  Colored operads were introduced in the 1970's for the purpose of studying
homotopy invariant algebraic structures on topological spaces. In this paper we
introduce colored operads in motivic stable homotopy theory. Our main
motivation is to uncover hitherto unknown highly structured properties of the
slice filtration. The latter decomposes every motivic spectrum into its slices,
which are motives, and one may ask to what extend the slice filtration
preserves highly structured objects such as algebras and modules. We use
colored operads to give a precise solution to this problem. Our approach makes
use of axiomatic setups which specialize to classical and motivic stable
homotopy theory. Accessible t-structures are central to the development of the
general theory. Concise introductions to colored operads and Bousfield
(co)localizations are given in separate appendices.
