Homotopy theory of symmetric powers

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.

25 pages. Comments and questions are very welcome. This article splits off for publication purposes the first 7 sections of arXiv:1410.5675v1. v2: Added the notion of a strongly admissibly generated model category. v3: Identical to the journal version except for formatting and style

Cited by in corpus (8)

• Enhancing the filtered derived category
• On rectification and enrichment of infinity properads
• Operads in Unstable Global Homotopy Theory
• Quillen cohomology of enriched operads
• Enveloping operads and applications
• The enriched Thomason model structure on 2-categories
• Admissible replacements for simplicial monoidal model categories
• Algebras for enriched $\infty$-operads