Homotopy theory of symmetric powers

Dmitri Pavlov, Jakob Scholbach

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

We introduce the symmetricity notions of symmetric h- monoidality, symmetroidality, and symmetric atness. As shown in our paper [PS14a], 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. This paper is also available at arXiv:1510.04969v3.

Original languageEnglish
Pages (from-to)359-397
Number of pages39
JournalHomology, Homotopy and Applications
Volume20
Issue number1
DOIs
StatePublished - 2018

Keywords

  • D-module
  • Model category
  • Operad
  • Symmetric atness
  • Symmetric h-monoidality
  • Symmetric power

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