We give necessary conditions for a category fibred in pseudo-abelian additive categories over the classifying topos of a profinite group to be a stack; these conditions are sufficient when the coefficients are Q-linear. This applies to pure motives over a field in the sense of Grothendieck, Deligne-Milne and André, to mixed motives in the sense of Nori and to several motivic categories considered in [13]. We also give a simple proof of the exactness of a sequence of motivic Galois groups under a Galois extension of the base field, which applies to all the above (Tannakian) situations. Finally, we clarify the construction of the categories of Chow-Lefschetz motives given in [14] and simplify the computation of their motivic Galois group in the numerical case.
Keywords: Stacks, motives, Tannakian categories
2020 MSC: 18F20, 18M25, 14C15
Theory and Applications of Categories, Vol. 44, 2025, No. 20, pp 588-616.
Published 2025-06-25.
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