## Skew category, Galois covering and smash product of a $k$-category

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- by Claude Cibils and Eduardo N. Marcos
- Proc. Amer. Math. Soc.
**134**(2006), 39-50 - DOI: https://doi.org/10.1090/S0002-9939-05-07955-4
- Published electronically: June 2, 2005
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## Abstract:

In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In the case of the smash product for an infinite group our construction specialized for a ring agrees with M. Beattie’s construction of a ring with local units. We recover in a categorical generalized setting the Duality Theorems of M. Cohen and S. Montgomery (1984), and we provide a unification with the results on coverings of quivers and relations by E. Green (1983). We obtain a confirmation in a quiver and relations-free categorical setting that both constructions are mutual inverses, namely the quotient of a free action category and the smash product of a graded category. Finally we describe functorial relations between the representation theories of a category and of a Galois cover of it.## References

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## Bibliographic Information

**Claude Cibils**- Affiliation: Institut de Mathématiques et Modélisation de Monpellier, Université de Montpellier 2, F–34095 Montpellier cedex 5, France
- MR Author ID: 49360
- ORCID: 0000-0003-3269-9525
- Email: Claude.Cibils@math.univ-montp2.fr
**Eduardo N. Marcos**- Affiliation: Departamento de Matemática, Universidade de São Paulo, IME-USP, Caixa Postal 66.281, São Paulo – SP, 05315–970, Brasil
- MR Author ID: 288969
- ORCID: 0000-0001-8514-1192
- Email: enmarcos@ime.usp.br
- Received by editor(s): December 22, 2003
- Received by editor(s) in revised form: August 26, 2004
- Published electronically: June 2, 2005
- Additional Notes: The second author thanks CNPq (Brazil) for financial support, in the form of a productivity scholarship. The authors thank the IME of the Universidade de São Paulo for support during the preparation of this work
- Communicated by: Martin Lorenz
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**134**(2006), 39-50 - MSC (2000): Primary 18A32, 16S35, 16G20
- DOI: https://doi.org/10.1090/S0002-9939-05-07955-4
- MathSciNet review: 2170541