nLab
simplicial operad

Contents

Definition

A simplicial operad is an operad over sSet. It has for each kk \in \mathbb{N} a simplicial set of kk-ary operations.

Structures on the category of simplicial operads

Model structure

There is a model category structure on the category of simplicial operads that makes them present (∞,1)-operads. See model structure on operads.

There is a Quillen equivalence between this model structure and the model structure on dendroidal sets, via the dendroidal homotopy coherent nerve.

Examples

Classes of examples

Specific examples

  • The Barratt-Eccles operad has, as a simplicial operad, a single color, and its simplicial set of nn-ary operations is the nerve N(Σ n//Σ n)N(\Sigma_n//\Sigma_n) of the action groupoid Σ n//Σ n\Sigma_n // \Sigma_n of the symmetric group Σ n\Sigma_n acting on itself. This is a contractible simplicial set in each degree, and, indeed, this operad is a resolution of Comm, which has the point in each degree.

Revised on March 7, 2012 02:13:29 by Urs Schreiber (82.169.65.155)