Integrals along bimonoid homomorphisms (Q2046252)

The notion of an integral of a bialgebra was introduced in [\textit{R. G. Larson} and \textit{M. E. Sweedler}, Am. J. Math. 91, 75--94 (1969; Zbl 0179.05803)] as a generalization of the Haar measure of a group, which has been used in the study of bialgebras or Hopf algebras [\textit{M. E. Sweedler}, Ann. Math. (2) 89, 323--335 (1969; Zbl 0174.06903); \textit{D. E. Radford}, Bull. Am. Math. Soc. 81, 1103--1105 (1975; Zbl 0326.16008); Am. J. Math. 98, 333--355 (1976; Zbl 0332.16007)]. The notion of a bialgebra was generalized to bimonoids in a symmetric monoidal category \(\mathcal{C}\) [\textit{M. Aguiar} and \textit{S. Mahajan}, Monoidal functors, species and Hopf algebras. Providence, RI: American Mathematical Society (AMS) (2010; Zbl 1209.18002); \textit{S. Mac Lane}, Categories for the working mathematician. New York, NY: Springer (1998; Zbl 0906.18001)], while the integral theory was generalized to the categorical setting to study bimonoids or Hopf monoids [\textit{Y. Bespalov} et al., J. Pure Appl. Algebra 148, No. 2, 113--164 (2000; Zbl 0961.16023)]. This paper introduces a notion of an integral along a bimonoid homomorphism in a symmetric monoidal category \(\mathcal{C}\), generalizing the notions of integral and cointegral of a bimonoid. The principal objective in this paper is to establish the following theorem. Theorem. Let \(A\) and \(B\) be bicommutative Hopf algebras in \(\mathcal{C}\) with \(\xi:A\rightarrow B\) a Hopf homomorphism. Then there exists a normalized generator integral \(\mu_{\xi}\) along \(\xi\) iff the following conditions are satisfied: \begin{itemize} \item[1.] The kernel Hopf monoid \(Ker(\xi)\) has a normalized integral. \item[2.] The cokernel Hopf monoid \(\mathrm{Cok}(\xi)\) has a normalized cointegral. \end{itemize} Moreover, if a normalized integral exists, then it is unique. Some applications are as follows. \begin{itemize} \item The author investigates the category \(\mathrm{Hopf}^{\mathrm{bc},\ast}(\mathcal{C})\) of bicommutative Hopf monoids with a normalized integral and cointegral, establishing that the category \(\mathrm{Hopf} ^{\mathrm{bc},\ast}(\mathcal{C})\) is an abelian subcategory of \(\mathrm{Hopf}^{\mathrm{bc}}(\mathcal{C})\) and closed under short exact sequents. \item The author introduces the notion of volume on an abelian category as a generalization of the dimension of vector spaces and the order of abelian groups, studying basic notions related with it. \item The author constructs an \(\mathrm{End}_{\mathcal{C}}(\boldsymbol{1} )\)-valued volume \(\mathrm{vol}^{-1}\) on the abelian category \(\mathrm{Hopf} ^{\mathrm{bc},\ast}(\mathcal{C})\), where \(\boldsymbol{1}\) is the unit object of \(\mathcal{C}\) and the endomorphism set \(\mathrm{End}_{\mathcal{C} }(\boldsymbol{1})\) is an abelian monoid induced by the symmetric monoidal structure of \(\mathcal{C}\). \item By using the volume \(\mathrm{vol}^{-1}\), the author introduces a notion of Fredholm homomorphisms between bicommutative Hopf monoids as an analogue of Fredholm operators [\textit{T. Kato}, Perturbation theory for linear operators. Berlin: Springer-Verlag (1995; Zbl 0836.47009)], investigating its index which is robust to some perturbations. A functorial assignment of integrals to Fredholm homomorphisms is constructed. \item The author expects that the result in this paper could be applied to topology. There is a topological invariant of \(3\)-manifolds induced by a finite-dimensional Hopf algebra, called the Kuperberg invariant [\textit{G. Kuperberg}, Int. J. Math. 2, No. 1, 41--66 (1991; Zbl 0726.57016); ``Non-involutory Hopf algebras and 3-manifold invariants'', Preprint, \url{arXiv:q-alg/9712047}]. \end{itemize} This paper gives a technical preliminary to the author's subsequent paper [\textit{M. Kim}, ``A pair of homotopy-theoretic version of TQFT's induced by a Brown functor'', Preprint, \url{arXiv:2006.10438}], which uses the results in this paper to give a generalization of the untwisted abelian Dijkgraaf-Witten theory [\textit{R. Dijkgraaf} and \textit{E. Witten}, Commun. Math. Phys. 129, No. 2, 393--429 (1990; Zbl 0703.58011); \textit{M. Wakui}, Osaka J. Math. 29, No. 4, 675--696 (1992; Zbl 0786.57008); \textit{D. S. Freed} and \textit{F. Quinn}, Commun. Math. Phys. 156, No. 3, 435--472 (1993; Zbl 0788.58013)] and the bicommutative Turaev-Viro TQFT [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009); \textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], giving a systematic way to construct a sequence of TQFT's from (co)homology theory. The TQFT's are constructed by using path-integral which is formulated by some integral along bimonoid homomorphisms.