Motivic slices and colored operads

TL;DR

This paper introduces colored operads within motivic stable homotopy theory to analyze the preservation of algebraic structures under the slice filtration, revealing new structured properties of motivic spectra.

Contribution

It develops a framework using colored operads and accessible t-structures to study how the slice filtration preserves algebraic and module structures in motivic spectra.

Findings

Colored operads effectively describe structured properties of motivic spectra.

The slice filtration preserves certain algebraic structures under specific conditions.

Axiomatic setups unify classical and motivic stable homotopy theories.

Abstract

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.