A descent theorem for pure motives
Résumé
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 $\mathbf{Q}$-linear. We use this to prove that pure motives \`a la Grothendieck (with rational coefficients) over a field form a stack for the \'etale topology; this holds more generally for several motivic categories considered in arXiv:1506.08386 [math.AG]. Finally, we clarify the construction of Chow-Lefschetz motives given in arXiv:2302.08327 [math.AG], and simplify the proof of the computation of the motivic Galois group of Lefschetz motives modulo numerical equivalence, given in loc. cit.