Motivic slices and coloured operads

Abstract

Coloured operads were introduced in the 1970s for the purpose of studying homotopy invariant algebraic structures on topological spaces. In this paper, we introduce coloured 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 extent the slice filtration preserves highly structured objects such as algebras and modules. We use coloured 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 coloured operads and Bousfield (co)localizations are given in separate appendices.