We introduce a notion of an integral along a bimonoid homomorphism as a simultaneous generalization of the integral and cointegral of bimonoids. The purpose of this paper is 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 dimension of vector spaces and the order of abelian groups. In applications, we show that there exists a nontrivial volume partially defined on a category of bicommutative Hopf monoids. The volume yields a notion of Fredholm homomorphisms between bicommutative Hopf monoids, which gives an analogue of the Fredholm index theory. This paper gives a technical preliminary of our subsequent paper about a construction of TQFT’s.

我们引入了沿双态同态的积分概念，作为双态的积分和协积分的同时推广。本文的目的是根据同构的核和方差，描述一个沿着Bimonoid同构的特定积分（称为归一化生成器积分）的存在。我们介绍一个关于阿贝尔类别的量的概念作为向量空间维数和阿贝尔群序的一般化。在应用程序中，我们表明存在一个非平凡的体积，该体积部分地定义在双交换Hopf单面体的类别上。该体积产生了双交换Hopf单边形之间的Fredholm同态的概念，从而给出了Fredholm指数理论的类似物。本文提供了我们后续论文中有关TQFT的结构的技术性初步介绍。