IINTEGRALS ALONG BIMONOID HOMOMORPHISMS
MINKYU KIMA bstract . We introduce a notion of an integral along a bimonoid homomorphism as a simul-taneous generalization of the integral and cointegral of bimonoids. The purpose of this paperis to characterize an existence of a specific integral, called a normalized generator integral ,along a bimonoid homomorphism in terms of the kernel and cokernel of the homomorphism.We introduce a notion of a volume on an abelian category as a generalization of the dimen-sion of vector spaces and the order of abelian groups. In applications, we show that there ex-ists a nontrivial volume partially defined on a category of bicommutative Hopf monoids. Thevolume yields a notion of Fredholm homomorphisms between bicommutative Hopf monoids,which gives an analogue of the Fredholm index theory. This paper gives a technical prelimi-nary of our subsequent paper about a construction of TQFT’s. C ontents
1. Introduction 2Acknowledgements 42. Integrals 42.1. Integrals of bimonoids 42.2. Integrals along bimonoid homomorphisms 52.3. Generator integrals 83. A refinement of the main result 83.1. Existence of a normalized integral 83.2. Compositions 93.3. Applications to bicommutative Hopf monoids 104. Some objects associated with action 124.1. Invariant object 124.2. Stabilized object 124.3. Stable monoidal structure 135. Normal homomorphism 146. Small bimonoid and integral 177. Integral along bimonoid homomorphism 227.1. Uniqueness of normalized integral 227.2. Proof of Theorem 3.1 238. Computation of
Int p ξ q a r X i v : . [ m a t h . QA ] F e b MINKYU KIM
Hopf bc p C q ntroduction The notion of integrals of a bialgebra is introduced by Larson and Sweedler [11]. It is ageneralization of the Haar measure of a group. The integral theory has been used to studybialgebras or Hopf algebras [11] [17] [16]. The notion of bialgebras are generalized to bi-monoids in a symmetric monoidal category C [1] [12]. The integral theory is generalized tothe categorical settings and used to study bimonoids or Hopf monoids [21].In this paper, we introduce a notion of an integral along a bimonoid homomorphism ina symmetric monoidal category C . For bimonoids A and B , an integral along a bimonoidhomomorphism ξ : A Ñ B is a morphism µ : B Ñ A satisfying some axioms (see Definition2.4). The integrals along bimonoid homomorphisms simultaneously generalize the notions ofintegral and cointegral of bimonoid : the notion of integral (cointegrals, resp.) of a bimonoidcoincides with that of integrals along the counit (unit, resp.).The purpose of this paper is to characterize the existence of a normalized generator integralalong a bimonoid homomorphism in terms of the kernel and cokernel of the homomorphism.The reader is referred to Definition 2.4 and 2.11 for the definition of normalized integrals andgenerator integrals respectively. If C satisfies some assumptions (see (Assumption 0,1,2) insubsubsection 3.3.1), then the existence of a normalized generator integral is characterized asfollows for a homomorphism between bicommutative Hopf monoids : Theorem 1.1.
Let A , B be bicommutative Hopf monoids in C and ξ : A Ñ B be a Hopfhomomorphism. Then there exists a normalized generator integral µ ξ along ξ if and only ifthe following conditions hold : (1) the kernel Hopf monoid Ker p ξ q has a normalized integral. (2) the cokernel Hopf monoid Cok p ξ q has a normalized cointegral.Moreover, if a normalized integral exists, then it is unique. It follows from Theorem 3.3.The assumptions on C automatically hold if C “ Vec b F , the tensor category of (possiblyinfinite-dimensional) vector spaces over a field F . See Example 3.8.We note that even if A , B are not bicommutative or it is not obvious whether C satisfies (As-sumption 0,1,2), we have analogous results under some assumptions on the homomorphism ξ . We give such refinement of the main result in subsection 3.1.1.In applications, we examine the category Hopf bc , ‹ p C q of bicommutative Hopf monoids witha normalized integral and cointegral. We prove that the category Hopf bc , ‹ p C q is an abeliansubcategory of Hopf bc p C q and closed under short exact sequences. See subsubsection 3.3.1 .We introduce a notion of volume on an abelian category A as a generalization of thedimension of vector spaces and the order of abelian groups. It is an invariant of objects By Theorem 6.13, the category
Hopf bc , ‹ p C q coincides with Hopf bc , bs p C q in subsubsection 3.3.1. NTEGRALS ALONG BIMONOID HOMOMORPHISMS 3 in A compatible with short exact sequences (see Definition 3.7). As another application to Hopf bc , ‹ p C q , we construct an End C p q -valued volume vol ´ on the abelian category A “ Hopf bc , ‹ p C q . Here is the unit object of C and the endomorphism set End C p q is an abelianmonoid induced by the symmetric monoidal structure of C .By using the volume vol ´ , we introduce a notion of Fredholm homomorphisms betweenbicommutative Hopf monoids as an analogue of the Fredholm operator [7] (see Definition14.3). We study its index which is robust to some finite perturbations (see Proposition 15.8).Furthermore, we construct a functorial assignment of integrals to Fredholm homomorphisms.This paper gives a technical preliminary of our subsequent paper. Indeed, we use the resultsin this paper to give a generalization of the untwisted abelian Dijkgraaf-Witten theory [5][20] [6] and the bicommutative Turaev-Viro TQFT [19] [3]. We will give a systematic way toconstruct a sequence of TQFT’s from (co)homology theory. The TQFT’s are constructed byusing path-integral which is formulated by some integral along bimonoid homomorphisms.As a corollary of our subsequent paper, if the volume vol ´ p A q of an object A in Hopf bc , ‹ p C q is invertible in End C p q , then the underlying object of A is dualizable in C and its categoricaldimension coincides with the inverse of vol ´ p A q . If C is a rigid symmetric monoidal categorywith split idempotents, then the inverse volume of any Hopf monoid is invertible [21]. It isnot obvious whether the inverse volume is invertible or not in general. Note that we do notassume a duality on objects of C .There is another approach to a generalization of (co)integrals of bimonoids. In [21],(co)integrals are defined by a universality. It is not obvious whether our integrals could begeneralized by universality.We expect that the result in this paper could be applied to topology through another ap-proach. There is a topological invariant of 3-manifolds induced by a finite-dimensional Hopfalgebra, called the Kuperberg invariant [9] [10]. In particular, if the Hopf algebra is involu-tory, then it is defined by using the normalized integral and cointegral of the Hopf algebra.The organization of this paper is as follows. In subsection 2.1, we review integrals of bi-monoids. In subsection 2.2, we introduce the notion of (normalized) integral along bimonoidhomomorphisms and give some basic properties. In subsection 2.3, we introduce a notion ofgenerator integral and give some basic properties. In section 3, we give a refinement of themain result without precise definitions. In subsection 4.1, 4.2, we introduce the notion of in-variant objects and stabilized objects respectively. In subsection 4.3, we introduce the notionof (co, bi) stable monoidal structure. In section 5, we introduce the notion of (co,bi)normalityof bimonoid homomorphisms and give some basic properties. In section 6, we introduce thenotion of (co, bi) small bimonoids and examine it in terms of an existence of normalized(co)integrals. In subsection 7.1, we prove the uniqueness of a normalized integral. In sub-section 7.2, we prove Theorem 3.1. In section 8, by using a normalized generator integral,we show an isomorphism between the set of endomorphisms on the unit object and theset of integrals. In subsection 9.1, we prove a key lemma for Theorem 3.2. In subsection9.2, we introduce two notions of (weakly) well-decomposable homomorphism and (weakly)Fredholm homomorphism, and prove Theorem 3.2. In section 10, we prove Theorem 3.4. Insubsection 11.1, we introduce the inverse volume of some bimonoids. In subsection 11.2, weintroduce the inverse volume of some bimonoid homomorphisms. In subsection 12, we proveTheorem 3.6. In subsection 13.1, we give some conditions where Ker p ξ q , Cok p ξ q inherits a(co)smallness from that of the domain and the target of ξ . In subsection 13.2, we prove Theo-rem 3.11. In section 14, we introduce the notion of volume on an abelian category and studybasic notions related with it. In subsection 15.1, we prove that the inverse volume is a volumeon the category of bicommutative Hopf monoids. In subsection 15.2, we construct functorial MINKYU KIM integrals for Fredholm homomorphisms. In appendix A, we give our convention for stringdiagrams and a brief review of monoids in a symmetric monoidal category.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. I ntegrals
Integrals of bimonoids.
In this subsection, we review the notion of integral of a bi-monoid and its basic properties.We give some remark on terminologies. The integral in this paper is called a Haar integral[2], [4], [13], an
Int p H q -based integral [21] or an integral-element [8]. The cointegral inthis paper is called an Int p H q -valued integral in [21] or integral-functional [8]. In fact, thosenotions introduced in [21], [8] are more general ones which are defined by a universality. Definition 2.1.
Let A be a bimonoid. A morphism ϕ : Ñ A is a left integral of A if it satisfya commutative diagram (1). We denote by Int l p A q the set of left integrals of A . A morphism ϕ : Ñ A is a right integral if it satisfy a commutative diagram (2). We denote by Int r p A q the set of right integrals of A . A morphism ϕ : Ñ A is an integral if it is a left integraland a right integral. A left (right) integral is normalized if it satisfies a commutative diagram(3). For a bimonoid A , we denote by σ A : Ñ A the normalized integral of A if exists. It isunique for A as we will discuss in this section.We define cointegral of a bimonoid in a dual way. Denote by Int r p A q , Int l p A q , Int p A q the setof right integrals, left integrals and integrals of A . We denote by Cont r p A q , Coint l p A q , Coint p A q the set of right cointegrals, left cointegrals and cointegrals of A .(1) b A A b AA b A ϕ b id A ϕ b (cid:15) A ∇ A r A (2) A b A b A b A A id A b ϕ(cid:15) A b ϕ ∇ A l A (3) A ϕ (cid:15) A Remark 2.2.
The commutative diagrams in Definition 2.1 can be understood by equations ofsome string diagrams in Figure 1 where the null diagram is the identity on the unit . Proposition 2.3.
Let A be a bimonoid in a symmetric monoidal category, C . If the bimonoidA has a normalized left integral σ and a normalized right integral σ , then σ “ σ and it is anormalized integral of the bimonoid A. In particular, if a normalized integral exists, then it isunique. NTEGRALS ALONG BIMONOID HOMOMORPHISMS 5 F igure Proof.
It is proved by definitions. It follows from more general results in Proposition 7.1. Infact, a normalized left (right) integral of A is a normalized left (right) integral along counit of A . (cid:3) Integrals along bimonoid homomorphisms.
In this subsection, we introduce the no-tion of an integral along a homomorphism and give its basic properties. They are definedfor bimonoid homomorphisms whereas the notion of (co)integrals is defined for bimonoids.In fact, it is a generalization of (co)integrals. See Proposition 2.7. We also give a typicalexample in Example 2.6.
Definition 2.4.
Let A , B be bimonoids in a symmetric monoidal category C and ξ : A Ñ B be a bimonoid homomorphism. A morphism µ : B Ñ A in C is a right integral along ξ ifthe diagrams (4), (5) commute. A morphism µ : B Ñ A in C is a left integral along ξ if thediagrams (6), (7) commute. A morphism µ : B Ñ A in C is an integral along ξ if it is a rightintegral along ξ and a left integral along ξ . An integral (or a right integral, a left integral) is normalized if the diagram (8) commutes.We denote by Int l p ξ q , Int r p ξ q , Int p ξ q the set of left integrals along ξ , the set of right integralsalong ξ , the set of integrals along ξ respectively.(4) B b A A b A AB b B B µ b id A id B b ξ ∇ A ∇ B µ (5) B B b B A b BA A b A ∆ B µ µ b id B ∆ A id A b ξ (6) A b B A b A AB b B B id A b µξ b id B ∇ A ∇ B µ MINKYU KIM (7)
B B b B B b AA A b A ∆ B µ id B b µ ∆ A ξ b id A (8) A BB A ξξ µ ξ
Remark 2.5.
The commutative diagrams in Definition 2.4 can be understood by using somestring diagrams in Figure 2. From now on, we freely use these string diagrams. The stringdiagram is explained briefly in appendix A. F igure Example 2.6.
Consider C “ Vec b F . Let G , H be (possibly infinite) groups and (cid:37) : G Ñ Hbe a group homomorphism. It induces a bialgebra homomorphism ξ “ (cid:37) ˚ : A Ñ B for thegroup Hopf algebras A “ F G and B “ F H. Suppose that the kernel Ker p (cid:37) q is finite. Put µ : B Ñ A by µ p h q “ ÿ (cid:37) p g q“ h g . (9) Then µ is an integral along the homomorphism ξ . If the order of Ker p (cid:37) q , say N, is coprime tothe characteristic of F , then N ´ ¨ µ is a normalized integral. Proposition 2.7.
Let A be a bimonoid in a symmetric monoidal category C . We have,Int r p (cid:15) A q “ Int r p A q , Int l p (cid:15) A q “ Int l p A q , (10) Int r p η A q “ Coint r p A q , Int l p η A q “ Coint l p A q . (11) NTEGRALS ALONG BIMONOID HOMOMORPHISMS 7
In particular, we have Int p (cid:15) A q “ Int p A q , (12) Int p η A q “ Coint p A q . (13) Under these equations, the normalized condition is preserved.Proof.
We only prove that
Int r p (cid:15) A q “ Int r p A q and leave the other parts to the readers.Suppose that µ P Int r p (cid:15) A q . Then by (4), we have ∇ A ˝ p µ b id A q “ r A ˝ p µ b (cid:15) A q , i.e. µ is aright integral of the bimonoid A .Suppose that σ P Int r p A q . Then σ satisfies the commutative diagram (4). On the otherhand, (5) is automatic since B “ .Note that µ P Int r p (cid:15) A q is normalized ,i.e. (cid:15) A ˝ µ ˝ (cid:15) A “ (cid:15) A , if and only if (cid:15) A ˝ µ “ id . (cid:3) Proposition 2.8.
If a bimonoid homomorphism ξ : A Ñ B is an isomorphism, then we have ξ ´ P E p ξ q . Here, E denotes either Int r , Int l or Int. In particular, id A P E p id A q for anybimonoid A.Proof. We only prove the case of E “ Int r and leave the other parts to the readers. Themorphism ξ ´ satisfies the axiom (4) by the following equalitites. ∇ A ˝ p ξ ´ b id A q “ ∇ A ˝ p ξ ´ b ξ ´ q ˝ p id B b ξ q (14) “ ξ ´ ˝ ∇ B ˝ p id B b ξ q . (15)Here we use the assumption that ξ is a bimonoid homomorphism. Similarly, (5) is verified.Hence, ξ ´ P Int r p ξ q . (cid:3) Proposition 2.9.
We have E p id q “ End C p q . Here, E denotes either Int r , Int l or Int.Proof. We only prove the case of E “ Int r and leave the other parts to the readers. For ϕ P End C p q , the morphism ϕ satisfies the axiom (4) with respect to ξ “ id : ∇ ˝ p ϕ b id q “ r ˝ p ϕ b id q (16) “ ϕ ˝ ∇ . (17)Similarly, the axiom (5) is verified. It implies that ϕ P Int r p id q . (cid:3) Proposition 2.10.
The composition of morphisms induces a map,E p ξ q ˆ E p ξ q Ñ E p ξ ˝ ξ q ; p µ , µ q ÞÑ µ ˝ µ . (18) Here, E denotes either Int r , Int l or Int.Proof. We only prove the case of E “ Int r . Let ξ : A Ñ B , ξ : B Ñ C be bimonoidhomomorphisms and µ P Int r p ξ q and µ P Int r p ξ q . The composition µ ˝ µ satisfies he axiom(5) as follows : ∇ A ˝ pp µ ˝ µ q b id A q “ ∇ A ˝ p µ b id A q ˝ p µ b id A q (19) “ µ ˝ ∇ B ˝ p µ b ξ q (20) “ µ ˝ µ ˝ ∇ C ˝ p id A b p ξ ˝ ξ qq . (21)It is similarly verified that the composition µ ˝ µ satisfies the axiom (5). Hence, we obtain µ ˝ µ P Int r p ξ ˝ ξ q . (cid:3) MINKYU KIM
Generator integrals.
In this subsection, we define a notion of a generator integral . Theterminology is motivated by Proposition 2.12, which says that it plays a role of generator of(co)integrals of bimonoids. In fact, in section 8, we will prove Theorem 8.5 which justify theterminology.
Definition 2.11.
Let µ be an integral along a bimonoid homomorphism ξ : A Ñ B . Theintegral µ is a generator if the following two diagrams below commute for any µ P Int r p ξ q Y Int l p ξ q :(22) B AA B µ µ ξ µ (23) B AA B µ µ ξ µ Proposition 2.12.
Recall Proposition 2.7. Let A be a bimonoid in a symmetric monoidalcategory C . Let σ be an integral along the counit (cid:15) A . The integral σ is a generator if andonly if for any σ P p
Int r p (cid:15) A q Y Int l p (cid:15) A qq “ p Int r p A q Y Int l p A qq σ “ p (cid:15) A ˝ σ q ¨ σ. (24) In particular, if an integral σ is normalized, then σ is a generator.Proof. Let σ be a generator. Then the commutative diagram (22) proves the claim.Let σ P Int l p (cid:15) A q “ Int l p A q . Suppose that σ “ p (cid:15) A ˝ σ q ¨ σ . Since σ is a left integral of A , we have p (cid:15) A ˝ σ q ¨ σ “ ∇ A ˝ p σ b σ q “ p (cid:15) A ˝ σ q ¨ σ . Hence, we obtain σ “ p (cid:15) A ˝ σ q ¨ σ ,which is equivalent with (23). We leave the proof for a right integral σ to the readers.We prove that if σ is normalized, then it is a generator. Let σ P Int r p A q . Then σ ˚ σ “p (cid:15) A ˝ σ q ¨ σ “ σ since σ is normalized. We also have σ ˚ σ “ p (cid:15) A ˝ σ q ¨ σ since σ is anintegral. Hence, we obtain σ “ p (cid:15) A ˝ σ q ¨ σ . We leave the proof for σ P Int l p A q to thereaders. It completes the proof. (cid:3) We also have a dual statement for cointegrals.
Remark 2.13.
There exists a bimonoid A with a generator integral which is not normalized.For example, finite-dimensional Hopf algebra which is not semi-simple is such an example.
Proposition 2.14.
Let ξ : A Ñ B be a bimonoid isomorphism. Recall that ξ ´ is an integralof ξ by Proposition 2.8. The integral ξ ´ is a generator.Proof. It is immediate from definitions. (cid:3)
3. A refinement of the main result
In this section, we give a refinement of the main results mentioned in section 1 and explaintheir relationships. The precise definitions and proofs follow from the following sections.3.1.
Existence of a normalized integral.
NTEGRALS ALONG BIMONOID HOMOMORPHISMS 9
Necessary conditions for existence of normalized integrals.
An existence of a normal-ized integral along a bimonoid homomorphism is related with the kernel and the cokernel ofthe bimonoid homomorphism. We give a necessary condition for a bimonoid homomorphismto have a normalized integral :
Theorem 3.1.
Let ξ : A Ñ B be a bimonoid homomorphism with a normalized integral along ξ . If the homomorphism ξ is conormal, then the kernel bimonoid Ker p ξ q has a normalizedintegral. We have a dual claim : if the homomorphism ξ is normal, then the cokernel bimonoid Cok p ξ q has a normalized cointegral. The proofs follow from Theorem 7.5.For the definition of (co)normality of homomorphisms, see section 5. We remark that Ournotion is implied by the Milnor-Moore’s definition if C “ Vec b F . Milnor and Moore definedthe notion of normality of morphisms of augmented algebras over a ring and normality ofmorphisms of augmented coalgebras over a ring (Definition 3.3, 3.5 [14]). They are definedby using the additive structure of the category Vec F . We introduce a weaker notion of nor-mality and conormality of bimonoid homomorphisms without assuming an additive categorystructure on C .3.1.2. Su ffi cient conditions for existence of a normalized generator integral. We introducea notion of a generator integral (see Definition 2.11). It is named after the property that itgenerates the set of integrals under some conditions (see Theorem 8.5). We study su ffi cientconditions for a normalized generator integral to exist.A bimonoid A is small if an invariant object and a stabilized object of any (left or right)action of A exist and the canonical morphism between them is an isomorphism (see Definition6.2). The notion of cosmall bimonoids is a dual notion of small bimonoids.A weakly well-decomposable homomorphism is a bimonoid homomorphism whose kernel,cokernel, coimage and image exist and satisfy some axioms (see Definition 9.6). A weaklywell-decomposable homomorphism ξ is weakly pre-Fredholm if the kernel bimonoid Ker p ξ q is small and the cokernel bimonoid Cok p ξ q is cosmall. Then a su ffi cient condition for anormalized generator integral to exist is given as follows : Theorem 3.2.
Let A , B be bimonoids in a symmetric monoidal category C and ξ : A Ñ B be aweakly well-decomposable homomorphism. If the homomorphism ξ is weakly pre-Fredholm,then there exists a unique normalized generator integral µ ξ : B Ñ A along ξ . The existence follows from Theorem 9.9 and the uniqueness follows from Proposition 7.1.Let C be a symmetric monoidal category where every idempotent in C is a split idempo-tent. It is possible to characterize the existence of a normalized generator integral by weaklypre-Fredholmness. Before we give our theorem, we introduce a notion. A bimonoid homo-morphism ξ is well-decomposable if ξ is binormal, the canonical homomorphism ker p ξ q isnormal, cok p ξ q is conormal and the induced homomorphism ¯ ξ : Coim p ξ q Ñ Im p ξ q is anisomorphism. Recall that the (co)smallness of a bimonoid is equivalent with the existence ofa normalized (co)integral if every idempotent in C is a split idempotent.From Theorem 3.1, 3.2 and 6.13, we obtain the following theorem. Theorem 3.3.
Suppose that every idempotent in C is a split idempotent. Let ξ be a well-decomposable bimonoid homomorphism. There exists a normalized generator integral µ ξ along ξ if and only if the homomorphism ξ is weakly pre-Fredholm. Note that if a normalizedintegral exists, then it is unique. Compositions.
Composition of integrals and homomorphisms.
We give a su ffi cient condition for acommutative square diagram to induce commutative integrals and homomorphisms. Theorem 3.4.
Let A , B , C , D be bimonoids. Consider a commutative diagram (25) of bi-monoid homomorphisms. Suppose that the bimonoid homomorphisms ϕ , ψ are weakly well-decomposable and weakly pre-Fredholm. Recall that there exist normalized generator inte-grals µ ϕ , µ ψ along ϕ, ψ respectively by Theorem 3.2. If (a) the induced bimonoid homomorphism ϕ : Ker p ϕ q Ñ Ker p ψ q has a section in C , (b) the induced bimonoid homomorphism ψ : Cok p ϕ q Ñ Cok p ψ q has a retract in C ,then we have µ ψ ˝ ψ “ ϕ ˝ µ ϕ . (25) A CB D ϕ ϕ ψψ The proof appears in section 10.
Remark 3.5.
We give a remark about assumptions (a), (b) in Theorem 3.4. Suppose thatthe symmetric monoidal category C satisfies (Assumption 0,1,2) in subsubsection 3.3.1. Con-sider bicommutative Hopf monoids A , B , C , D and pre-Fredholm homomorphisms ϕ, ψ . Inparticular, Ker p ϕ q , Ker p ψ q , Cok p ϕ q , Cok p ψ q are small and cosmall. If the induced bimonoidhomomorphism ϕ is an epimorphism in Hopf bc p C q , then the assumption (a) is immediate.In fact, the normalized generator integral along the homomorphism ϕ , which exists due toTheorem 3.2, is a section of ϕ . See Lemma. Dually, if the induced bimonoid homomorphism ψ is a monomorphism in Hopf bc p C q , then the assumption (b) is immediate. Especially, by(Assumption 2), the conditions (a), (b) are equivalent with an exactness of the induced chaincomplex below where p ϕ, ϕ q “ p ϕ b ϕ q ˝ ∆ A and ψ ´ ψ “ ∇ D ˝ p ψ b p S C ˝ ψ qq :A p ϕ,ϕ q ÝÑ B b C ψ ´ ψ ÝÑ D (26)3.2.2. Composition of integrals.
Recall that integrals are preserved under compositions byProposition 2.10. Nevertheless, such a composition does not preserve normalized integrals.By considering normalized generator integrals rather than normalized integrals, one can de-duce that they are preserved up to a scalar . Here, a scalar formally means an endomorphismon the unit object . Under some assumptions on the homomorphisms ξ, ξ , we determine thescalar as follows. Theorem 3.6.
Let A , B , C be bimonoids. Let ξ : A Ñ B, ξ : B Ñ C be bimonoid homo-morphism. Suppose that the homomorphisms ξ, ξ , ξ ˝ ξ are well-decomposable and weaklypre-Fredholm. Recall that there exist normalized generator integrals µ ξ , µ ξ , µ ξ ˝ ξ along thebimonoid homomorphisms ξ, ξ , ξ ˝ ξ respectively by Theorem 3.2. Then there exists a unique λ P End C p q such that µ ξ ˝ µ ξ “ λ ¨ µ ξ ˝ ξ . (27)The existence follows from Theorem 12.1 and the uniqueness follows form Theorem 8.5.In fact, the endomorphism λ coincides with x cok p ξ q ˝ ker p ξ qy P End C p q . The symbol x´y represents an invariant of bimonoid homomorphisms from a bimonoid with a normalizedintegral to a bimonoid with a normalized cointegral (see Definition 11.5). In Theorem 3 . Ker p ξ q has a normalized integral and the cokernel bimonoid Cok p ξ q hasa normalized cointegral since we assume that ξ, ξ are weakly pre-Fredholm.3.3. Applications to bicommutative Hopf monoids.
NTEGRALS ALONG BIMONOID HOMOMORPHISMS 11
Volume on
Hopf bc , bs p C q . We introduce a notion of a volume on A for an arbitraryabelian category A as follows. Definition 3.7.
For an abelian monoid M , an M-valued volume on the abelian category A is an assignment of v p A q P M to an object A of A which satisfies(1) For a zero object 0 of A , the corresponding element v p q P M is the unit 1 of theabelian monoid M .(2) For an exact sequence 0 Ñ A Ñ B Ñ C Ñ A , we have v p B q “ v p A q ¨ v p C q .For a bimonoid A with a normalized integral and cointegral, we define vol ´ p A q by x id A y ,which we call an inverse volume of A (see Definition 11.1). As an application of the resultsin the previous subsection, we show that the inverse volume gives a volume on some abeliancategory. Consider the following assumptions on C . ‚ (Assumption 0) The category C has any equalizer and coequalizer. ‚ (Assumption 1) The monoidal structure of C is bistable. ‚ (Assumption 2) The category Hopf bc p C q is an abelian category.Here, (co, bi)stability of the monoidal structure of C is introduced in Definition 4.4. Example 3.8.
Note that the assumptions on C automatically hold if C “ Vec b F , the cat-egory of (possibly, infinite-dimensional) vector spaces over a field F . The (Assumption 1)follows from Proposition 4.6 (see Example 4.7). The (Assumption 2) follows from the factthat Hopf bc p Vec b F q is an abelian category by Corollary 4.16 in [18] or Theorem 4.3 in [15] . Denote by
Hopf bc , bs p C q the category of bicommutative bismall Hopf monoids. Then thecategory Hopf bc , bs p C q is an abelian category (see Proposition 15.4). Moreover, the inversevolume gives a volume on the abelian category Hopf bc , bs p C q : Theorem 3.9.
Under the (Assumption 0,1,2), the assignment vol ´ of inverse volume givesan End C p q -valued volume on the abelian category, Hopf bc , bs p C q . The proof appears in subsection 15.1.3.3.2.
Fredholm homomorphism.
Let B be an abelian category and A be an abelian subcat-egory. Let v be an M -valued volume on A , not necessarily on B . Suppose that A is closed in B under short exact sequences : Definition 3.10.
Let B be an abelian category and A be a abelian subcategory. The abeliansubcategory A is closed under short exact sequences if A , C are objects of A and B is anobject of B for a short exact sequence 0 Ñ A Ñ B Ñ C Ñ B , then B is an object of A .By regarding objects of A with invertible volume as “finite-dimensional objects”, we de-fine a notion of Fredholm morphisms in B and its index which is an invariant respectingcompositions and robust to finite perturbations (see Definition 14.3). It generalizes the Fred-holm index of Fredholm operator in the algebraic sense. We give an analogue of the Fredholmindex based on bicommutative Hopf monoids by applying the following theorem : Theorem 3.11.
The category
Hopf bc , bs p C q of bismall bicommutative Hopf monoids is closedunder short exact sequences in Hopf bc p C q . The proof appears in subsection 13.2.By Theorem 3.11, the notion of Fredholm homomorphisms between bicommutative Hopfmonoids and their index are defined. As a final application, we construct a functorial assign-ment of integrals to Fredholm homomorphisms (see subsection 15.2). The reason that we consider a monoid M , not a group is that we deal with infinite dimension or infinite order uniformly.
4. S ome objects associated with action
Invariant object.
In this subsection, we define a notion of an invariant object of a(co)action. It is a generalization of the invariant subspace of a group action.
Definition 4.1.
Let C be a symmetric monoidal category. Let p A , α, X q be a left action in C . A pair p α zz X , i q is an invariant object of the action p A , α, X q if it satisfies the followingaxioms : ‚ α zz X is an object of C . ‚ i : α zz X Ñ X is a morphism in C . ‚ The diagram commutes where τ is the trivial action :(28) A b X XA b p α zz X q α zz X ατ i b id A i ‚ It is universal : If a morphism ξ : Z Ñ X satisfies a commutative diagram,(29) A b X XA b Z Z ατξ b id A ξ then there exists a unique morphism ¯ ξ : Z Ñ α zz X such that i ˝ ¯ ξ “ ξ .In an analogous way, we define invariant object of a left (right) coactions.4.2. Stabilized object.
In this subsection, we define a notion of a stabilized object of anaction (coaction, resp.). It is enhanced to a functor from the category of (co)actions if thesymmetric monoidal category C has every coequalizer (equalizer, resp.). Definition 4.2.
We define a stabilized object of a left action p A , α, X q in C by a coequalizerof following morphisms where τ A , X is the trivial action in Definition A.1.(30) A b X X ατ A , X We denote it by α z X . Analogously, we define a stabilized object of a right action p X , α, A q by a coequalizer of α and τ X , A . We denote it by X { α .We define a stabilized object of a left coaction p B , β, Y q in C by an equalizer of followingmorphisms where τ A , X is the trivial action in Definition A.1.(31) Y B b Y βτ B , Y We denote it by β { Y . Analogously, we define a stabilized object of a right coaction p Y , β, B q by an equalizer of α and τ Y , B . We denote it by Y z β . Proposition 4.3.
The assignments of stabilized objects to (co)actions have the following func-toriality : (1)
Suppose that the category C has any coequalizers. The assignment p A , α, X q ÞÑ α z Xgives a symmetric comonoidal functor from
Act l p C q to C . Analogouly, the assignment p X , α, A q ÞÑ X { α gives a SCMF from Act r p C q to C . (2) Suppose that the category C has any equalizers. The assignment p A , α, X q ÞÑ α { Xgives a symmetric monoidal functor from
Coact l p C q to C . Analogously, the assign-ment p X , α, A q ÞÑ X z α gives a SMF from Coact r p C q to C . NTEGRALS ALONG BIMONOID HOMOMORPHISMS 13
Proof.
The functoriality follows from the universality of coequalizers and equalizers. Weonly consider the first case. It is necessary to construct structure maps of a symmetricmonoidal functor. Let us prove the first claim.Let p , τ, q be the unit object of the symmetric monoidal category, Act l p C q , i.e. the trivialaction of the trivial bimonoid on the object . Then we have a canonical morphism Φ : τ z Ñ , in particular an isomprhism.Let O “ p A , α, X q , O “ p A , α , X q be left actions in C , i.e. objects of Act l p C q . Denote by p A b A , β, X b X q “ p A , α, X q b p A , α , X q P Act l p C q . We construct a morphism Ψ O , O : β zp X b X q Ñ p α z X q b p α z X q : The canonical projections induce a morphism ξ : X b X Ñp α z X q b p α z X q . The morphism ξ coequalizes β : p A b A q b p X b X q Ñ X b X and thetrivial action of A b A due to the definitions of α z X and α z X . Thus, we obtain a canonicalmorphism Ψ O , O : β zp X b X q Ñ p α z X q b p α z X q .Due to the universality of coequalizers and the symmetric monoidal structure of C , Φ , Ψ O , O give structure morphisms for a symmetric monoidal functor p A , α, X q ÞÑ α z X .We leave it to the readers the proof of other part. (cid:3) Stable monoidal structure.
In this subsection, we define a (co)stability and bistabilityof the monoidal structure of a symmetric monoidal category. We assume that C is a symmetricmonoidal category with arbitrary equalizer and coequalizer. Definition 4.4.
Recall that the assignments of stabilized objects to actions (coactions, resp.)are symmetric comonoidal functors (symmetric monoidal functors, resp.) by Proposition4.3. The monoidal structure of C is stable if the assignments of stabilized objects to actions, Act l p C q Ñ C and Act r p C q Ñ C , are strongly symmetric monoidal functors. The monoidalstructure of C is costable if the assignments of stabilized objects to coactions, Coact l p C q Ñ C and Coact r p C q Ñ C , are SSMF’s. The monoidal structure of C is bistable if the monoidalstructure is stable and costable. Lemma 4.5.
Let Λ , Λ be small categories. Let F : Λ Ñ C , F : Λ Ñ C be functors withcolimits lim ÝÑ Λ F and lim ÝÑ Λ F respectively. Suppose that the functor F p λ qbp´q preserves smallcolimits for any object λ of Λ and so does the functor p´q b lim ÝÑ F . Then the exterior tensorproduct F b F : Λ ˆ Λ Ñ C has a colimit lim ÝÑ Λ ˆ Λ F b F , and we have lim ÝÑ Λ ˆ Λ F b F – lim ÝÑ Λ F b lim ÝÑ Λ F .Proof. Let X be an object of C and g λ,λ : F p λ q b F p λ q Ñ X be a family of morphisms for λ P Λ , λ P Λ such that g λ ,λ ˝ p F p ξ q b F p ξ qq “ g λ ,λ where ξ : λ Ñ λ , ξ : λ Ñ λ are morphisms in Λ , Λ respectively. By the first assumption, the object F p λ q b lim ÝÑ F isa colimit of F p λ q b F p´q for arbitrary object λ P Λ . We obtain a unique morphism g λ : F p λ q b lim ÝÑ F Ñ X such that g λ ˝ p id F p λ q b π λ q “ g λ,λ for every object λ P Λ . By theuniversality of colimits, the family of morphisms g λ is, in fact, a natural transformation. Bythe second assumption, lim ÝÑ F b lim ÝÑ F is a colimit of the functor F p´q b lim ÝÑ F . Hence, thefamily of morphisms g λ for λ P Λ induces a unique morphism g : lim ÝÑ F b lim ÝÑ F Ñ X suchthat g ˝ p π λ b id lim ÝÑ F q “ g λ . Above all, for objects λ P Λ , λ P Λ , we have g ˝ p π λ b π λ q “ g ˝ p π λ b id lim ÝÑ F q ˝ p id F p λ q b π λ q “ g λ ˝ p id F p λ q b π λ q “ g λ,λ .We prove that such a morphism g that g ˝ p π λ b π λ q “ g λ,λ is unique. Let g : lim ÝÑ F b lim ÝÑ F Ñ X be a morphism such that g ˝ p π λ b π λ q “ g λ,λ . Denote by h “ g ˝ p π λ b id lim ÝÑ F q and h “ g ˝ p π λ b id lim ÝÑ F q . Then we have h ˝ p id F p λ q b π λ q “ g λ,λ “ h ˝ p id F p λ q b π λ q by definitions. Since F p λ q b lim ÝÑ F is a colimit of the functor F p λ q b F p´q by the firstassumption, we see that h “ h . Equivalently, we have g ˝ p π λ b id lim ÝÑ F q “ g ˝ p π λ b id lim ÝÑ F q .Since lim ÝÑ F b lim ÝÑ F is a colimit of the functor F p´q b lim ÝÑ F by the second assumption, wesee that g “ g by the universality. It completes the proof. (cid:3) Proposition 4.6.
Suppose that the functor Z b p´q preserves coequalizers (equalizers resp.)for arbitrary object Z P C . Then the monoidal structure of C is stable (costable, resp.).Proof. Note that since C is a symmetric monoidal category, the functor p´q b Z preservescoequalizers (equalizers resp.) for arbitrary object Z P C by the assumption. We prove thestability and leave the proof o the costability to the readers.Let p A , α, X q , p B , β, Y q be left actions in C . Denote by α z X , β z Y their stabilized objectsas before. By the assumption, we can apply Lemma 4.5. By Lemma 4.5, p α z X b β z Y q is acoequalizer of morphisms α ˜ b β , α ˜ b τ B , τ A ˜ b β , τ A ˜ b τ B . Here, ˜ b is defined in Definition A.1. Itsu ffi ces to show that a coequalizer of α ˜ b β , α ˜ b τ B , τ A ˜ b β , τ A ˜ b τ B coincides with the stabilizedobject p α ˜ b β qzp X b Y q , i.e. a coequalizer of α ˜ b β , τ A ˜ b τ B .Let π : X b Y Ñ p α ˜ b β qzp X b Y q be the canonical projection. The unit axiom of the action β induces the following commutative diagram :(32) A b B b X b Y X b YA b B b X b Y α ˜ b τ B id A bp η B ˝ (cid:15) B qb id X b Y α ˜ b β Hence, we have π ˝ p α ˜ b τ B q “ π ˝ p α ˜ b β q ˝ p id A b p η B ˝ (cid:15) B q b id X b Y q “ π ˝ p τ A ˜ b τ B q ˝ p id A bp η B ˝ (cid:15) B q b id X b Y q “ π ˝ p τ A ˜ b τ B q . We obtain π ˝ p α ˜ b τ B q “ π ˝ p τ A ˜ b τ B q . Likewise, we have π ˝ p τ A ˜ b β q “ π ˝ p τ A ˜ b τ B q .Let g : X b Y Ñ Z be a morphism which coequalizes α ˜ b β , α ˜ b τ B , τ A ˜ b β , τ A ˜ b τ B . Since themorphism g coequalizes α ˜ b β , τ A ˜ b τ B , there exists a unique morphism g : p α ˜ b β qzp X b Y q Ñ Z such that g ˝ π “ g . Above all, p α ˜ b β qzp X b Y q is a coequalizer of α ˜ b β , α ˜ b τ B , τ A ˜ b β , τ A ˜ b τ B . (cid:3) Example 4.7.
Consider the symmetric monoidal category,
Vec b F , the category of vectorspaces over F and linear homomorphisms. Note that a coequalizer (an equalizer, resp.)of two morphisms in the category Vec F is obtained via a cokernel (a kernel, resp.) of theirdi ff erence morphism. A functor V b p´q preserves coequazliers and equazliers since it is anexact functor for any linear space V. Hence, by Proposition 4.6, the monoidal structure ofthe symmetric monoidal category, Vec b F , is bistable.
5. N ormal homomorphism
In this section, we define a notion of normality , conormality and binormality of bimonoidhomomorphisms. We prove that every homomorphism between bicommutative Hopf monoidsis binormal under some assumptions on the symmetric monoidal category C . Definition 5.1.
Let D be a category with a zero object, i.e. an initial object which is simul-taneously a terminal object. Let A , B be objects of D and ξ : A Ñ B be a morphism in D . A cokernel of ξ is given by a pair p Cok p ξ q , cok p ξ qq of an object Cok p ξ q and a morphism cok p ξ q : B Ñ Cok p ξ q , which gives a coequalizer of ξ : A Ñ B and 0 : A Ñ B in D .A kernel of ξ is given by a pair p Ker p ξ q , ker p ξ qq of an object Ker p ξ q and a morphism ker p ξ q : Ker p ξ q Ñ A , which gives an equalizer of ξ : A Ñ B and 0 : A Ñ B in D . Definition 5.2.
Let A , B be bimonoids in a symmetric monoidal category C and ξ : A Ñ B bea bimonoid homomorphism. We define a left action p A , α Ñ ξ , B q and a right action p B , α Ð ξ , A q by the following compositions : α Ñ ξ : A b B ξ b id B Ñ B b B ∇ B Ñ B , (33) α Ð ξ : B b A id B b ξ Ñ B b B ∇ B Ñ B . (34) NTEGRALS ALONG BIMONOID HOMOMORPHISMS 15
We define a left coaction p A , β Ñ ξ , B q and a right coaction p B , β Ð ξ , A q by the following compo-sitions : β Ñ ξ : A ∆ A Ñ A b A ξ b id A Ñ B b A , (35) β Ð ξ : A ∆ A Ñ A b A id A b ξ Ñ A b B . (36) Definition 5.3.
Let A , B be bimonoids in a symmetric monoidal category C . A bimonoidhomomorphism ξ : A Ñ B is normal if there exists a bimonoid structure on the stabilizedobjects α Ñ ξ z B , B { α Ð ξ such that the canonical morphisms π : B Ñ α Ñ ξ z B , ˜ π : B Ñ B { α Ð ξ are bimonoid homomorphisms and the pairs p α Ñ ξ z B , π q , p B { α Ð ξ , ˜ π q give cokernels of ξ in Bimon p C q .A conormal bimonoid homomorphism is defined in a dual way by using the coactions β Ð ξ , β Ñ ξ instead of α Ñ ξ , α Ð ξ . A bimonoid homomrphism ξ : A Ñ B is binormal if it is normaland conormal in Bimon p C q . Remark 5.4.
We use the terminology normal due to the following reason. If C “ Sets ˆ ,then a Hopf monoid in that symmetric monoidal category is given by a group. For a group Hand its subgroup G, one can determine a set H { G which is a candidate of a cokernel of theinclusion. The set H { G plays a role of cokernel group if and only if the image G is a normalsubgroup of H. In this example, the normality defined in this paper means that the set H { Gis a cokernel group of the inclusion G Ñ H. Proposition 5.5.
Let A be a bimonoid. The identity homomorphism id A : A Ñ A is binormal.Proof.
We prove that the identity homomorphism id A is normal. The counit (cid:15) A : A Ñ on A induces gives a coequalizer of the regular action α Ñ id A : A b A Ñ A and the trivial action τ : A b A Ñ A . In particular, we have a natural isomorphism α Ñ id A z A – . We give abimonoid structure on α Ñ id A by the isomorphism. Moreover the counit (cid:15) A : A Ñ is obviouslya cokernel of the identity homomorphism id A in the category of bimonoids Bimon p C q . Thus,the identity homomorphsim id A is normal. In a dual way, the identity homomorphsim id A isconormal, so that binormal. (cid:3) Proposition 5.6.
Let A , B be Hopf monoids in a symmetric monoidal category C . Let ξ : A Ñ B be a bimonoid homomorphism. If the homomorphism ξ is normal, then a cokernel p Cok p ξ q , cok p ξ qq in the category of bimoniods Bimon p C q is a cokernel in the category of Hopfmonoids Hopf p C q .Proof. Since cok p ξ q ˝ S B ˝ ξ “ cok p ξ q ˝ ξ ˝ S A is trivial, the anti-homomorphism cok p ξ q ˝ S B induces an anti-homomorphism S : Cok p ξ q Ñ Cok p ξ q such that S ˝ cok p ξ q “ cok p ξ q ˝ S B .We claim that S gives an antipode on the bimonoid C “ Cok p ξ q . It su ffi ces to prove that ∇ C ˝ p S b id C q ˝ ∆ C “ η C ˝ (cid:15) C “ ∇ C ˝ p id C b S q ˝ ∆ C . Since p α Ñ ξ z B , π q , p B { α Ð ξ , ˜ π q givecokernels, the canonical morphism cok p ξ q is an epimorphism in C by the universality ofstabilized objects. Hence, it su ffi ces to prove that ∇ C ˝ p S b id C q ˝ ∆ C ˝ cok p ξ q “ η C ˝ (cid:15) C ˝ cok p ξ q “ ∇ C ˝ p id C b S q ˝ ∆ C ˝ cok p ξ q . We prove the first equation by using the fact that cok p ξ q : B Ñ Cok p ξ q “ C is a bimonoid homomorphism. ∇ C ˝ p S b id C q ˝ ∆ C ˝ cok p ξ q “ ∇ C ˝ p S b id C q ˝ p cok p ξ q b cok p ξ qq ˝ ∆ B , (37) “ ∇ C ˝ pp S ˝ cok p ξ qq b cok p ξ qq ˝ ∆ B , (38) “ ∇ C ˝ pp cok p ξ q ˝ S B q b cok p ξ qq ˝ ∆ B , (39) “ ∇ C ˝ p cok p ξ q b cok p ξ qq ˝ p S B b id B q ˝ ∆ B , (40) “ cok p ξ q ˝ ∇ B ˝ p S B b id B q ˝ ∆ B , (41) “ cok p ξ q ˝ η B ˝ (cid:15) B , (42) “ η C ˝ (cid:15) C ˝ cok p ξ q . (43)The second equation is proved similarly. It completes the proof. (cid:3) Proposition 5.7.
Suppose that the monoidal structure of C is stable (costable, resp.). Then ev-ery bimonoid homomorphism between bicommutative bimonoids is normal (conormal, resp.)and its cokernel (kernel, resp.) is a bicommutative bimonoid. In particular, if the monoidalstructure of C is bistable, then every bimonoid homomorphism between bicommutative bi-monoids is binormal.Proof. We prove that if the monoidal structure of C is stable, then every bimonoid homo-morphism between bicommutative bimonoids is normal and its cokernel is a bicommutativebimonoid. Let A , B be bicommutative bimonoids in a symmetric monoidal category C and ξ : A Ñ B be a bimonoid homomorphism. Note that the left action p A , α Ñ ξ , B q has a naturalbicommutative bimonoid structure in the symmetric monoidal category Act l p C q , the categoryof left actions in C . The symmetric monoidal category structure on Act l p C q is described inDefinition A.1. In fact, it is due to the commutativity of B : We explain the monoid structureof p A , α Ñ ξ , B q here. Since B is a bicommutative bimonoid, ∇ B : B b B Ñ B is a bimonoid ho-momorphism. In particular, ∇ B is compatible with the action α Ñ ξ , i.e. the following diagramcommutes.(44) p A b A q b p B b B q B b BA b B B α Ñ ξ ˜ b α Ñ ξ ∇ A b ∇ B ∇ B α Ñ ξ Since η B : Ñ B is a bimonoid homomorphism, the following diagram commutes.(45) b A b B B – η A b η B η B α Ñ ξ Hence, they induce a monoid structure on p A , α Ñ ξ , B q in the symmetric monoidal category Act l p C q . Likewise, p A , α Ñ ξ , B q has a comonoid structure in Act l p C q : The comultiplicationson A , B induces a comultiplication on p A , α Ñ ξ , B q due to following diagram commutes.(46) p A b A q b p B b B q B b BA b B B α Ñ ξ ˜ b α Ñ ξ α Ñ ξ ∆ A b ∆ B ∆ B In fact, we do not need any commutativity or cocommutativity of A , B to prove the commuta-tivity of the diagram. The counits on A , B induce a counit on p A , α Ñ ξ , B q due to the following NTEGRALS ALONG BIMONOID HOMOMORPHISMS 17 commutativity diagram.(47) b A b B B – (cid:15) A b (cid:15) B α Ñ ξ (cid:15) B Since the morphisms ∆ A , ∇ A , (cid:15) A , η A and the morphisms ∆ B , ∇ B , (cid:15) B , η B give bicommutative bi-monoid structure on A , B respectively, the above monoid structure and comonoid structure on p A , α Ñ ξ , B q give a bicommutative bimonoid structure on p A , α Ñ ξ , B q .Since the monoidal structure of C is stable by the assumption, the assignment of stabilizedobjects to actions is a strongly symmetric monoidal functor by definition. The bicommutativebimonoid structure on p A , α Ñ ξ , B q is inherited to its stabilized object α Ñ ξ z B . We consider α Ñ ξ z B as a bicommutative bimonoid by the inherited structure.The canonical morphism π : B Ñ α Ñ ξ z B is a bimonoid homomorphism with respect to thebimonoid structure on α Ñ ξ z B described above. In fact, the commutative diagram (48) inducesa bimonoid homomorphism p , α Ñ η B , B q Ñ p A , α Ñ ξ , B q between bicommutative bimonoids inthe symmetric monoidal category Act l p C q .(48) BA B η B η A id B ξ By the stability of the monoidal structure of C again, we obtain a bimonoid homomorphism, B – α Ñ η B z B Ñ α Ñ ξ z B . (49)It coincides with the canonical projection π : B Ñ α Ñ ξ z B by definitions.All that remain is to show that the pair p α Ñ ξ z B , π q is a cokernel of the bimonoid homo-morphism ξ in Bimon p C q in the sense of Definition 5.1. Let C be another bimonoid and ϕ : B Ñ C be a bimonoid homomorphism such that ϕ ˝ ξ “ η C ˝ (cid:15) A . It coequazlies the action α Ñ ξ : A b B Ñ B and the trivial action τ A , B : A b B Ñ B so that it induces a unique mor-phism ¯ ϕ : α Ñ ξ z B Ñ C such that ¯ ϕ ˝ π “ ϕ . We prove that ¯ ϕ is a bimonoid homomorphism.Note that the counit (cid:15) A : A Ñ and the homomorphism ϕ : B Ñ C induces a bimonoidhomomorphism p A , α Ñ ξ , B q Ñ p , α Ñ η C , C q . By the stability of the monoidal structure of C again, it induces a bimonoid homomorphism α Ñ ξ z B Ñ α Ñ η C z C – C which coincides with ¯ ϕ .It completes the proof. (cid:3) Corollary 5.8.
Suppose that the monoidal structure of C is stable (costable, resp.). Let A , Bbe bicommutative Hopf monoids and ξ : A Ñ B be a bimonoid homomorphism. Then acokernel (kernel, resp.) of ξ in Bimon p C q is a cokernel (kernel, resp.) of ξ in Hopf bc p C q .Proof. Suppose that the monoidal structure of C is stable. Let A , B be bicommutative Hopfmonoids and ξ : A Ñ B be a bimonoid homomorphism. By Proposition 5.7, the homomor-phism ξ is normal and its cokernel is a bicommutative bimonoid. By Proposition 5.6, thecokernel of ξ is a bicommutative Hopf monoid. (cid:3)
6. S mall bimonoid and integral
In this section, we introduce a notion of (co,bi)small bimonoids . We study the relationshipbetween existence of normalized (co)integrals and (co)smallness of bimonoids.
Definition 6.1.
Let C be a symmetric monoidal category. Let p A , α, X q be a left action in thesymmetric monoidal category C . Recall the invariant object α zz X and the stabilized object α z X of the left action p A , α, X q . We define a morphism α γ : α zz X Ñ α z X in C by composingthe canonical morphisms X Ñ α z X and α zz X Ñ X . Likewise, we define γ α : X {{ α Ñ X { α for a right action p X , α, A q , β γ : β { Y Ñ β {{ Y for a left coaction p B , β, Y q , γ β : Y z β Ñ Y zz β for a right coaction p Y , β, B q . Definition 6.2.
Let C be a symmetric monoidal category. Recall Definition 5.2. A bimonoid A in the symmetric monoidal category C is small if ‚ For every left action p A , α, X q , an invariant object α zz X and a stabilized object α z X exist. Furthermore, the canonical morphism α γ : α zz X Ñ α z X is an isomorphism. ‚ For every right action p X , α, A q , an invariant object X {{ α and a stabilized object X { α exist. Furthermore, the canonical morphism γ α : X {{ α Ñ X { α is an isomorphism.A bimonoid A in the symmetric monoidal category C is cosmall if ‚ For every left coaction p B , β, Y q , an invariant object β {{ Y and a stabilized object β { Y exist. Furthermore, the canonical morphism β γ : β z Y Ñ β zz Y is an isomorphism. ‚ For every right coaction p Y , β, B q , an invariant object Y zz β and a stabilized object Y z β exist. Furthermore, the canonical morphism γ β : Y { β Ñ Y {{ β is an isomorphism.A bimonoid A is bismall if the bimonoid A is small and cosmall.We use subscript ‘bs’ to denote ‘bismall’. For example, Hopf bs p C q is a full subcategory of Hopf p C q formed by bismall Hopf monoids. Remark 6.3.
In general, the morphism α γ : α zz X Ñ α z X (also, β γ, γ α , γ β ) in Definition 6.1is not an isomorphism. We give three examples as follows. Example 6.4.
Let p A , α, X q be a left action where A “ X “ F G and α is the multiplication.There exists an invariant object α zz F G and a stabilized object α z F G given by α zz F G “ t λ ÿ g P G g ; λ P F u (50) α z F G “ F G { p g „ e q (51) Here, e P G denotes the unit of G and F G { p g „ e q means the quotient space of F G bythe given relation. Then we see that the morphism α γ is zero while α zz F G, α z F G are 1-dimensional.
Definition 6.5.
Let C be a category. A morphism p : X Ñ X is an idempotent if p ˝ p “ p .A retract of an idempotent p is given by p X p , ι, π q where ι : X p Ñ X , π : X Ñ X p aremorphisms in C such that π ˝ ι “ id X p and ι ˝ π “ p . If an idempotent p has a retract, then p is called a split idempotent . Proposition 6.6.
Let C be a category and p : X Ñ X be an idempotent. Suppose that thereexists an equalizer of the identity id X and p and a coequalizer of the identity id X and p. Thenthe idempotent p is a split idempotent.Proof. Denote by e : E Ñ X an equalizer of the identity id X and the morphism p : X Ñ X .Denote by c : X Ñ C a coequalizer of the identity id X and the morphism p : X Ñ X .We claim that c ˝ e : K Ñ E is an isomorphism and p E , e , p c ˝ e q ´ ˝ c q is a retract of theidempotent p .Note that the morphism p equalizes the identity id X and the morphism p due to p ˝ p “ p .The morphism p induces a unique morphism p : X Ñ E such that e ˝ p “ p . Note thatthe morphism p coequalizes the identity id X and the morphism p due to p ˝ p “ p . Themorphism p induces a unique morphism p : C Ñ E such that p ˝ c “ p . Then p is aninverse of the composition c ˝ e so that c ˝ e is an isomorphism. NTEGRALS ALONG BIMONOID HOMOMORPHISMS 19
We prove that p E , e , p c ˝ e q ´ ˝ c q is a retract of the idempotent p . It follows from pp c ˝ e q ´ ˝ c q˝ e “ id K and e ˝ pp c ˝ e q ´ ˝ c q “ p . The latter one follows from the above discussion that p c ˝ e q ´ “ p and e ˝ p ˝ c “ e ˝ p “ p . (cid:3) Proposition 6.7.
Let p A , α, X q be a left action in a symmetric monoidal category C with aninvariant object α zz X and a stabilized object α z X. Suppose that the morphism α γ : α zz X Ñ α z X is an isomorphism. Then the endomorphism p : X Ñ X defined by following compositionis a split idempotent. α p “ ˆ X ι Ñ α z X α γ ´ Ñ α zz X π Ñ X ˙ . (52) Here, ι, π are the canonical morphisms.Proof.
We prove that p is an idempotent on X . It follows from p ˝ p “ ι ˝ α γ ´ ˝ π ˝ ι ˝ α γ ´ ˝ π “ ι ˝ α γ ´ ˝ α γ ˝ α γ ´ ˝ π “ ι ˝ α γ ´ ˝ π “ p .We prove that p α z X , ι ˝ α γ ´ , π q give a retract of the idempotent p . By definition, we have ι ˝ α γ ´ ˝ π “ p . Moreover, we have π ˝ ι ˝ α γ ´ “ α γ ˝ α γ ´ “ id α z X . (cid:3) Lemma 6.8.
Let A be a bimonoid in a symmetric monoidal category C . Suppose that for theregular left action p A , α Ñ id A , A q , an invariant object α Ñ id A zz A and a stabilized object α Ñ id A z A existand the canonical morphism α Ñ idA γ : α Ñ id A zz A Ñ α Ñ id A z A is an isomorphism. Then the bimonoidA has a normalized left integral.Proof.
Let A be a bimonoid. Suppose that the bimonoid A is small. Consider a left action p A , α, A q in C where α “ α Ñ id A “ ∇ A : A b A Ñ A is the regular left action. Since A is small, theinvariant object α zz A and the stabilized object α z A exist and the morphism α γ : α zz A Ñ α z A is an isomorphism. Let p : A Ñ A be a composition of A π Ñ α z A α γ ´ Ñ α zz A ι Ñ A where π , ι are canonical morphisms. We prove that σ “ p ˝ η A : Ñ A is a normalized right integral.We claim that (cid:15) A ˝ p “ (cid:15) . Then (cid:15) A ˝ σ “ (cid:15) A ˝ η A “ id which is the axiom (3) : Notethat the canonical morphism π : A Ñ α z A coequalizes the regular left action α and the trivialleft action. The counit morphism (cid:15) A induces a unique morphism ¯ (cid:15) A : α z A Ñ such that¯ (cid:15) A ˝ π “ (cid:15) A . We obtain following commutative diagram so that (cid:15) A ˝ p “ (cid:15) .(53) A α z A α zz A AA p π (cid:15) A α γ ´ ¯ (cid:15) A ιι (cid:15) A (cid:15) A π We claim that ∇ A ˝ p id A b p q “ r A ˝ p (cid:15) A b p q : A b A Ñ A . Then by composing id A b η A : A b Ñ A b A we see that σ “ p ˝ η A satisfies the axiom (2) : In fact, we have ∇ A ˝ p id A b ι q “ (cid:15) A b ι : A b p α zz A q Ñ A by definition of ι : α zz A Ñ A . Thus, we have ∇ A ˝p id A b p q “ ∇ A ˝p id A b ι q˝p id A bp α γ ´ ˝ π qq “ p (cid:15) A b ι q˝p id A bp α γ ´ ˝ π qq “ r A bp (cid:15) A b p q .Above all, the morphism σ “ p ˝ η A : Ñ A is a normalized right integral of A . (cid:3) Remark 6.9.
In Lemma 6.8, we show that a bimonoid A has a normalized left integral undersome assumptions on the bimonoid A. Similarly, a bimonoid has a normalized right integralif A satisfies similar assumptions on the regular right action. Especially, if the bimonoid Ais small, then the bimonoid A has a normalized left integral and a normalized right integral.We also have a dual statement.
Definition 6.10.
Let p A , α, X q be a left action in a symmetric monoidal category C . For amorphism a : Ñ A in C , we define an endomorphism L α p a q : X Ñ X by a composition, X l ´ X Ñ b X a b id X Ñ A b X α Ñ X . (54)Let p Y , β, B q be a right coaction in C . For a morphism b : B Ñ in C , we define anendomorphism R β p b q : Y Ñ Y by a composition, Y β Ñ Y b B id Y b b Ñ Y b r Y Ñ Y . (55) Proposition 6.11.
Let p A , α, X q be a left action in C . Then a P Mor C p , A q ÞÑ L α p a q P End C p X q is a homomorphism. Here, the monoid End C p X q consists of endomorphisms on X :L α p a ˚ a q “ L α p a q ˝ L α p a q , a , a P Mor C p , A q . (56) Likewise, for a right coaction p Y , β, B q , the assignment b P Mor C p B , q ÞÑ R β p b q P End C p Y q is a homomorphism : R β p b ˚ b q “ R β p b q ˝ R β p b q , b , b P Mor C p B , q (57) Proof.
It follows from the associativity of an action and a coaction. (cid:3)
Proposition 6.12.
Let A be a small bimonoid in a symmetric monoidal category C . Let p A , α, X q be a left action in C . Recall Lemma 6.8, then we have a normalized integral σ A ofA. The induced morphism L α p σ A q is a split idempotent. Moreover we have α p “ L α p σ A q where α p is given in Proposition 6.7.Proof. The morphsim L α p σ A q is an idempotent by Proposition 6.11 and σ A ˚ σ A “ σ A . σ A ˚ σ A “ σ A follows from the normality of σ A .Let α zz X be an invariant object and α z X be a stabilized object of the left action p A , α, X q .Denote by ι : α zz X Ñ X and π : X Ñ α z X the canonical morphisms. We claim that themorphism ι gives an equalizer of L α p σ A q and id X , and the morphism π gives a coequalizer of L α p σ A q and id X . Then the idempotent L α p σ A q is a split idempotent by Proposition 6.6.We prove that the morphism ι gives an equalizer of L α p σ A q and id X . Note that L α p σ A q ˝ ι “ id X ˝ ι since the integral σ A is normalized. We prove the universality. Suppose that f : Z Ñ X equalizes L α p σ A q and id X , i.e. L α p σ A q ˝ f “ f . Then α ˝ p id A b f q “ τ A , X ˝ p id A b f q by Figure3. By definition of the invariant object α zz X , f induces a unique morphism f : Z Ñ α zz X such that ι ˝ f “ f . F igure π gives a coequalizer of L α p σ A q and id X . Note that π ˝ L α p σ A q and π ˝ id X since the integral σ A is normalized. We prove the universality. Suppose that g : X Ñ Z coequalizes L α p σ A q and id X , i.e. g ˝ L α p σ A q “ g . Then g ˝ α “ g ˝ τ A , X by Figure4. By definition of the stabilzed object α z X , the morphism g induces a unique morphism g : α z X Ñ Z such that g ˝ π “ g .All that remain is to prove that α p “ L α p σ A q . Note that p α zz X , ι, α γ ´ ˝ π q gives a retract ofthe idempotent of L α p σ A q . See the proof of Proposition 6.6. Hence, L α p σ A q “ ι ˝p α γ ´ ˝ π q “ α p . It completes the proof. (cid:3) NTEGRALS ALONG BIMONOID HOMOMORPHISMS 21 F igure Theorem 6.13.
Let C be a symmetric monoidal category. Suppose that every idempotentin C is a split idempotent. A bimonoid A in C is small if and only if the bimonoid A has anormalized integral.Proof. By Proposition 2.3, Lemma 6.8, and Remark 6.9, if a bimonoid A is small, then A hasa normalized integral.Suppose that a bimonoid A has a normalized integral σ A . Let p A , α, X q be a left action in C .Let us write p “ L α p σ A q : X Ñ X . By Proposition 6.11, we have p ˝ p “ L α p σ A q ˝ L α p σ A q “ L α p σ A ˚ σ A q “ L α p σ A q “ p since σ A is a normalized integral of A . In other words, themorphsim p is an idempotent on X . By the assumption, there exists a retract p X p , ι, π q of theidempotent p : X Ñ X . We claim that,(1) The morphism π : X Ñ X p gives a stabilized object α z X of the left action p A , α, X q .(2) The morphism ι : X p Ñ X gives an invariant object α zz X of the left action p A , α, X q .Then the canonical morphism α γ : α zz X Ñ α z X coincides with π ˝ ι “ id X p so that α γ is anisomorphism. It completes the proof.We prove the first claim. Suppose that a morphism f : X Ñ Y coequalizes the action α : A b X Ñ X and the trivial action τ A , X : A b X Ñ X , i.e. f ˝ α “ f ˝ τ A , X . We set f “ f ˝ ι : X p Ñ Y . Then we have f ˝ π “ f ˝ ι ˝ π “ f ˝ p “ f ˝ L α p σ A q “ f ˝ α ˝p σ A b id X q . By f ˝ α “ f ˝ τ A , X , we obtain f ˝ π “ f ˝ τ A , X ˝ p σ A b id X q “ f since σ A is a normalized integral.Moreover, if we have f ˝ π “ f for a morphism f : X p Ñ Y , then f “ f ˝ π ˝ ι “ f ˝ ι “ f .Above all, the morphism π : X Ñ X p gives a stabilized object α z X of the left action p A , α, X q .We prove the second claim. The following diagram commutes :(58) A b X XA b X p X p ατ A , Xp id A b ι ι It follows from Figure 5. We prove the universality of an invariant object. Suppose thata morphism g : Z Ñ X satisfies α ˝ p id A b g q “ τ A , X ˝ p id A b g q : A b Z Ñ X . Put g “ π ˝ g : Z Ñ X p : Z Ñ X p . We have ι ˝ g “ ι ˝ π ˝ g “ p ˝ g “ α ˝ p σ A b id X q ˝ g “ τ A , X ˝ p σ A b id X q ˝ g “ g since σ A is the normalized integral. If for a morphism g : Z Ñ X p we have ι ˝ g “ g , then we have g “ π ˝ ι ˝ g “ π ˝ g “ g . It proves the universality of aninvariant object ι : X p Ñ X . (cid:3) Corollary 6.14.
Let C be a symmetric monoidal category. Suppose that every idempotentin C is a split idempotent. A bimonoid A in C is bismall if and only if A has a normalizedintegral and a normalized cointegral.Proof. We have a dual statement of Theorem 6.13. The dual statement and Theorem 6.13complete the proof. (cid:3) F igure Corollary 6.15.
Suppose that every idempotent in C is a split idempotent. The full subcat-egory of (co)small bimonoids in a symmetric monoidal category C forms a sub symmetricmonoidal category of Bimon p C q . In particular, the full subcategory of bismall bimonoids ina symmetric monoidal category C forms a sub symmetric monoidal category of Bimon p C q .Proof. We prove the claim for small cases and leave the second claim to the readers. ByTheorem 6.13, small bimonoids A , B have nomalized integrals σ A , σ B . Then a morphism σ A b σ B : – b Ñ A b B is verified to give a morphism of the bimonoid A b B bydirect calculation. Hence the bimonoid A b B possesses a normalized integral so that A b B is small by Theorem 6.13. It completes the proof. (cid:3)
7. I ntegral along bimonoid homomorphism
Uniqueness of normalized integral.
In this subsection, we prove the uniqueness ofnormalized integrals along homomorphisms. It is a generalization of the uniqueness of nor-malized (co)integrals of bimonoids in Proposition 2.3.
Proposition 7.1.
Let ξ : A Ñ B be a bimonoid homomorphism. Suppose that µ P Int r p ξ q , µ P Int l p ξ q are normalized. Then we have µ “ µ P Int p ξ q . (59) In particular, a normalized integral along ξ is unique if exists.Proof. It is proved by two equations µ “ µ ˝ ξ ˝ µ and µ “ µ ˝ ξ ˝ µ . The former claimfollows from (Figure 6) and the latter claim follows from (Figure 7). It completes the proof. (cid:3) Corollary 7.2.
Let ξ : A Ñ B a bimonoid homomorphism. If µ P Int p ξ q is normalized, thenwe have ‚ µ ˝ ξ ˝ µ “ µ . ‚ µ ˝ ξ : A Ñ A is an idempotent on A. ‚ ξ ˝ µ : B Ñ B is an idempotent on B.Proof.
By direct verification, µ “ µ ˝ ξ ˝ µ is an integral along ξ . Also, µ is normalizedsince ξ ˝ µ ˝ ξ “ ξ ˝ µ ˝ ξ ˝ µ ˝ ξ “ ξ by the normality of µ . By Proposition 7.1, we have NTEGRALS ALONG BIMONOID HOMOMORPHISMS 23 F igure igure µ “ µ . It completes the proof of the first claim. The other claims are immediate from thefirst claim. (cid:3) Proof of Theorem 3.1.
An existence of a normalized integral along a homomorphism ξ is related with an existence of a normlaized integral of Ker p ξ q and a cointegral Cok p ξ q . Inthis subsection, we prove Theorem 7.5 which implies Theorem 3.1. We define an integralˇ F p µ q of Ker p ξ q from an integral µ along ξ when ξ is conormal. Furthermore, if the integral µ is normalized, then the integral ˇ F p µ q is normalized. Lemma 7.3.
Let µ P Int r p ξ q . Then µ ˝ η B : Ñ A equalizes the homomorphism ξ and thetrivial homomorphism, i.e. ξ ˝ p µ ˝ η B q “ η B ˝ (cid:15) A ˝ p µ ˝ η B q .Proof. It is verified by Figure 8. (cid:3)
Definition 7.4.
Let ξ : A Ñ B be a bimonoid homomorphism and µ P Int r p ξ q . If ξ isconormal, a morphism ˇ F p µ q : Ñ Ker p ξ q is defined as follows. By Lemma 7.3, µ ˝ η B isdecomposed into(60) ϕ Ñ A z β Ð ξ Ñ A . Since ξ is conormal, A z β Ð ξ gives a kernel bimonoid of ξ , Ker p ξ q so that the morphism ϕ defines ˇ F p µ q : Ñ Ker p ξ q .If ξ is normal, we define a morphism ˆ F p µ q : Cok p ξ q Ñ in an analogous way, i.e. (cid:15) A ˝ µ is decomposed into(61) B Ñ Cok p ξ q ˆ F p µ q Ñ . F igure Theorem 7.5.
Let A , B be bimonoids and ξ : A Ñ B be a bimonoid homomorphism Let µ P Int r p ξ q . (1) Suppose that ξ is conormal. Then the morphism ˇ F p µ q : Ñ Ker p ξ q is defined and itis a right integral of Ker p ξ q . If the integral µ along ξ is normalized, then the integral ˇ F p µ q is normalized. (2) Suppose that ξ : A Ñ B is normal. Then the morphism ˆ F p µ q : Cok p ξ q Ñ is definedand it is a right cointegral of Cok p ξ q . If the integral µ along ξ is normalized, then thecointegral ˆ F p µ q is normalized.Proof. We only prove the first part. For simplicity, let us write j “ ker p ξ q : Ker p ξ q Ñ A . Weprove that ∇ Ker p ξ q ˝ p ˇ F p µ q b id Ker p ξ q q “ ˇ F p µ q b (cid:15) Ker p ξ q . Due to the universality of kernels, itsu ffi ces to show that j ˝ ∇ Ker p ξ q ˝ p ˇ F p µ q b id Ker p ξ q q “ j ˝ p ˇ F p µ q b (cid:15) Ker p ξ q q . See Figure 9.Let us prove that ˇ F p µ q is normalized if µ is normalized. It is shown by the following directcalculation : (cid:15) Ker p ξ q ˝ ˇ F p µ q “ (cid:15) A ˝ ker p ξ q ˝ ˇ F p µ q (62) “ (cid:15) A ˝ µ ˝ η B (63) “ (cid:15) B ˝ ξ ˝ µ ˝ ξ ˝ η A (64) “ (cid:15) B ˝ ξ ˝ η A p µ : normalized q (65) “ id (66) F igure (cid:3)
8. C omputation of
Int p ξ q In this section, we compute
Int p ξ q by using ˇ F , ˆ F in Definition 7.4. The main result inthis subsection is that if ξ has a normalized generator integral, then Int p ξ q is isomorphic to End C p q , the endomorphism set of the unit P C . NTEGRALS ALONG BIMONOID HOMOMORPHISMS 25
Definition 8.1.
Let A , B be bimonoids and ξ : A Ñ B be a bimonoid homomorphism witha kernel bimonoid Ker p ξ q . Let ϕ P Mor C p , Ker p ξ qq and µ P Int r p ξ q . We define ϕ ˙ µ P Mor C p B , A q by ϕ ˙ µ def . “ ˆ B l ´ B Ñ b B ϕ b id B Ñ Ker p ξ q b B ker p ξ qb µ Ñ A b A ∇ A Ñ A ˙ (67) µ ¸ ϕ def . “ ˆ B r ´ B Ñ B b id B b ϕ Ñ B b Ker p ξ q µ b ker p ξ q Ñ A b A ∇ A Ñ A ˙ (68) Remark 8.2.
The definitions of ϕ ˙ µ and µ ¸ ϕ can be understood via some string diagramsin Figure 10. F igure Proposition 8.3.
Let µ P Int r p ξ q . Then we have ‚ ϕ ˙ µ P Int r p ξ q . ‚ µ ¸ ϕ “ p (cid:15) Ker p ξ q ˝ ϕ q ¨ µ P Int r p ξ q .Proof. For simplicity we denote j “ ker p ξ q : Ker p ξ q Ñ A . We show that ϕ ˙ µ P Int r p ξ q .The axiom (4) is verified by Figure 11. The axiom (5) is verified by Figure 12. Note that thetarget of ϕ needs to be Ker p ξ q to verify Figure 12.We show that µ ¸ ϕ “ p (cid:15) Ker p ξ q ˝ ϕ q ¨ µ P Int r p ξ q . The equation is verified by Figure 13.Since µ P Int r p ξ q , µ ¸ ϕ lives in Int r p ξ q . F igure (cid:3) Lemma 8.4.
Let ξ : A Ñ B be a bimonoid homomorphism which is conormal. Let µ be agenerator integral along ξ . For an integral µ P Int p ξ q , we have ˇ F p µ q ˙ µ “ µ . (69) F igure igure In particular, if a bimonoid homomorphism ξ has a generator integral, then ˇ F : Int p ξ q Ñ Int p Ker p ξ qq is injective.Proof. It follows from Figure 14. (cid:3)
Theorem 8.5.
Let ξ : A Ñ B be a bimonoid homomorphism which is either conormalor normal. Let µ be a normalized generator integral along ξ . Then the map End C p q Ñ Int p ξ q ; λ ÞÑ λ ¨ µ is a bijection.Proof. We only prove the statement for conormal ξ . It su ffi ces to replace ˇ F p µ q with ˆ F p µ q fornormal ξ and other discussion with a dual one.We claim that Int p ξ q Ñ End C p q ; µ ÞÑ (cid:15) Ker p ξ q ˝ ˇ F p µ q gives an inverse map. It su ffi ces toprove that µ “ ` (cid:15) Ker p ξ q ˝ ˇ F p µ q ˘ ¨ µ and (cid:15) Ker p ξ q ˝ ˇ F p λ ¨ µ q “ λ . The latter one follows from (cid:15) Ker p ξ q ˝ ˇ F p µ q “ id which is nothing but the normality of ˇ F p µ q by Theorem 7.5. We show theformer one by calculating ˇ F p µ q ˙ µ in a di ff erent way. It follows from Figure 15. By Lemma NTEGRALS ALONG BIMONOID HOMOMORPHISMS 27 F igure igure F p µ q ˙ µ “ µ , so that µ “ ` (cid:15) Ker p ξ q ˝ ˇ F p µ q ˘ ¨ µ . (cid:3)
9. E xistence of normalized generator integral
In this section, we give a su ffi cient condition for a normalized generator integral alonga homomorphism exists in Theorem 9.9. By Proposition 7.1, such a normalized generatorintegral is unique.9.1. Key Lemma.Lemma 9.1.
Let A , B be bimonoids. Let ξ : A Ñ B be a bimonoid homomorphism. (1)
Suppose that A is small. In particular, the canonical morphism ξ γ : α Ñ ξ zz B Ñ α Ñ ξ z Bis an isomorphism. Here, the left action α Ñ ξ is defined in Definition 5.2. Let µ “ ˆ α Ñ ξ z B p ξ γ q ´ Ñ α Ñ ξ zz B Ñ B ˙ . (70) If α Ñ ξ z B has a bimonoid structure such that the canonical morphism π : B Ñ α Ñ ξ z Bis a bimonoid homomorphism, then we have ‚ µ P Int r p π q . In particular, Int r p π q ‰ H . ‚ π ˝ µ “ id α Ñ ξ z B . In particular, the right integral µ is normalized. ‚ By Remark 6.9, the bimonoid A has a normalized integral σ A . We have, µ ˝ π “ L α Ñ ξ p σ A q . (71) If B is commutative, then µ P Int l p π q , in particular, µ P Int p π q ‰ H . We have ananalogous statement for the right action p B , α Ð ξ , A q . (2) Suppose that B is cosmall. In particular, the canonical morphism γ ξ : A z β Ð ξ Ñ A zz β Ð ξ is an isomorphism. Here, the right coaction β Ð ξ is defined in Definition 5.2. Let µ “ ˆ A Ñ A zz β ξ p γ ξ q ´ Ñ A z β ξ ˙ . (72) If A z β ξ has a bimonoid structure such that the canonical morphism ι : A z β Ñ A is abimonoid homomorphism, then we have ‚ µ P Int l p ι q . In particular, Int l p ι q ‰ H . ‚ µ ˝ ι “ id A z β Ð ξ . In particular, the left integral µ is normalized. ‚ By Remark 6.9, the bimonoid B has a normalized cointegral σ B . We have, ι ˝ µ “ R β Ð ξ p σ B q . (73) If A is cocommutative, then µ P Int r p ι q , in particular, µ P Int p ι q ‰ H . We have ananalogous statement for the left coaction p B , β Ñ ξ , A q .Proof. We prove the first claim here and leave the second claim to the readers. Recall Lemma6.8 that a small bimonoid A has a normalized integral. We denote the normalized integral by σ A : Ñ A .We prove that µ satisfies the axiom (4). Denote by j : α Ñ ξ zz B Ñ B the canonical mor-phism. Since γ “ ξ γ is an isomorphism, it su ffi ces to show that ∇ B ˝ pp µ ˝ γ q b id B q “ µ ˝ ∇ α Ñ ξ z B ˝ p γ b π q . It is verified by Figure 16.F igure µ satisfies the axiom (5). Due to the universality of π : B Ñ α Ñ ξ z B , itsu ffi ces to show that p µ b id α Ñ ξ z B q ˝ ∆ α Ñ ξ z B ˝ π “ p id B b π q ˝ ∆ B ˝ µ ˝ π . It is verified byFigure 17. Thus, we obtain µ P Int r p π q .The claim π ˝ µ id α Ñ ξ z B follows from π ˝ µ “ ξ γ ˝ p ξ γ q ´ “ id α Ñ ξ z B .The claim µ ˝ π “ L α Ñ ξ p σ A q follows from the definition of α Ñ ξ and Proposition 6.12.From now on, we suppose that B is commutative and show that µ P Int l p π q . We provethat µ satisfies the axiom (6). Since γ “ ξ γ is an isomorphism, it su ffi ces to show that NTEGRALS ALONG BIMONOID HOMOMORPHISMS 29 F igure ∇ B ˝ p id B b p µ ˝ γ qq “ µ ˝ ∇ α Ñ ξ z B ˝ p π b γ q . It is verified by Figure 18. We need thecommutativity of B here. F igure We prove that µ satisfies the axiom (7). Due to the universality of π : B Ñ α Ñ ξ z B , itsu ffi ces to show that p id α Ñ ξ z B b µ q ˝ ∆ α Ñ ξ z B ˝ π “ p π b id q ˝ ∆ B ˝ µ ˝ π . It is verified by Figure19. F igure (cid:3) Definition 9.2.
Let A , B be bimonoids in a symmetric monoidal category C and ξ : A Ñ B be a bimonoid homomorphism. Suppose that the bimonoid A is small and ξ is normal.By Lemma 9.1, there exists a normalized right integral along the homomorphism cok p ξ q : B Ñ Cok p ξ q . Analogously, there also exists a normalized left integral along cok p ξ q since thehomomorphism ξ is normal. By Proposition 7.1, these coincide to each other. Denote thenormalized integral by ˜ µ cok p ξ q P Int p cok p ξ qq .Suppose that B is cosmall and ξ is conormal. Analogously, by Lemma 9.1, we define anormalized integral ˜ µ ker p ξ q P Int p ker p ξ qq . Lemma 9.3.
Let A , B be bimonoids and ξ : A Ñ B be a bimonoid homomorphism. Supposethat A is small and the homomorphism ξ is normal. Then we havecok p ξ q ˝ ˜ µ cok p ξ q “ id Cok p ξ q (74) ˜ µ cok p ξ q ˝ cok p ξ q “ L α Ñ ξ p σ A q (75) “ R α Ð ξ p σ A q (76) In particular, cok p ξ q has a section in C .Suppose that B is cosmall and the canonical morphism ξ is conormal. Then we have, ˜ µ ker p ξ q ˝ ker p ξ q “ id Ker p ξ q (77) ker p ξ q ˝ ˜ µ ker p ξ q “ R β Ð ξ p σ B q (78) “ L β Ñ ξ p σ B q (79) In particular, ker p ξ q has a retract in C .Proof. It follows from the definitions of ˜ µ cok p ξ q , ˜ µ ker p ξ q and Lemma 9.1. (cid:3) NTEGRALS ALONG BIMONOID HOMOMORPHISMS 31
Proof of Theorem 3.2.
In this subsection, we prove Theorem3.2 which follows fromTheorem 9.9.
Definition 9.4.
Let A , B be bimonoids and ξ : A Ñ B be a bimonoid homomorphism with akernel bimonoid Ker p ξ q . Suppose that Ker p ξ q is small and the canonical morphism ker p ξ q : Ker p ξ q Ñ A is normal. We define a normalized integral along coim p ξ q “ cok p ker p ξ qq : A Ñ Coim p ξ q by ˜ µ cok p ζ q in Definition 9.2 where ζ “ ker p ξ q . We denote ˜ µ cok p ζ q by ˜ µ coim p ξ q P Int p coim p ξ qq .Analogously we define ˜ µ im p ξ q : Let A , B be bimonoids and ξ : A Ñ B be a bimonoidhomomorphism with a cokernel bimonoid Cok p ξ q . Suppose that Cok p ξ q is cosmall and thecanonical morphism ker p ξ q : Ker p ξ q Ñ A is conormal. We define a normalized integralalong im p ξ q “ ker p cok p ξ qq : A Ñ Im p ξ q by ˜ µ ker p ζ q in Definition 9.2 where ζ “ cok p ξ q . Wedenote ˜ µ ker p ζ q by ˜ µ im p ξ q P Int p im p ξ qq . Lemma 9.5.
Let A , B be bimonoids and ξ : A Ñ B be a bimonoid homomorphism with akernel Ker p ξ q . Suppose that the kernel bimonoid Ker p ξ q is small and the canonical morphismker p ξ q : Ker p ξ q Ñ A is normal. Then we havecoim p ξ q ˝ ˜ µ coim p ξ q “ id Coim p ξ q (80) ˜ µ coim p ξ q ˝ coim p ξ q “ L α Ñ ker p ξ q p σ Ker p ξ q q (81) “ R α Ð ker p ξ q p σ Ker p ξ q q (82) In particular, coim p ξ q has a section in C .An analogous statement for Im p ξ q holds : Let A , B be bimonoids and ξ : A Ñ B be abimonoid homomorphism with a cokernel bimonoid Cok p ξ q . Suppose that Cok p ξ q is cosmalland the canonical morphism cok p ξ q : B Ñ Cok p ξ q is conormal. Then we have, ˜ µ im p ξ q ˝ im p ξ q “ id Im p ξ q (83) im p ξ q ˝ ˜ µ im p ξ q “ R β Ð cok p ξ q p σ Cok p ξ q q (84) “ L β Ñ cok p ξ q p σ Cok p ξ q q (85) In particular, im p ξ q has a retract in C .Proof. It follows from Lemma 9.3. (cid:3)
Definition 9.6.
Let A , B be bimonoids. A bimonoid homomorphism ξ : A Ñ B is weaklywell-decomposable if following conditions hold : ‚ Ker p ξ q , Cok p ξ q , Coim p ξ q , Im p ξ q exist in Bimon p C q . ‚ ker p ξ q : Ker p ξ q Ñ A is normal and cok p ξ q : B Ñ Cok p ξ q is conormal. ‚ ¯ ξ : Coim p ξ q Ñ Im p ξ q is an isomorphism.A bimonoid homomorphism ξ : A Ñ B is well-decomposable if following conditions hold: ‚ ξ is binormal. In particular, Ker p ξ q , Cok p ξ q exist in Bimon p C q . ‚ ker p ξ q : Ker p ξ q Ñ A is normal and cok p ξ q : B Ñ Cok p ξ q is conormal. In particular, Coim p ξ q , Im p ξ q exist. ‚ ¯ ξ : Coim p ξ q Ñ Im p ξ q is an isomorphism. Definition 9.7.
Let ξ : A Ñ B be a weakly well-decomposable homomorphism. The homo-morphism ξ is weakly pre-Fredholm if the kernel bimonoid Ker p ξ q is small and the cokernelbimonoid Cok p ξ q is cosmall. Recall Definition 9.4. For a weakly pre-Fredholm homomor-phism ξ : A Ñ B , we define µ ξ def . “ ˜ µ coim p ξ q ˝ ¯ ξ ´ ˝ ˜ µ im p ξ q : B Ñ A . (86) The homomorphism ξ is pre-Fredholm if if both of the kernel bimonoid Ker p ξ q and the cok-ernel bimonoid Cok p ξ q are bismall. Proposition 9.8.
Let A be a bimonoid. (1)
The unit η A : Ñ A and the counit (cid:15) A : A Ñ are well-decomposable. (2) The unit η A is weakly pre-Fredholm if and only if A is cosmall. Then µ η A in Definition9.7 is well-defined and we have µ η A “ σ A . (3) The counit (cid:15) A is weakly pre-Fredholm if and only if A is small. Then µ (cid:15) A in Definition9.7 is well-defined and we have µ (cid:15) A “ σ A .Proof. We prove that η A is well-decomposable and leave the proof of (cid:15) A to the readers. Notethat the unit bimonoid is bismall since it has a normalized (co)integral. The bimonoidhomomorphism η A is normal due to the canonical isomorphism α η A z A Ð A “ Cok p η A q . Thebimonoid homomorphism η A is conormal due to the canonical isomorphism z β η A Ñ “ Ker p η A q . Moreover, ker p η A q : Ker p η A q “ Ñ and cok p η A q : A Ñ Cok p η A q “ A arenormal and conormal due to Proposition 5.5. The final axiom is verified since ¯ η A : “ Coim p η A q Ñ Im p η A q “ is the identity.The morphism µ η A is a normalized integral by the following Theorem 9.9. By Proposition7.1, we obtain µ η A “ σ A . (cid:3) Theorem 9.9.
Let A , B be bimonoids and ξ : A Ñ B be a weakly well-decomposable ho-momorphism. If the homomorphism ξ is weakly pre-Fredholm, then the morphism µ ξ is anormalized generator integral along ξ .Proof. Recall that ˜ µ coim p ξ q P Int p coim p ξ qq , ˜ µ im p ξ q P Int p im p ξ qq by Definition 9.4. By Proposi-tion 2.8, ¯ ξ ´ P Int p ¯ ξ q . By Proposition 2.10, µ ξ is an integral along ξ since µ ξ is defined to bea composition of ˜ µ coim p ξ q , ˜ µ im p ξ q , ¯ ξ ´ .Note that µ ξ ˝ ξ “ ˜ µ coim p ξ q ˝ coim p ξ q . In fact, by Lemma 9.5, we have µ ξ ˝ ξ “ ` ˜ µ coim p ξ q ˝ ¯ ξ ´ ˝ ˜ µ im p ξ q ˘ ˝ p im p ξ q ˝ ¯ ξ ˝ coim p ξ qq (87) “ ˜ µ coim p ξ q ˝ ¯ ξ ´ ˝ ¯ ξ ˝ coim p ξ q (88) “ ˜ µ coim p ξ q ˝ coim p ξ q (89)We prove that the integral µ ξ is normalized, i.e. ξ ˝ µ ξ ˝ ξ “ ξ . By Lemma 9.5, we have˜ µ coim p ξ q ˝ coim p ξ q “ L α Ñ ker p ξ q p σ Ker p ξ q q . Then the claim ξ ˝ µ ξ ˝ ξ “ ξ follows from Figure 20where we put j “ ker p ξ q . F igure µ ξ is a generator. We first prove that µ ξ ˝ ξ ˝ µ “ µ for any µ P Int l p ξ q Y Int r p ξ q . By Lemma 9.5, we have ˜ µ coim p ξ q ˝ coim p ξ q “ R α Ð ker p ξ q p σ Ker p ξ q q . Weobtain µ ξ ˝ ξ ˝ µ “ µ for arbitrary µ P Int l p ξ q from Figure 21 where we put j “ ker p ξ q . NTEGRALS ALONG BIMONOID HOMOMORPHISMS 33 F igure µ ξ ˝ ξ ˝ µ “ µ for arbitrary µ P Int r p ξ q by using ˜ µ coim p ξ q ˝ coim p ξ q “ L α Ñ ker p ξ q p σ Ker p ξ q q in Lemma 9.5.All that remain is to prove that µ ˝ ξ ˝ µ ξ “ µ for any µ P Int l p ξ q Y Int r p ξ q . Note thatwe have ξ ˝ µ ξ “ im p ξ q ˝ ˜ µ im p ξ q by Lemma 9.5. We prove that µ ˝ im p ξ q ˝ ˜ µ im p ξ q “ µ forarbitrary µ P Int l p ξ q . By Lemma 9.5, we have im p ξ q ˝ ˜ µ im p ξ q “ R β Ð cok p ξ q p σ Cok p ξ q q . Then the claim µ ˝ im p ξ q˝ ˜ µ im p ξ q “ µ follows from Figure 22. Analogously, we prove that µ ˝ im p ξ q˝ ˜ µ im p ξ q “ µ for arbitrary µ P Int r p ξ q by using im p ξ q ˝ ˜ µ im p ξ q “ L β Ñ cok p ξ q p σ Cok p ξ q q in Lemma 9.5. It completesthe proof. F igure (cid:3)
10. P roof of T heorem Lemma 10.1.
Consider the following commutative diagram of bimonoid homomorphisms.Suppose that ϕ, ψ are weakly well-decomposable and weakly pre-Fredholm.A CB D ϕ ϕ ψψ Then we have ψ ˝ p ϕ ˝ µ ϕ q ˝ ϕ “ ψ ˝ p µ ψ ˝ ψ q ˝ ϕ . In particular, if ϕ is an epimorphism in C and ψ is a monomorphism in C , then ϕ ˝ µ ϕ “ µ ψ ˝ ψ .Proof. Since µ ϕ is normalized, we have, ψ ˝ ϕ ˝ µ ϕ ˝ ϕ “ ψ ˝ ϕ ˝ µ ϕ ˝ ϕ (90) “ ψ ˝ ϕ. (91) Since µ ψ is normalized, we have ψ ˝ µ ψ ˝ ψ ˝ ϕ “ ψ ˝ µ ψ ˝ ψ ˝ ϕ (92) “ ψ ˝ ϕ . (93)It completes the proof. (cid:3) Proof of Theorem 3.4
By Theorem 9.9, the morphisms µ ϕ , µ ψ in Definition 9.7 are thenormalized generator integrals. Note that the homomorphisms in the above diagram are de-composed into following diagram. A CCoim p ϕ q Coim p ψ q Im p ϕ q Im p ψ q B D ϕ coim p ϕ q ϕ coim p ψ q ¯ ϕ ˜ µ coim p ϕ q ¯ ψ ˜ µ coim p ψ q ψ im p ϕ q im p ψ q ψ ˜ µ im p ϕ q ˜ µ im p ψ q By Lemma 10.1, we have ϕ ˝ ˜ µ coim p ϕ q ˝ ¯ ϕ ´ “ ¯ ψ ´ ˝ ˜ µ im p ψ q ˝ ψ . Here, we use the fact that coim p ϕ q is an epimorphism in C and im p ψ q is a monomorphism in C by Lemma 9.5. Thus,we have coim p ψ q ˝ ϕ ˝ ˜ µ coim p ϕ q ˝ ¯ ϕ ´ “ ¯ ψ ´ ˝ ˜ µ im p ψ q ˝ ψ ˝ im p ϕ q .We claim that(1) ˜ µ coim p ψ q ˝ coim p ψ q ˝ ϕ ˝ ˜ µ coim p ϕ q “ ϕ ˝ ˜ µ coim p ϕ q .(2) ˜ µ im p ψ q ˝ ψ ˝ im p ϕ q ˝ ˜ µ im p ϕ q “ ˜ µ im p ψ q ˝ ψ .By these claims, we have µ ψ ˝ ψ “ ˜ µ coim p ψ q ˝ ¯ ψ ´ ˝ ˜ µ im p ψ q ˝ ψ (94) “ ˜ µ coim p ψ q ˝ ¯ ψ ´ ˝ ˜ µ im p ψ q ˝ ψ ˝ im p ϕ q ˝ ˜ µ im p ϕ q (95) “ ˜ µ coim p ψ q ˝ coim p ψ q ˝ ϕ ˝ ˜ µ coim p ϕ q ˝ ¯ ϕ ´ ˝ ˜ µ im p ϕ q (96) “ ϕ ˝ ˜ µ coim p ϕ q ˝ ¯ ϕ ´ ˝ ˜ µ im p ϕ q (97) “ ϕ ˝ µ ϕ . (98)It su ffi ces to prove the above claims.From now on, we show the first claim. We use the hypothesis to prove ϕ ˝ ker p ϕ q ˝ σ Ker p ϕ q “ ker p ψ q ˝ σ Ker p ψ q . Since ϕ “ ϕ | Ker p ϕ q : Ker p ϕ q Ñ Ker p ψ q has a section in C ,we have ϕ ˝ σ Ker p ϕ q “ σ Ker p ψ q by Lemma 11.7. Hence, we obtain ϕ ˝ ker p ϕ q ˝ σ Ker p ϕ q “ ker p ψ q ˝ ϕ ˝ σ Ker p ϕ q “ ker p ψ q ˝ σ Ker p ψ q .Recall that ˜ µ coim p ψ q ˝ coim p ψ q : C Ñ C coincides with the action by ker p ψ q˝ σ Ker p ψ q : Ñ C by Lemma 9.5. Then Figure 23 completes the proof of the first claim.Dually we can prove the second claim. Here, we use the section of ψ : Cok p ϕ q Ñ Cok p ψ q and apply Lemma 11.7 again. It completes the proof.11. I nverse volume Inverse volume of bimonoid.
In this subsection, we introduce a notion of inversevolume vol ´ p A q of a bimonoid A with a normalized integral and a normalized cointegral.It gives an invariant of such bimonoids by Proposition 11.4. By Remark 6.9, it defines aninvariant of bismall bimonoids. NTEGRALS ALONG BIMONOID HOMOMORPHISMS 35 F igure Definition 11.1.
Let A be a bimonoid with a normalized integral σ A : Ñ A and a nor-malized cointegral σ A : A Ñ . An inverse volume of the bimonoid A is an endomorphism vol ´ p A q : Ñ in C , defined by a compostiion, vol ´ p A q def . “ σ A ˝ σ A . (99) Definition 11.2.
A bimonoid
A has a finite volume if A has a normalized integral and anormalized cointegral, and its inverse volume vol ´ p A q : Ñ is invertible. Example 11.3.
Consider the symmetric monoidal category, C “ Vec b F . Let G be a finitegroup. Suppose that the characteristic of F is not a divisor of the order G of G. Thenthe induced Hopf monoid A “ F G in
Vec b F has a normalized integral σ A and a normalizedcointegral σ A . In particular, σ A : F Ñ F G ; 1 ÞÑ p7 G q ´ ÿ g P G g , (100) σ A : F G Ñ F ; g ÞÑ δ e p g q , (101) give a normalized integral and a normalized cointegral of A “ F G respectively.. Then wehave vol ´ p F p G qq : F Ñ F ; 1 ÞÑ p7 G q ´ . (102) Proposition 11.4.
Let A , B be bimonoids with a normalized integral and a normalized coin-tegral. ‚ For the unit bimonoid, we have vol ´ p q “ id . ‚ A bimonoid isomorphism A – B implies vol ´ p A q “ vol ´ p B q . ‚ vol ´ p A b B q “ vol ´ p A q ˝ vol ´ p B q “ vol ´ p B q ˝ vol ´ p A q . ‚ If A _ is a dual bimonoid of the bimonoid A, then the bimonoid A _ has a normalizedintegral and a normalized cointegral and we havevol ´ p A _ q “ vol ´ p A q . (103) Proof.
Since σ “ σ “ id , we have vol ´ p q “ id .If A – B as bimonoids, then their normalized (co)integrals coincide via that isomorphismdue to their uniqueness. Hence, we have vol ´ p A q “ σ A ˝ σ A “ σ B ˝ σ B “ vol ´ p B q .Since σ A b B “ σ A b σ B : Ñ A b B and σ A b B : σ A b σ B : A b B Ñ , we have vol ´ p A b B q “ vol ´ p A q ˚ vol ´ p B q “ vol ´ p A q ˝ vol ´ p B q “ vol ´ p B q ˝ vol ´ p A q . By direct calculations, the following morphisms give a normalized integral and a normal-ized cointegral on the dual bimonoid A _ : σ A _ “ ´ coev A Ñ A _ b A id A _ b σ A Ñ A _ b – A _ ¯ (104) σ A _ “ ´ A _ – b A _ σ A b id A _ Ñ A b A _ ev A Ñ ¯ (105)It implies that σ A _ ˝ σ A _ “ σ A ˝ σ A since l A ˝ p ev A b id A q ˝ p id A b coev A q ˝ r A “ id A . (cid:3) Inverse volume of homomorphisms.Definition 11.5.
Let A be a bimonoid with a normalized integral σ A and B be a bimnoidwith a normalized cointegral σ B . For a bimonoid homomorphism ξ : A Ñ B , we define amorphism x ξ y : Ñ by x ξ y def . “ σ B ˝ ξ ˝ σ A . (106) Remark 11.6.
Since x id A y “ vol ´ p A q by definitions, x´y is an extended notion of the inversevolume in Definition 11.1. On the other hand, for some special ξ , we can compute x ξ y froman inverse volume. See Proposition 11.9. Lemma 11.7.
Let A , B be bimonoids. Let σ A be a normalized integral of A. Let ξ : A Ñ B bea bimonoid homomorphism. If there exists a morphism ξ : B Ñ A in C such that ξ ˝ ξ “ id A ,then ξ ˝ σ A is a normalized integral of B.Proof. The morphism ξ ˝ σ A : Ñ B is a right integral due to Figure 24. It can be verifiedto be a left integral in a similar way. Moreover, it is normalized since we have (cid:15) ξ ˝ ξ ˝ σ A “ (cid:15) A ˝ σ A “ id . F igure (cid:3) Proposition 11.8.
Let ξ : A Ñ B be a bimonoid homomorphism. Suppose that every idem-potent in the symmetric monoidal category C is a split idempotent. If the bimonoid A is smalland there exists a morphism ξ : B Ñ A in C such that ξ ˝ ξ “ id A , then the bimonoid B issmall.Proof. It is immediate from Lemma 11.7 and Theorem 6.13. (cid:3)
Proposition 11.9.
Let ξ : A Ñ B be a bimonoid homomorphism. Suppose that a kernel bimo-niod Ker p ξ q , a cokernel bimonoid Cok p ξ q , a coimage bimonoid Coim p ξ q , an image bimonoidIm p ξ q exist. Suppose that Ker p ξ q is small and Cok p ξ q is cosmall. Suppose that the canonicalhomomorphism ker p ξ q : Ker p ξ q Ñ A is normal and cok p ξ q : B Ñ Cok p ξ q is conormal. Thenfor the canonical homomorphism ¯ ξ : Coim p ξ q Ñ Im p ξ q , we have, x ξ y “ x ¯ ξ y . (107) In particular, if ¯ ξ is an isomorphism, then we have x ξ y “ x ¯ ξ y “ vol ´ p Coim p ξ qq “ vol ´ p Im p ξ qq . NTEGRALS ALONG BIMONOID HOMOMORPHISMS 37
Proof.
It su ffi ces to prove that x ξ y “ x ¯ ξ y . Since x ξ y “ σ B ˝ ξ ˝ σ A “ σ B ˝ im p ξ q ˝ ¯ ξ ˝ coim p ξ q ˝ σ A , it su ffi ces to show that coim p ξ q ˝ σ A “ σ Coim p ξ q and σ B ˝ im p ξ q “ σ Im p ξ q . Themorphism coim p ξ q ( im p ξ q , resp.) has a section (retract, resp.) in C by Lemma 9.5. Hence, thecompositions coim p ξ q ˝ σ A ( σ B ˝ im p ξ q , resp.) are normalized integrals by Lemma 11.7. Itcompletes the proof. (cid:3)
12. P roof of T heorem Theorem 12.1.
Let A , B , C be bimonoids. Let ξ : A Ñ B, ξ : B Ñ C be bimonoid homomor-phism. Suppose that ‚ ξ is normal, ξ is conormal. The composition ξ ˝ ξ is either conormal or normal. ‚ µ, µ are normalized integrals along ξ, ξ respectively. µ is a normalized integralalong ξ ˝ ξ , which is a generator.Recall that the cokernel bimonoid Cok p ξ q has a normalized cointegral and the kernel bi-monoid Ker p ξ q has a normalized integral by Theorem 7.5. Then we have, µ ˝ µ “ x cok p ξ q ˝ ker p ξ qy ¨ µ . (108) Proof.
By Proposition 2.10, µ ˝ µ is an integral along the composition ξ ˝ ξ . By Theorem8.5, there exists a unique λ P End C p q such that µ ˝ µ “ λ ¨ µ since ξ ˝ ξ is either conormalor normal.We have (cid:15) A ˝ µ ˝ η C “ id due to the following computation : (cid:15) A ˝ µ ˝ η C “ p (cid:15) C ˝ ξ ˝ ξ q ˝ µ ˝ p ξ ˝ ξη A q (109) “ (cid:15) C ˝ p ξ ˝ ξ ˝ µ ˝ ξ ˝ ξ q ˝ η A (110) “ (cid:15) C ˝ p ξ ˝ ξ q ˝ η A p µ : normalized q (111) “ id (112)Hence it su ffi ces to calculate (cid:15) A ˝ µ ˝ µ ˝ η C to know λ . Since ξ is conormal, we have amorphism ˇ F p µ q such that µ ˝ η C “ ker p ξ q ˝ ˇ F p µ q (see Definition 7.4). Since ξ is normal,we have a morphism ˆ F p µ q such that (cid:15) A ˝ µ “ ˆ F p µ q ˝ cok p ξ q . Since the integrals µ, µ arenormalized, ˇ F p µ q and ˆ F p µ q are normalized integrals by Theorem 7.5. By using our notations,ˇ F p µ q “ σ Ker p ξ q and ˆ F p µ q “ σ Cok p ξ q . Therefore, we have (cid:15) A ˝ µ ˝ µ ˝ η C “ σ Cok p ξ q ˝ cok p ξ q ˝ ker p ξ q ˝ σ Ker p ξ q “ x cok p ξ q ˝ ker p ξ qy by definitions. It completes the proof. (cid:3) Corollary 12.2.
Let A , B be bimonoids and ξ : A Ñ B be a bimonoid homomorphism.Suppose that ‚ ξ is normal. ‚ µ is a normalized integral along ξ , σ B is a normalized integral of B, and σ A is anormalized integral of A which is a generator.Then we have µ ˝ σ B “ vol ´ p Cok p ξ qq ¨ σ A . (113) We have an analogous statement. Suppose that ‚ ξ is conormal. ‚ µ is a normalized integral along ξ , σ A is a normalized cointegral of A, and σ B is anormalized integral of B which is a generator.Then we have σ A ˝ µ “ vol ´ p Ker p ξ qq ¨ σ B . (114) Proof.
We prove the first claim. We replace ξ, ξ in Theorem 12.1 with ξ, (cid:15) B in the aboveassumption. Then the assumption in Theorem 12.1 is satisfied.We prove the second claim. We replace ξ, ξ in Theorem 12.1 with η A , ξ in the aboveassumption. Then the assumption in Theorem 12.1 is satisfied. (cid:3) Corollary 12.3.
Let A , B be bimonoids and ξ : A Ñ B be a bimonoid homomorphism.Suppose that ‚ ξ is binormal. ‚ There exists a normalized integral along ξ . ‚ A, B are bismall ‚ The normalized integral σ A of A is a generator. The normalized cointegral σ B of B isa generator.Then we have vol ´ p Cok p ξ qq ˝ vol ´ p A q “ vol ´ p Ker p ξ qq ˝ vol ´ p B q . (115) Proof.
Since A , B are bismall, the counit (cid:15) A and the unit η B are pre-Fredholm. Since the counit (cid:15) A and the unit η B are well-decomposable, the normalized integral σ A of A and te normalizedcointegral σ B of B are generators by Theorem 9.9. Hence, the assumptions in Corollary 12.2are satisfied. By Corollary 12.2, we obtain µ ξ ˝ σ B “ vol ´ p Cok p ξ qq ¨ σ A , (116) σ A ˝ µ ξ “ vol ´ p Ker p ξ qq ¨ σ B . (117)Hence, we obtain vol ´ p Cok p ξ qq ¨ σ A ˝ σ A “ vol ´ p Ker p ξ qq ¨ σ B ˝ σ B , which is equivalentwith (115). (cid:3) Proof of Theorem 3.6
It is a corollary of Theorem 12.1. Since ξ, ξ , ξ ˝ ξ are well-decomposable, in particular weakly well-decomposable, and weakly pre-Fredholm, we ob-tain normalized generator integrals µ ξ , µ ξ , µ ξ ˝ ξ by Theorem 9.9. Since ξ, ξ , ξ ˝ ξ are well-decomposable, they satisfy the first assumption in Theorem 12.1. By Theorem Theorem 9.9,the integrals µ “ µ ξ , µ “ µ ξ , µ “ µ ξ ˝ ξ satisfy the second assumption in Theorem 12.1.13. I nduced bismallness In this section, we assume that every idempotent in a symmetric monoidal category C is asplit idempotent.13.1. Bismallness of (co)kernels.
In this subsection, we give some conditions where
Ker p ξ q , Cok p ξ q inherits a (co)smallness from that of the domain and the target of ξ . Proposition 13.1.
Let ξ : A Ñ B be a bimonoid homomorphism. Suppose that A is small, Bis cosmall. If ξ is normal, then Cok p ξ q is cosmall. If ξ is conormal, then Ker p ξ q is small.Proof. We only prove the first claim. Let ξ be normal. We have Cok p ξ q “ α Ñ ξ z B . Thereexists a normalized cointegral of B since B is cosmall by Corollary 6.14. We denote it by σ B : B Ñ . Put σ “ σ B ˝ ˜ µ cok p ξ q : Cok p ξ q “ α Ñ ξ z B Ñ . Note that σ P Int r p η α Ñ ξ z B q due toProposition 2.10. In other words, σ is a right cointegral of Cok p ξ q “ α Ñ ξ z B .We prove that σ is normalized. Let π : B Ñ α Ñ ξ z B be the canonical morphism. We have σ ˝ η α Ñ ξ z B “ σ B ˝ ˜ µ cok p ξ q ˝ η α Ñ ξ z B “ σ B ˝ ˜ µ cok p ξ q ˝ π ˝ η B . σ ˝ η α Ñ ξ z B “ id follows from˜ µ cok p ξ q ˝ π “ L α Ñ ξ p σ A q in Lemma 9.1 (1), and (cid:15) A ˝ σ A “ id . Hence, σ is a normalized rightcointegral of α Ñ ξ z B “ Cok p ξ q .Analogously, we use Cok p ξ q “ B { α Ð ξ to verify an existence of a normalized left cointegralof Cok p ξ q . By Proposition 2.3, the cokernel Cok p ξ q has a normalized cointegral. By Corollary6.14, the cokernel bimonoid Cok p ξ q is cosmall. (cid:3) NTEGRALS ALONG BIMONOID HOMOMORPHISMS 39
Proposition 13.2.
Let A , B be bimonoids. Let ξ : A Ñ B be a bimonoid homomorphism. IfA, B are small and ξ is normal, then Cok p ξ q is small. If A, B are cosmall and ξ is conormal,then Ker p ξ q is cosmall.Proof. We only prove the first claim. The small bimonoid B has a unique normalized integral σ B : Ñ B by Corollary 6.14. By Definition 9.2, a normalized integral ˜ µ cok p ξ q P Int p cok p ξ qq exists. By Lemma 9.3, ˜ µ cok p ξ q is a section of cok p ξ q in C . By Lemma 11.7, cok p ξ q ˝ σ B is anormalized integral of Cok p ξ q . By Corollary 6.14, Cok p ξ q is small. (cid:3) Corollary 13.3.
Let A , B be bimonoids. Let ξ : A Ñ B be a well-decomposable homomor-phism. If A is small and B is cosmall, then the homomorphism ξ is weakly pre-Fredholm. Ifboth of A, B are bismall, then the homomorphism ξ is pre-Fredholm.Proof. Suppose that A is a small bimonoid and B is a cosmall bimonoid. Since ξ is well-decomposable, the cokernel bimonoid Cok p ξ q is cosmall and the kernel biomonoid Ker p ξ q issmall by Proposition 13.1.Suppose that both of A , B are bismall bimonoids. Then the homomorphism ξ is weaklypre-Fredholm by the above discussion. Moreover, the cokernel bimonoid Cok p ξ q is small andkernel bimonoid Ker p ξ q is cosmall by Proposition 13.2. (cid:3) Proof of Theorem 3.11.
In this subsection, we discuss some conditions for (co)smallnessof a bimonoid to be inherited from an exact sequence.
Lemma 13.4.
Let A , B , C be bimonoids. Let ι : B Ñ A be a normal homomorphism and π : A Ñ C be a homomorphism. Suppose that the following sequence is exact :B ι Ñ A π Ñ C Ñ (118) Here, the exactness means that π ˝ ι is trivial and the induced homomorphism Cok p ι q Ñ C isan isomorphism. If the bimonoids B, C are small, then A is small.Proof.
It su ffi ces to prove that A has a normalized integral by Corollary 6.14. We denote by σ C the normalized integral of C . Since B is small and ι is normal, we have a normalizedintegral ˜ µ cok p ι q along cok p ι q (see Definition 9.2). Since the induced homomorphism Cok p ι q Ñ C is isomorphism by the assumption, we have a normalized integral ˜ µ π along π . Then thecomposition ˜ µ π ˝ σ C : Ñ A gives an integral of A by Proposition 2.10. Moreover ˜ µ π ˝ σ C isnormalized since (cid:15) A ˝ ˜ µ π ˝ σ C “ (cid:15) C ˝ π ˝ ˜ µ π ˝ σ C “ (cid:15) C ˝ σ C “ id by Lemma 9.1. It completesthe proof. (cid:3) Proposition 13.5.
Let A , B , C , C be bimonoids. Let ι : B Ñ A be a normal homomorphism, π : C Ñ C be a conormal homomorphism and π : A Ñ C be a homomorphism. Supposethat the following sequence is exact :B ι Ñ A π Ñ C π Ñ C (119) Suppose that Cok p ι q Ñ Ker p π q is an isomorphism. If the bimonoids B , C are small and thebimonoid C is cosmall, then the bimonoid A is small.Proof. By the assumption, we obtain an exact sequence in the sense of Lemma 13.4, B ι Ñ A ¯ π Ñ Ker p π q Ñ . (120)Note that Ker p π q is small by Proposition 13.1. Since ι is normal and B , Ker p π q are small,the bimonoid A is small due to Lemma 13.4. (cid:3) We have dual statements as follows. For convenience of the readers, we give them withoutproof.
Lemma 13.6.
Let A , B , C be bimonoids. Let ι : B Ñ A be a homomorphism and π : A Ñ Cbe a conormal homomorphism. Suppose that the following sequence is exact. Ñ B ι Ñ A π Ñ C (121) Here, the exactness means that π ˝ ι is trivial and the induced morphism B Ñ Ker p ξ q is anisomorphism. If π is conormal and the bimonoids B, C are cosmall, then A is cosmall. Proposition 13.7.
Let A , B , B , C be bimonoids. Let ι : B Ñ B be a normal homomorphism, π : A Ñ C be a conormal homomorphism, and ι : B Ñ A be a homomorphism. Suppose thatthe following sequence is exact. B ι Ñ B ι Ñ A π Ñ C (122) Suppose that Cok p ι q Ñ Ker p π q is an isomorphism. If the bimonoid B are small and thebimonoids B , C is cosmall, then the bimonoid A is cosmall.proof of Theorem 3.11
Consider an exact sequence in
Hopf bc p C q where B “ “ C . B ι Ñ B ι Ñ A π Ñ C π Ñ C (123)By Proposition 5.7, any morphism in Hopf bc p C q is binormal. By Corollary 5.8, a cokernel(kernel, resp.) as a bimonoid is a cokernel (cokernel, resp.) as a bicommutative Hopf monoid.Hence, the assumptions in Proposition 13.5, 13.7 are deduced from the assumption in thestatement. By Proposition 13.5, 13.7, we obtain the result.14. V olume on abelian category In this section, we study the volume on an abelian category . For the definition, see Defini-tion 3.7.14.1.
Basic properties.Proposition 14.1.
An M-valued volume v on an abelian category A is an isomorphism in-variant. In other words, if objects A , B of A are isomorphic to each other, then we havev p A q “ v p B q .Proof. If we choose an isomorphism between A and B , then we obtain an exact sequence0 Ñ A Ñ B Ñ Ñ
0. By the second axiom in Definition 3.7, we obtain v p B q “ v p A q ¨ v p q .Since v p q “ v p A q “ v p B q . (cid:3) Proposition 14.2.
An M-valued volume v on an abelian category A is compatible with thedirect sum ‘ on the abelian category A . In other words, for objects A , B of A , we havev p A ‘ B q “ v p A q ¨ v p B q .Proof. Note that we have an exact sequence 0 Ñ A Ñ A ‘ B Ñ B Ñ
0. By the secondaxiom in Definition 3.7, we obtain v p A ‘ B q “ v p A q ¨ v p B q . (cid:3) Fredholm index.
In this subsection, we introduce a notion of index of morphisms inan abelian category.
Definition 14.3.
Let B be an abelian category and A be its abelian subcategory closed undershort exact sequences. Let M be an abelian monoid and v be an M -valued volume on A . Fortwo objects A , B of B , a morphism f : A Ñ B is Fredholm with respect to the volume v if Ker p f q and Cok p f q are essentially objects of A and the volumes v p Ker p f qq , v p Cok p f qq P M are invertible. For a Fredholm morphism f : A Ñ B , we define its Fredholm index by Ind B , A , v p f q def . “ v p Cok p f qq ¨ v p Ker p f qq ´ P M . (124) NTEGRALS ALONG BIMONOID HOMOMORPHISMS 41
Lemma 14.4.
Let A be an object of B . The identity Id A on A is Fredholm. We haveInd B , A , v p Id A q “ P M.Proof.
It follows from the fact that
Ker p Id A q “ “ Cok p Id A q whose volume is the unit1 P M . (cid:3) Lemma 14.5.
Let f : A Ñ B and g : B Ñ C be morphisms in B . If the morphismsf , g are Fredholm, then the composition g ˝ f is Fredholm. We have Ind B , A , v p g ˝ f q “ Ind B , A , v p g q ¨ Ind B , A , v p f q P M.Proof.
We use the exact sequence 0 Ñ Ker p f q Ñ Ker p g ˝ f q Ñ Ker p g q cok p f q˝ ker p g q Ñ Cok p f q Ñ Cok p g ˝ f q Ñ Cok p g q Ñ
0. Since v p Ker p g qq P M is invertible, any subobject of Ker p g q hasan invertible volume. The volume v p Ker p cok p f q ˝ ker p g qqq P M is invertible. By the in-duced exact sequence 0 Ñ Ker p f q Ñ Ker p g ˝ f q Ñ Ker p cok p f q ˝ ker p g qq Ñ
0, we see that v p Ker p g ˝ f qq P M is invertible. Likewise, v p Cok p g ˝ f qq is invertible. Hence, the composition g ˝ f is Fredholm with respect to the volume v . By repeating the second axiom of volumes inDefinition 3.7, we obtain v p Ker p f qq ¨ v p Ker p g qq ¨ v p Cok p g ˝ f qq “ v p Ker p g ˝ f qq ¨ v p Cok p f qq ¨ v p Cok p g qq . (125)It proves that Ind B , A , v p g ˝ f q “ Ind B , A , v p g q ¨ Ind B , A , v p f q P M . (cid:3) Definition 14.6.
Let B be an abelian category and A be an abelian subcategory which isclosed under short exact sequences. Let v be an M -valued volume on A . We define a category A Fr as a subcategory of A formed by any Fredholm homomorphisms. It is a well-definedcategory due to Lemma 14.4, 14.5. Proposition 14.7.
Every morphism f : A Ñ B between objects with invertible volumes isFredholm. Then we have Ind B , A , v p f q “ v p B q ˝ v p A q ´ P M . (126) Proof.
If objects A , B of A have invertible volumes, then for a morphism f : A Ñ B its kerneland cokernel have invertible volumes due to the second axiom in Definition 3.7.By the exact sequence 0 Ñ Ker p f q Ñ A f Ñ B Ñ Cok p f q Ñ
0, we have v p B q¨ v p Ker p f qq “ v p A q ¨ v p Cok p f qq . We obtain Ind B , A , v p f q “ v p B q ˝ v p A q ´ . (cid:3) Finite perturbation.
In this subsection, consider an abelian category B and its abeliansubcategory A closed under short exact sequences. See Definition3.10. Let v be an M -valuedvolume on the abelian category A where M is an abelian monoid. Definition 14.8.
Let f be a morphism in B . A morphism f in B is finite with respect to thevolume v if the value of the image of f (equivalently, the coimage of f ) by v is invertible in M .In other words, the image Im p f q is essentially an object of A and the volume v p Im p f qq P M is invertible. Proposition 14.9 (Invariance of index under finite perturbations) . Let f , k : A Ñ B be mor-phisms in B . If the morphism f is Fredholm and the morphism k is finite with respect to thevolume v, then the morphism p f ` k q : A Ñ B is Fredholm with respect to the volume v.Moreover, we have Ind B , A , v p f ` k q “ Ind B , A , v p f q P M . (127) Proof.
Denote by C the (co)image of the morphism k : A Ñ B . Note that p f ` k q is decom-posed into following morphisms : A p id A ‘ coim p k qq˝ ∆ A ÝÑ A ‘ C f ‘ id C ÝÑ B ‘ C ∇ B ˝p id B ‘ im p k qq ÝÑ B . (128) Since the volume v p C q P M is invertible, the morphisms p id A ‘ coim p k qq ˝ ∆ A and ∇ B ˝ p id B ‘ im p k qq are Fredholm with respect to the volume v . Since the morphism f is Fredholm withrespect to the volume v , so the morphism f ‘ id C is. By Lemma 14.5, p f ` k q is Fredholmand, Ind B , A , v p f ` k q (129) “ Ind B , A , v p ∇ B ˝ p id B ‘ im p k qqq ¨ Ind B , A , v p f ‘ id C q ¨ Ind B , A , v pp id A ‘ coim p k qq ˝ ∆ A q . (130)Note that Ind B , A , v p f ‘ id C q “ Ind B , A , v p f q . Moreover we have Ind B , A , v p ∇ B ˝ p id B ‘ im p k qqq ¨ Ind B , A , v pp id A ‘ coim p k qq ˝ ∆ A q “ v p C q ´ ¨ v p C q “ (cid:3)
15. A pplications to the category
Hopf bc p C q In this section, we give an application of the previous results to the category of bicom-mutative Hopf monoids
Hopf bc p C q . From now on, we assume the (Assumption 0,1,2) insubsubsection 3.3.1. Before we go into details, we give two remarks about the assumptions. Remark 15.1.
We remark a relationship between the assumptions. (Assumption 0,1) impliesthat the category
Hopf bc p C q is an pre-abelian category i.e. an additive category with arbi-trary kernel and cokernel. Under (Assumption 0,1), (Assumption 2) is equivalent with thefundamental theorem on homomorphisms. Remark 15.2.
We need those (Assumption 0,1,2) because we use the following properties : (1)
By (Assumption 0), every idempotent in C is a split idempotent due to Proposition6.6. By Corollary 6.14, a bimonoid A in C is bismall if and only if A has a normal-ized integral and a normalized cointegral. By Corollary 6.15, the full subcategoryof bismall bimonoids in the symmetric monoidal category C gives a sub symmetricmonoidal category of Bimon p C q . (2) We need (Assumption 1) to make use of Proposition 5.7, i.e. every homomorphism in
Hopf bc p C q is binormal. (3) Recall Definition 9.6. Furthermore, due to (Assumption 0, 1), every homomorphismin
Hopf bc p C q is well-decomposable by definition. (4) From (Assumption 2), we obtain the following exact sequence : For bicommutativeHopf monoids A , B , C in C and homomorphisms ξ : A Ñ B, ξ : B Ñ C, we have anexact sequence, Ñ Ker p ξ q Ñ Ker p ξ ˝ ξ q Ñ Ker p ξ q Ñ Cok p ξ q Ñ Cok p ξ ˝ ξ q Ñ Cok p ξ q Ñ (131) Note that until this subsection, we use the notation Ker p ξ q , Cok p ξ q for the kerneland cokernel in Bimon p C q . See Definition 5.1. In (131), Ker p ξ q , Cok p ξ q denote akernel and a cokernel in Hopf bc p C q . In fact, these coincide with each other due to(Assumption 1) and Corollary 5.8. Proof of Theorem 3.9.
In this subsection, we prove Theorem 3.9 which follows fromTheorem 15.6.
Proposition 15.3.
Let A , B , C be bicommutative Hopf monoids. Let ξ : A Ñ B, ξ : B Ñ Cbe bimonoid homomorphism. If the bimonoid homomorphisms ξ, ξ are pre-Fredholm, thenthe composition ξ ˝ ξ is pre-Fredholm. Moreover we have,vol ´ p Ker p ξ qq ˝ vol ´ p Ker p ξ qq “ x cok p ξ q ˝ ker p ξ qy ˝ vol ´ p Ker p ξ ˝ ξ qq , (132) vol ´ p Cok p ξ qq ˝ vol ´ p Cok p ξ qq “ x cok p ξ q ˝ ker p ξ qy ˝ vol ´ p Cok p ξ ˝ ξ qq . (133) Proof.
Recall that we have an exact sequence (131). By Theorem 3.11, the Hopf monoids
Cok p ξ ˝ ξ q , Ker p ξ ˝ ξ q are bismall since the Hopf monoids Ker p ξ q , Ker p ξ q and cokernels Cok p ξ q , Cok p ξ q are bismall. Hence, the composition ξ ˝ ξ is pre-Fredholm. NTEGRALS ALONG BIMONOID HOMOMORPHISMS 43
We prove the first equation. Denote by ϕ “ cok p ξ q ˝ ker p ξ q : Ker p ξ q Ñ Cok p ξ q . Fromthe exact sequence (131), we obtain an exact sequence, Ñ Ker p ξ q Ñ Ker p ξ ˝ ξ q Ñ Ker p ξ q Ñ Im p ϕ q Ñ (134)We apply Corollary 12.3 by assuming A , B , ξ in Corollary 12.3 are Ker p ξ ˝ ξ q , Ker p ξ q andthe homomorphism Ker p ξ ˝ ξ q Ñ Ker p ξ q . In fact, the first assumption in Corollary 12.3follows from (Assumption 1). The second and fourth assumptions in Corollary 12.3 followsfrom Theorem 9.9. The third assumption is already proved as before. Then we obtain, vol ´ p Ker p ξ qq ˝ vol ´ p Ker p ξ qq “ vol ´ p Im p ϕ qq ˝ vol ´ p Ker p ξ ˝ ξ qq . (135)By Proposition 11.9, we have x ϕ y “ vol ´ p Im p ϕ qq so that it completes the first equation. Thesecond equation is proved analogously. (cid:3) Proposition 15.4.
The subcategory
Hopf bc , bs p C q is an abelian subcategory of the abeliancategory Hopf bc p C q .Proof. Let A , B be bicommutative bismall Hopf monoids. Let ξ : A Ñ B be a bimonoidhomomorphism, i.e. a morphism in Hopf bc p C q . We have an exact sequence, Ñ Ñ Ker p ξ q ker p ξ q Ñ A ξ Ñ B . (136)Due to (Assumption 1) and (Assumption 2), we can apply Theorem 3.11. By Theorem 3.11,the kernel Hopf monoid Ker p ξ q is bismall. Analogously, the cokernel Hopf monoid Cok p ξ q is bismall. It completes the proof. (cid:3) Definition 15.5.
Let
End C p q be the set of endomorphism on the unit object . Note thatthe composition induces an abelian monoid structure on the set End C p q . We denote by M C the smallest submonoid of End C p q containing f P End C p q such that f “ vol ´ p A q or f ˝ vol ´ p A q “ id “ vol ´ p A q ˝ f for some bicommutative bismall Hopf monoid A . Denoteby M ´ C the submonoid consisting of invertible elements in the monoid M C , i.e. M ´ C “ M C X Aut C p q . Theorem 15.6.
The assignment vol ´ of inverse volumes is a M C -valued volume on theabelian category Hopf bc , bs p C q .Proof. Put v “ vol ´ . The unit Hopf monoid is a zero object of Hopf bc , bs p C q . By the firstpart of Proposition 11.4, we have v p q “ vol ´ p q P M C is the unit of M C .Let Ñ A f Ñ B g Ñ C Ñ be an exact sequence in the abelian category A “ Hopf bc , bs p C q .We apply the first equation in Theorem 15.3 by considering ξ “ g and ξ “ (cid:15) C . In fact, B , C , are bismall bimonoids, the homomorphisms g and (cid:15) C are pre-Fredholm. We obtain vol ´ p Ker p g qq ˝ vol ´ p Ker p (cid:15) C qq “ x cok p g q ˝ ker p (cid:15) C qy ˝ vol ´ p Ker p (cid:15) B qq . (137)By the exactness, we have A – Ker p g q and Cok p g q – . Moreover we have Ker p (cid:15) C q – C and Ker p (cid:15) B q – B . Hence, we obtain x cok p g q ˝ ker p (cid:15) C qy “ id so that vol ´ p A q ¨ vol ´ p C q “ vol ´ p B q . It completes the proof. (cid:3) Functorial integral.Definition 15.7. (1) Recall Definition 14.3. For two bicommutative Hopf monoids A , B ,a bimonoid homomorphism ξ : A Ñ B is Fredholm if it is Fredholm with respectto the inverse volume vol ´ . In other words, the homomorphism ξ is pre-Fredholm,and its kernel Hopf monoid and cokernel Hopf monoid have finite volumes. For aFredholm homomorphism ξ : A Ñ B between bicommutative Hopf monoids, wedenote by Ind p ξ q def . “ Ind B , A , v p ξ q for B “ Hopf bc p C q , A “ Hopf bc , bs p C q , M “ M C and v “ vol ´ . (2) We denote by Hopf bc , Fr p C q the category consisting of Fredholm homomorphisms be-tween bicommutative Hopf monoids. If one recalls Definition 14.6, then the sub-category Hopf bc , Fr p C q of Hopf bc p C q by Hopf bc , Fr p C q def . “ A Fr for B “ Hopf bc p C q , A “ Hopf bc , bs p C q , M “ M C and v “ vol ´ . We give a symmetric monoidal structureon Hopf bc , Fr p C q from that of Hopf bc p C q .(3) Let ξ : A Ñ B be a homomorphism between bicommutative Hopf monoids. Thehomomorphism ξ is finite if the morphism ξ in Hopf bc is finite with respect to thevolume vol ´ . See Definition 14.8. Proposition 15.8. (1)
For a bicommutative Hopf monoid A, the identity id A is Fredholmand we have Ind p id A q “ id P M ´ C . (2) For Fredholm homomorphisms ξ : A Ñ B and ξ : B Ñ C between bicommutativeHopf monoids, the composition ξ ˝ ξ is Fredholm and we have Ind p ξ ˝ ξ q “ Ind p ξ q ˝ Ind p ξ q P M ´ C . (3) For a Fredholm homomorphism ξ : A Ñ B and a finite homomorphism (cid:15) : A Ñ B,the convolution ξ ˚ (cid:15) is Fredholm and we have Ind p ξ ˚ (cid:15) q “ Ind p ξ q P M ´ C .Proof. The first part follows from Lemma 14.4. The second part follows from Lemma 14.5.The third part follows from Proposition 14.9. (cid:3)
Definition 15.9.
We define a 2-cochain ω C of the symmetric monoidal category Hopf bc , Fr p C q with coe ffi cients in the abelian group M ´ C . Let ξ : A Ñ B , ξ : B Ñ C be composableFredholm homomorphisms between bicommutative Hopf monoids. We define ω C p ξ, ξ q def . “ x cok p ξ q ˝ ker p ξ qy P M ´ C . (138) Proposition 15.10.
The 2-cochain ω C is a 2-cocycle.Proof. The 2-cocycle condition is immediate from the associativity of compositions. In fact, µ ξ ˝ p µ ξ ˝ µ ξ q “ p µ ξ ˝ µ ξ q ˝ µ ξ implies, p ω C p ξ, ξ q ˝ ω C p ξ ˝ ξ, ξ qq ¨ µ ξ ˝ ξ ˝ ξ “ p ω C p ξ , ξ q ˝ ω C p ξ, ξ ˝ ξ qq ¨ µ ξ ˝ ξ ˝ ξ . (139)Here, we use Theorem 12.1 where the assumptions in Theorem are deduced from (Assump-tion 0, 1). By Theorem 8.5, we obtain ω C p ξ, ξ q ˝ ω C p ξ ˝ ξ, ξ q “ ω C p ξ , ξ q ˝ ω C p ξ, ξ ˝ ξ q . (140)It proves that the 2-cochain ω C is a 2-cocycle.Moreover we have ω C p id B , ξ q “ “ ω C p ξ, id A q by definitions. Hence, the 2-cocycle ω C isnormalized. It completes the proof. (cid:3) Definition 15.11.
We define a 2-cohomology class o C P H nor p Hopf bc , Fr p C q ; M ´ C q by the classof the 2-cocycle ω C . Proposition 15.12.
We have o C “ P H nor p Hopf bc , Fr p C q ; M ´ C q . In particular, the induced2-cohomology class o C P H nor p Hopf bc , Fr p C q ; Aut C p qq by M ´ C Ă Aut C p q is trivial.Proof. Choose υ defined by υ p ξ q “ vol ´ p Ker p ξ qq . Then the first equation in Theorem 15.3proves the claim. (cid:3) Definition 15.13 (Functorial integral) . Let υ be a normalized 1-cochain with coe ffi cients inthe abelian group Aut C p q such that δ υ “ ω C . Let ξ : A Ñ B be a Fredholm bimonoidhomomorphism between bicommutative Hopf monoids. Recall µ ξ in Definition 9.7. Wedefine a morphism ξ ! : B Ñ A by ξ ! def . “ υ p ξ q ´ ¨ µ ξ . (141) NTEGRALS ALONG BIMONOID HOMOMORPHISMS 45
Proposition 15.14.
Let A be a bicommutative Hopf monoid. Note that the identity id A isFredholm. We have, p id A q ! “ id A . (142) Proof.
It follows from υ p id A q “ id . (cid:3) Proposition 15.15.
Let A , B , C be bicommutative Hopf monoids. Let ξ : A Ñ B , ξ : B Ñ Cbe bimonoid homomorphisms. If ξ, ξ are Fredholm, then the composition ξ ˝ ξ is Fredholmand we have p ξ ˝ ξ q ! “ ξ ! ˝ ξ ! . (143) Proof.
By Theorem 15.3, we have p ξ ˝ ξ q ! “ υ p ξ ˝ ξ q ´ ¨ µ ξ ˝ ξ (144) “ ` υ p ξ ˝ ξ q ´ ˝ ω p ξ , ξ q ´ ˘ ¨ p µ ξ ˝ µ ξ q (145) “ ` υ p ξ q ´ ˝ υ p ξ q ´ ˘ ¨ p µ ξ ˝ µ ξ q (146) “ ξ ! ˝ ξ ! . (147) (cid:3) Definition 15.16.
We define a normalized 1-cochain υ with coe ffi cients in M ´ C . For aFredholm homomorphism ξ , we define υ p ξ q def . “ vol ´ p Ker p ξ qq . We define another nor-malized 1-cochain υ with coe ffi cients in M ´ C by υ p ξ q def . “ vol ´ p Cok p ξ qq . They satisfy δ υ “ ω C “ δ υ . Theorem 15.17.
Consider υ “ υ ( υ “ υ , resp.) in Definition 15.13. Let A , B , C , D bebicommutative Hopf monoids. Consider a commutative diagram of Fredholm bimonoid ho-momorphisms. Suppose that ‚ the induced bimonoid homomorphism Ker p ϕ q Ñ Ker p ψ q is an isomorphism (an epi-morphism resp.) in Hopf bc p C q . ‚ the induced bimonoid homomorphism Cok p ϕ q Ñ Cok p ψ q is a monomorphism (anisomorphism, resp.) in Hopf bc p C q .Then we have ϕ ˝ ϕ ! “ ψ ! ˝ ψ . A CB D ϕ ϕ ψψ Proof.
We prove the case υ “ υ and leave to the readers the case υ “ υ . Note that thereexists a section of the induced bimonoid homomorphism ϕ : Ker p ϕ q Ñ Ker p ψ q in C since ϕ is an isomorphism in Hopf bc p C q , in particular in C . Moreover, the induced morphism ψ : Cok p ϕ q Ñ Cok p ψ q has a retract in C . In fact, since ψ is a monomorphism, there existsa morphism ξ in Hopf bc p C q such that ker p ξ q “ ψ . By Lemma 9.3, ˜ µ ker p ξ q ˝ ψ “ id Cok p ϕ q .By Theorem 3.4, we have µ ψ ˝ ψ “ ϕ ˝ µ ϕ . Since υ p ϕ q “ vol ´ p Ker p ϕ qq , υ p ψ q “ vol ´ p Ker p ψ qq and ϕ is an isomorphism, we have υ p ϕ q “ υ p ψ q . By definitions, we obtain ψ ! ˝ ψ “ ϕ ˝ ϕ ! . (cid:3) A. N otations
This section gives our convention about notations. The reader is referred to some introduc-tory books for category theory or (Hopf) monoid theory [12] [1].
We denote by the unit object of a monoidal category C , by b the monoidal operation, by r a : a b Ñ a the right unitor and by l a : b a Ñ a the left unitor. String diagrams.
We explain our convention to represent string diagrams . It is convenientto use string diagrams to discuss equations of morphisms in a symmetric monoidal category C . It is based on finite graphs where for each vertex v the set of edges passing through v has a partition by, namely, incoming edges and outcoming edges . For example, a morphism f : x Ñ y in C is represented by (1) in Figure 25. In this example, the underlying graph hasone 2-valent vertex and two edges. If there is no confusion from the context, we abbreviatethe objects as (2) in Figure 25. For another example, a morphism g : a b b Ñ x b y b z isrepresented by (3) in Figure 25. F igure C bygluing two string diagrams. For example, if h : x Ñ y , k : a Ñ b are morphisms, then werepresent h b k : x b a Ñ y b b by (1) in Figure 26.We represent the composition of morphisms by connecting some edges of string diagrams.For example, if q : x Ñ y and p : y Ñ z are morphisms, we represent their composition p ˝ q : x Ñ z by (2) in Figure 26. F igure s x , y : x b y Ñ y b x which is a natural isomorphism is denoted by (1) inFigure 27.The edge colored by the unit object of the symmetric monoidal category C is abbreviated.For example, a morphism u : Ñ a is denoted by (2) in Figure 27 and a morphism v : b Ñ is denoted by (3) in Figure 27. F igure Monoid.
The notion of monoid in a symmetric monoidal category is a generalization of thenotion of monoid which is a set equipped with a unital and associative product. Furthermore,it is a generalization of the notion of algebra . We use the notations ∇ : A b A Ñ A and NTEGRALS ALONG BIMONOID HOMOMORPHISMS 47 η : Ñ A to represent the multiplication and the unit. On the one hand, the comonoidis a dual notion of the monoid. We use the notations ∆ : A Ñ A b A and (cid:15) : A Ñ torepresent the comultiplication and the counit. Figure 28 denotes the structure morphisms asstring diagrams.The notions of bimonoid and Hopf monoid are defined as an object of C equipped with amonoid structure and a comonoid structure which are subject to some axioms. We denote by Bimon p C q , Hopf p C q the categories of bimonoids and Hopf monoids respectively.F igure Action.
We give some notations about actions in a symmetric monoidal category. Thenotations related with coaction is defined similarly.
Definition A.1.
Let X be an object of C , A be a bimonoid, and α : A b X Ñ X be a morphismin C . A triple p A , α, X q is a left action in C if following diagrams commute :(148) A b A b X A b XA b X X id A b α ∇ A b id X αα (149) b X A b XX l X η A b id X α Let p A , α, X q , p A , α , X q be left actions in a symmetric monoidal category C . A pair p ξ , ξ q : p A , α, X q Ñ p A , α , X q is a morphism of left actions if ξ : A Ñ A is a monoid homomor-phism and ξ : X Ñ X is a morphism in C which intertwines the actions.Left actions in C and morphisms of left actions form a category which we denote by Act l p C q . The symmetric monoidal category structures of C and Bimon p C q induce a sym-metric monoidal category on Act l p C q by p A , α, X q b p A , α , X q def . “ p A b A , α ˜ b α , X b X q .Here, α ˜ b α : p A b A q b p X b X q Ñ X b X is defined by composing A b A b X b X id A b s A , X b id X ÝÑ A b X b A b X α b α ÝÑ X b X . (150)We define a right action in a symmetric monoidal category C and its morphism similarly.Note that for a right action, we use the notation p X , α, A q where A is a bimonoid and X is an object on which A acts. We denote by Act r p C q the category of right actions and theirmorphisms. It inherits a symmetric monoidal category structure from that of C and Bimon p C q . Let A be a bimonoid in a symmetric monoidal category C and X be an object of C . A leftaction p A , τ A , X , X q is trivial if τ A , X : A b X (cid:15) A b id X Ñ b X l X Ñ X . (151)We also define a trivial right action analogously. We abbreviate τ “ τ A , X if there is noconfusion. R eferences [1] Marcelo Aguiar and Swapneel Arvind Mahajan. Monoidal functors, species and Hopf algebras , vol-ume 29. American Mathematical Society Providence, RI, 2010.[2] Benjamin Balsam and Alexander Kirillov Jr. Kitaev’s lattice model and Turaev-Viro TQFTs. arXivpreprint arXiv:1206.2308 , 2012.[3] John Barrett and Bruce Westbury. Invariants of piecewise-linear 3-manifolds.
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