Galois coverings, Morita Equivalence and Smash Extensions of Categories over a Field

Algebras over a field $k$ generalize to categories over $k$ in order to considers Galois coverings. Two theories presenting analogies, namely smash extensions and Galois coverings with respect to a finite group are known to be different. However we prove in this paper that they are Morita equivalent. For this purpose we need to describe explicit processes providing Morita equivalences of categories which we call contraction and expansion. A structure theorem is obtained: composition of these processes provides any Morita equivalence up to equivalence, a result which is related with the karoubianisation (or idempotent completion) and additivisation of a $k$-category.

2000 Mathematics Subject Classification: 16W50, 18E05, 16W30, 16S40, 16D90.

Keywords and Phrases: Hopf algebra, Galois covering, $k$-category, Morita theory, smash product, completion, karoubianisation.

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