nLab
sequence operad
Contents
Idea
The sequence operad is an E-infinity operad , valued in cochain complexes , that acts on simplicial cochains of any simplicial set .

Furthermore, this operad has an explicit presentation in terms of a countable collection of generators and relations.

The operations in this operad are generalized Steenrod cup-$i$ products (denoted by $x \cup_i y$ ).

Definition
The sequence operad $Seq$ is the suboperad? of the endomorphism operad of the simplicial cochains functor

$S^* \colon sSet \to coCh$

such that $Seq(n)$ is the subobject of $Hom((S^*)^{\otimes n},S^*)$ consisting of Steenrod’s generalized cup products .

Main property
Theorem 2.15 in McClure and Smith Cochain shows that $Seq$ is an E-infinity operad in cochain complexes .

By definition, the operad $Seq$ acts on simplicial cochains of any simplicial set , in particular, it acts on singular cochains of a topological space .

This provides a simple and concrete model of the E-infinity algebra structure on simplicial cochains .

Generalizations to $E_n$ -algebras
One can easily identify a suboperad $Seq_n$ of the operad $Seq$ by imposing a simple combinatorial condition on the multiindices of generalized cup products (Definition 3.2 in Cochain ).

The resulting operad is an E_n-operad , i.e., it is weakly equivalent to the singular cochains on the little $n$ -cubes operad (Theorem 3.5 in Cochain ).

References
Cochain?

James E. McClure , Jeffrey H. Smith ,

Multivariable cochain operations and little $n$ -cubes , arXiv:math/0106024v3 .

Created on February 22, 2020 at 13:37:57.
See the history of this page for a list of all contributions to it.