Homology, Homotopy and Applications

Volume 20 (2018)

Number 1

Homotopy theory of symmetric powers

Pages: 359 – 397

DOI: https://dx.doi.org/10.4310/HHA.2018.v20.n1.a20


Dmitri Pavlov (Faculty of Mathematics, University of Regensburg, Germany; and Department of Mathematics and Statistics, Texas Tech University, Lubbock, Tx., U.S.A.)

Jakob Scholbach (Mathematical Institute, University of Münster, Germany)


We introduce the symmetricity notions of symmetric $h$-monoidality, symmetroidality, and symmetric flatness. As shown in our paper “Admissibility and rectification of colored symmetric operads” [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.


model category, operad, symmetric power, symmetric flatness, symmetric $h$-monoidality, $D$-module

2010 Mathematics Subject Classification

Primary 18D50, 55P48. Secondary 18D20, 18G55, 55P43, 55U35.

Received 25 November 2016

Received revised 27 October 2017

Published 21 March 2018