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Integrals Along Bimonoid Homomorphisms

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Abstract

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.

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Notes

  1. By Theorem 6.13, the category \({\mathsf {Hopf}}^{{\mathsf {bc}},\star } ( {\mathcal {C}} )\) coincides with \({\mathsf {Hopf}}^{{\mathsf {bc,bs}}} ( {\mathcal {C}} )\) in Sect. 15.

  2. The reason that we consider a monoid M, not a group is that we deal with infinite dimension or infinite order uniformly.

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Acknowledgements

The author appreciates Christine Vespa who read this paper carefully and gave helpful comments. The author was supported by FMSP, a JSPS Program for Leading Graduate Schools in the University of Tokyo, and JPSJ Grant-in-Aid for Scientific Research on Innovative Areas Grant Number JP17H06461.

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  1. Graduate School of Mathematical Sciences, University of Tokyo, Tokyo, Japan

    Minkyu Kim

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  1. Minkyu Kim
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Correspondence to Minkyu Kim.

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Communicated by Wendy Lowen.

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Kim, M. Integrals Along Bimonoid Homomorphisms. Appl Categor Struct 29, 577–627 (2021). https://doi.org/10.1007/s10485-020-09627-5

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