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

Authors: Claude Cibils and Eduardo N. Marcos
Journal: Proc. Amer. Math. Soc. 134 (2006), 39-50
MSC (2000): Primary 18A32, 16S35, 16G20
Published electronically: June 2, 2005
MathSciNet review: 2170541
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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.

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Additional Information

Claude Cibils
Affiliation: Institut de Mathématiques et Modélisation de Monpellier, Université de Montpellier 2, F–34095 Montpellier cedex 5, France

Eduardo N. Marcos
Affiliation: Departamento de Matemática, Universidade de São Paulo, IME-USP, Caixa Postal 66.281, São Paulo – SP, 05315–970, Brasil

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
Article copyright: © Copyright 2005 American Mathematical Society

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