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
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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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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DOI: https://doi.org/10.1007/s10485-020-09627-5