  With representation-theoretic applications in mind, we construct a formalism
of reduced motives with integral coefficients. These are motivic sheaves from
which the higher motivic cohomology of the base scheme has been removed. We
show that reduced stratified Tate motives satisfy favorable properties
including weight and t-structures. We also prove that reduced motives on
cellular (ind-)schemes unify various approaches to mixed sheaves in
representation theory, such as Soergel-Wendt's semisimplified Hodge motives,
Achar-Riche's complexes of parity sheaves, as well as Ho-Li's recent category
of graded $\ell$-adic sheaves.
