ PI p r E q in Theorem 6.13, the compositionU ˝ Z ˝ Φ , coincides with the TVBW TQFT associated with the category Rep p A q . Here,U : C k Ñ Vec fin k denotes the forgetful functor.Proof. We sketch the proof. If we denote by G the set of group-like elements of A , then wehave an isomorphism A – kG since k is algebraically closed and A is finite-dimensional,semisimple. The construction of TVBW TQFT implies that it is isomorphic to DWFQ TQFTassociated with Hom p G , k ˚ q – ˆ G , i.e. the Pontryagin dual of G . By subsubsection 6.3.1, itis isomorphic to our TQFT induced by the first singular homology theory with coe ffi cients H p´ ; A q since we have k ˆ G – kG – A . (cid:3) A. P rojective symmetric monoidal functor
Fix a symmetric monoidal category E satisfying the following assumptions :(1) The endomorphism set of the unit object End E p q consists of automorphisms, i.e. End E p q “ Aut E p q .(2) For any morphism f : x Ñ y in E , the scalar-multiplication λ P End E p q ÞÑ λ ¨ f P Mor E p x , y q (induced by the monoidal structure) is injective.The typical example of such E in this paper is the symmetric monoidal category Hopf bc , vol k defined in subsubsection 2.2. Definition A.1.
Let D be a symmetric monoidal category. Consider a triple F “ p F o , F m , Ψ q satisfying followings :(1) F o assigns an object F o p x q of E to an object x of D .(2) F m assigns a morphism F m p f q : F o p x q Ñ F o p y q of E to a morphism f : x Ñ y of D .(3) F m p Id x q “ Id F o p x q .(4) For composable morphisms f , g of D , there exists λ P End E p E q such that p F m p g q ˝ F m p f qq “ λ ¨ F m p g ˝ f q .(5) Ψ is a natural isomorphism Ψ x , x : F o p x q b F o p x q Ñ F o p x b x q which is compatiblewith the unitors, associators and symmetry of D , E .We refer to such an assignment F as a projective symmetric monoidal functor from D to E .As a notation, we denote by F : D Ñ E . By an abuse of notations, we also denote by F o p x q “ F p x q and F m p f q “ F p f q . Definition A.2.
Let F , F : D Ñ E be projective symmetric monoidal functors. Then a natural isomorphism Φ : F Ñ F in the projective sense is given as follows :(1) For any object x of D , we have an isomorphism Φ p x q : F p x q Ñ F p x q in E .(2) For a morphism f : x Ñ y of D , there exists λ, λ P End E p E q such that λ ¨ F p f q ˝ Φ p x q “ λ ¨ Φ p y q ˝ F p f q .If there exists a projective natural isomorphism from F to F , we denote by F – proj F .A natural isomorphism Φ : F Ñ F of projective symmetric monoidal functors gives anatural isomorphism in the strong sense if the second condition holds for λ “ λ “ Id E . Inthat case, we write F – F . FAMILY OF TQFT’S ASSOCIATED WITH HOMOLOGY THEORY 23
Definition A.3.
Let F : D Ñ E be a projective symmetric monoidal functor. We definea 2-cochain ω p F q P C p D ; Aut E p qq as follows. For a 2-simplex p f , g q of D , we define ω p F qp f , g q P Aut E p q by F p g q ˝ F p f q “ ω p F qp f , g q ¨ F p g ˝ f q . (63)Then the assignment ω p F q is a well-defined 2-cochain with coe ffi cients in Aut E p q . Proposition-Definition A.4.
Let F : D Ñ E be a projective symmetric monoidal functor.Then the 2-cochain ω p F q is a normalized 2-cocycle. We define O p F q def . “ r ω p F qs P H nor p D ; Aut E p qq . (64) Proof.
The 2-cocycle condition is immediate from the associativity of compositions : By thefollowing equality, we obtain ω p f , f q ¨ ω p f ˝ f , f q ¨ F p f ˝ f ˝ f q “ ω p f , f q ¨ ω p f , f ˝ f q ¨ F p f ˝ f ˝ f q . F p f q ˝ p F p f q ˝ F p f qq “ p F p f q ˝ F p f qq ˝ F p f q (65)By the assumption on E in the beginning of this section, we obtain ω p f , f q ¨ ω p f ˝ f , f q “ ω p f , f q ¨ ω p f , f ˝ f q . (66)It proves that the 2-cochain ω is a 2-cocycle.Moreover we have ω p F qp Id y , f q “ “ ω p F qp f , Id x q for any morphism f : x Ñ y since wehave F p Id x q “ Id F p x q for arbitrary object x of D . Hence, the 2-cocycle ω p F q is normalized.It completes the proof. (cid:3) Definition A.5.
Suppose that the automorphism group of the unit object
Aut E p q is an abeliangroup. Let F : D Ñ E be a projective symmetric monoidal functor. We define a set Θ p F q asfollows : Θ p F q def . “ t θ P C p D ; Aut E p qq ; δ θ “ ω p F qu . (67)An element θ P Θ p F q is called a complementary 1-cochain for a projective symmetricmonoidal functor F . Note that any θ P Θ p F q is normalized. It is obvious that O p F q “ P H nor p D ; Aut E p qq if and only if Θ p F q ‰ H . Definition A.6.
Let F : D Ñ E be a projective symmetric monoidal functor. For θ P Θ p F q , we define a lift of F by a complementary 1-cochain θ as a symmetric monoidal functor p θ ´ ¨ F q : D Ñ E given by p θ ´ ¨ F qp x q def . “ F p x q (68) p θ ´ ¨ F qp f q def . “ θ p f q ´ ¨ F p f q . (69)The assignment p θ ´ ¨ F q is verified to be a symmetric monoidal functor by definitions. Proposition A.7.
Let F : D Ñ E be a projective symmetric monoidal functor. The inducedobstruction is trivial, i.e. O p F q “ if and only if there exists a symmetric monoidal functorF : D Ñ E such that F – proj F .Proof. Suppose that O p F q “ P H p D ; Aut E p qq . By definition of o p F q , we choose anormalized 1-cochain θ P Θ p F q . We define a symmetric monoidal functor F “ p θ ´ ¨ F q We have a natural isomorphism F Ñ F between projective symmetric monoidal functors.In fact, the identity Id F p x q : F p x q Ñ F p x q “ F p x q for any object x of D gives a naturalisomorphism between projective symmetric monoidal functors. It completes the proof. (cid:3) R eferences [1] Benjamin Balsam and Alexander Kirillov Jr. Kitaev’s lattice model and Turaev-Viro TQFTs. arXivpreprint arXiv:1206.2308 , 2012.[2] Yuri Bespalov, Thomas Kerler, Volodymyr Lyubashenko, and Vladimir Turaev. Integrals for braided Hopfalgebras. Journal of Pure and Applied Algebra , 148(2):113–164, 2000.[3] Oliver Buerschaper, Juan Mart´ın Mombelli, Matthias Christandl, and Miguel Aguado. A hierarchy oftopological tensor network states.
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