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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

Eilenberg-Watts calculus for finite categories and a bimodule Radford $ S^4$ theorem


Authors: Jürgen Fuchs, Gregor Schaumann and Christoph Schweigert
Journal: Trans. Amer. Math. Soc. 373 (2020), 1-40
MSC (2010): Primary 16T05; Secondary 18D10, 16Gxx
DOI: https://doi.org/10.1090/tran/7838
Published electronically: October 1, 2019
MathSciNet review: 4042867
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Abstract: We obtain Morita invariant versions of Eilenberg-Watts type theorems, relating Deligne products of finite linear categories to categories of left exact as well as of right exact functors. This makes it possible to switch between different functor categories as well as Deligne products, which is often very convenient. For instance, we can show that applying the equivalence from left exact to right exact functors to the identity functor, regarded as a left exact functor, gives a Nakayama functor. The equivalences of categories we exhibit are compatible with the structure of module categories over finite tensor categories. This leads to a generalization of Radford's $ S^4$-theorem to bimodule categories. We also explain the relation of our construction to relative Serre functors on module categories that are constructed via inner Hom functors.


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Additional Information

Jürgen Fuchs
Affiliation: Teoretisk fysik, Karlstads Universitet, Universitetsgatan 21, S–65188 Karlstad, Sweden; Fachbereich Mathematik, Universität Hamburg, Bereich Algebra und Zahlentheorie, Bundesstraße 55, D–20146 Hamburg, Germany

Gregor Schaumann
Affiliation: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A–1090 Wien, Austria

Christoph Schweigert
Affiliation: Fachbereich Mathematik, Universität Hamburg, Bereich Algebra und Zahlentheorie, Bundesstraße 55, D–20146 Hamburg, Germany

DOI: https://doi.org/10.1090/tran/7838
Received by editor(s): January 19, 2017
Received by editor(s) in revised form: December 22, 2018
Published electronically: October 1, 2019
Additional Notes: The first author was supported by VR under project No. 621-2013-4207.
The second author was supported by Nils Carqueville’s project P 27513-N27 of the Austrian Science Fund.
The third author was partially supported by the Collaborative Research Centre 676 “Particles, Strings and the Early Universe - The Structure of Matter and Space-Time” and by the RTG 1670 “Mathematics Inspired by String Theory and Quantum Field Theory”.
Article copyright: © Copyright 2019 Jürgen Fuchs, Gregor Schaumann, and Christoph Schweigert