Théorie tannakienne non commutative
Résumé
Inspired by a recent paper by Deligne [2], we extend the Tannaka-Krein duility results (over a field) to the non-commutative situation. To be precise, we establish a 1-1 corresponde:ice between 'tensorial autonomous categories' equipped with a 'fibre functor' (i. e. tannakian categories without the commutativity condition on the tensor product), and 'quantum groupoids' (as defined by Maltsiniotis, [9]) which are 'transitive' (7.1.). When the base field is perfect, a quantum groupoid over Spec B is transitive iff it is projective and faithfully fiat over B⊗k B. Moreover, the fibre functor is unique up to 'quantum isomorphism' (7.6.). Actually, we show Tannaka-Krein duality results in the more general setting where there is no monoidal structure on the category (and the functor); the algebraic object corresponding to such a category is a 'semi-transitive' coalgebroid (5.2. and 5.8.).