nLab
simplicial operad

Definition

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

• Every operad over Set may be regarded as a simplicial operad, via the discrete object embedding $\mathrm{Disc}:\mathrm{Set}\to \mathrm{sSet}$.

• Every topological operad induces a simplicial operad by applying the singular simplicial complex functor $\mathrm{Sing}:\mathrm{Top}\to \mathrm{sSet}$ degreewise.

• Every dendroidal set induces a simplicial operad via the right adjoint to the dendroidal homotopy coherent nerve.

Specific examples

• The Barratt-Eccles operad has, as a simplicial operad, a single color, and its simplicial set of $n$-ary operations is the nerve $N({\Sigma }_n//{\Sigma }_n)$ of the action groupoid ${\Sigma }_n//{\Sigma }_n$ of the symmetric group ${\Sigma }_n$ 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.

Related concepts

• topological operad

• dg-operad

• dendroidal set