nLab sequence operad

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 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 here).

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 here).

References

• James E. McClure, Jeffrey H. Smith,

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