Symmetric powers in abstract homotopy categories

TL;DR

This paper investigates symmetric powers within abstract homotopy categories, demonstrating their ability to preserve key homotopy types in motivic theories, and establishing categorical lambda-structures in these contexts.

Contribution

It introduces the preservation of Nisnevich and etale homotopy types by symmetric powers in motivic homotopy categories, extending the understanding of symmetric powers in both unstable and stable settings.

Findings

Symmetric powers preserve weak equivalences in motivic homotopy theories.

Established categorical lambda-structures on motivic homotopy categories.

Demonstrated preservation of homotopy types under symmetric powers in various topologies.

Abstract

We study symmetric powers in the homotopy categories of abstract closed symmetric monoidal model categories, in both unstable and stable settings. As an outcome, we prove that symmetric powers preserve the Nisnevich and etale homotopy type in the unstable and stable motivic homotopy theories of schemes over a base. More precisely, if f is a weak equivalence of motivic spaces, or a weak equivalence between positively cofibrant motivic spectra, with respect to the Nisnevich or etale topology on schemes, then all symmetric powers Sym^n(f) are weak equivalences too. This gives left derived symmetric powers which aggregate into a categorical lambda-structures on the corresponding motivic homotopy categories of schemes over a base.