In mathematics, a motivic sheaf is a motivic-cohomology counterpart of an l-adic sheaf. It was first introduced by Morel and VoevodskyVladimir Voevodsky, A1-homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), no. Extra Vol. I, 1998, pp. 579–604. and was later developed by J. Ayoub,Joseph Ayoub, A guide to (étale) motivic sheaves, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 1101–1124 Deniz-Charles Cisinski, F. Déglise, F. Morel, and others. For Nori motives, the first construction is due to D. Arapura. In practice, a motivic sheaf is sometimes used instead of an l-adic sheaf because the former’s cycle-theoretic nature may be important. In the language of Ayoub,§ 1.4.1. of
References
Further reading Adeel Khan, Motivic sheaves on algebraic stacks https://mathoverflow.net/questions/422870/what-is-motivic-sheaf-intuitively