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COHOMOLOGICAL MACKEY 2-FUNCTORS

Part of: Connections with homological algebra and category theory Homotopy theory Special categories

Published online by Cambridge University Press:  18 August 2022

Paul Balmer Open the ORCID record for Paul Balmer [Opens in a new window]  and
Ivo Dell’Ambrogio
Paul Balmer*
Affiliation:
UCLA Mathematics Department, Los Angeles, CA 90095-1555, USA
Ivo Dell’Ambrogio
Affiliation:
Univ. Lille, CNRS, UMR 8524—Laboratoire Paul Painlevé, F-59000 Lille, France ([email protected])
*
[email protected]
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Abstract

We show that the bicategory of finite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in [2], obtained by modding out the so-called cohomological relations. This categorifies Yoshida’s theorem for ordinary cohomological Mackey functors and provides a direct connection between Mackey 2-motives and the usual blocks of representation theory.

Keywords

Mackey functorsMackey 2-functorsMackey 2-motivesYoshidaBisetsPermutation modulesDecategorification

MSC classification

Primary: 20J05: Homological methods in group theory
Secondary: 18B40: Groupoids, semigroupoids, semigroups, groups (viewed as categories) 55P91: Equivariant homotopy theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 1 , January 2024 , pp. 279 - 309
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

First-named author partially supported by NSF grant DMS-1901696.

Second-named author partially supported by Project ANR ChroK (ANR-16-CE40-0003) and Labex CEMPI (ANR-11-LABX-0007-01).

References

Balmer, P., ‘Stacks of group representations’, J. Eur. Math. Soc. (JEMS) 17(1) (2015), 189–228.CrossRefGoogle Scholar
Balmer, P. and Dell’Ambrogio, I., Mackey 2-functors and Mackey 2-motives (European Mathematical Society (EMS), Zürich, 2020).CrossRefGoogle Scholar
Balmer, P. and Dell’Ambrogio, I., ‘Green equivalences in equivariant mathematics’, Math. Ann. 379(3–4) (2021), 1315–1342.CrossRefGoogle Scholar
Bénabou, J., ‘Distributors at work’, Course notes’, 2000, http://www.entretemps.asso.fr/maths/Distributors.pdf.Google Scholar
Bouc, S., ‘The $p$ -blocks of the Mackey algebra’, Algebr. Represent. Theory 6(5) (2003), 515–543.CrossRefGoogle Scholar
Bouc, S., Biset Functors for Finite Groups, Lecture Notes in Mathematics, vol. 1990 (Springer-Verlag, Berlin, 2010).CrossRefGoogle Scholar
I. Dell’Ambrogio, ‘Green 2-functors’, Trans. Amer. Math. Soc. (to appear), Preprint, 2021.CrossRefGoogle Scholar
Dell’Ambrogio, I., ‘Axiomatic representation theory of finite groups by way of groupoids’, in Equivariant Topology and Derived Algebra, London Math. Soc. Lecture Note Ser., vol. 474 (Cambridge Univ. Press, Cambridge, 2022), 39–99.Google Scholar
Dell’Ambrogio, I., ‘On Krull–Schmidt bicategories’, Theory Appl. Categ. 38(8) (2022), 232–256.Google Scholar
Dell’Ambrogio, I. and Huglo, J., ‘On the comparison of spans and bisets’, Cahiers Topologie Géom. Différentielle Catég. 62(1) (2021), 63–104.Google Scholar
Greenlees, J. P. C. and May, J. P., ‘Generalized Tate cohomology’, Mem. Amer. Math. Soc. 113(543) (1995), viii+178.Google Scholar
Huglo, J., Functorial precomposition with applications to Mackey functors and biset functors. PhD thesis, Université de Lille 1, 2019, https://math.univ-lille1.fr/~dellambr/TheseVersionFinaleHUGLO.pdf.Google Scholar
Linckelmann, M., The Block Theory of Finite Group Algebras , vol. II, London Mathematical Society Student Texts, vol. 92 (Cambridge University Press, Cambridge, 2018).Google Scholar
Maillard, J., ‘A categorification of the Cartan–Eilenberg formula’, Adv. Math. 396 (2022), 1081–1087.CrossRefGoogle Scholar
Miller, H., ‘The Burnside bicategory of groupoids’, Bol. Soc. Mat. Mex. (3) 23(1) (2017), 173–194.CrossRefGoogle Scholar
Oda, F., Tagegahara, Y. and Yoshida, T., ‘Crossed Burnside rings and cohomological Mackey 2-motives’, Preprint, 2022, arXiv:2201.04744.Google Scholar
Webb, P., ‘Two classifications of simple Mackey functors with applications to group cohomology and the decomposition of classifying spaces’, J. Pure Appl. Algebra 88(1–3) (1993), 265–304.CrossRefGoogle Scholar
Webb, P., ‘A guide to Mackey functors’, in Handbook of Algebra , vol. 2 (North-Holland, Amsterdam, 2000), 805–836.CrossRefGoogle Scholar
Yoshida, T., ‘On $G$ -functors. II. Hecke operators and $G$ -functors’, J. Math. Soc. Japan 35(1) (1983), 179–190.CrossRefGoogle Scholar
Yoshida, T., ‘Crossed $G$ -sets and crossed Burnside rings’, Sūrikaisekikenkyūsho Kōkyūroku, 991 (1997), 1–15.Google Scholar