Homotopy theory of symmetric powers

TL;DR

This paper introduces new symmetricity properties crucial for the homotopy theory of symmetric operads and their algebras, enabling model structures and Quillen equivalences across various categories.

Contribution

It defines symmetric h-monoidality, symmetroidality, and symmetric flatness, and demonstrates how to transfer these properties to complex model categories like symmetric spectra.

Findings

Symmetric h-monoidality enables model structures for operad algebras.

Symmetric flatness ensures Quillen equivalences between operad categories.

Tools are provided to extend properties from basic to advanced model categories.

Abstract

We introduce the symmetricity notions of symmetric h-monoidality, symmetroidality, and symmetric flatness. As shown in our paper arXiv:1410.5675, these properties lie at the heart of the homotopy theory of colored symmetric operads and their algebras. In particular, the former property can be seen as the analog of Schwede and Shipley's monoid axiom for algebras over symmetric operads and allows one to equip categories of such algebras with model structures, whereas the latter ensures that weak equivalences of operads induce Quillen equivalences of categories of algebras. We discuss these properties for elementary model categories such as simplicial sets, simplicial presheaves, and chain complexes. Moreover, we provide powerful tools to promote these properties from such basic model categories to more involved ones, such as the stable model structure on symmetric spectra.