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 i products (denoted by x iyx \cup_i y).

Definition

The sequence operad SeqSeq is the suboperad? of the endomorphism operad of the simplicial cochains functor

S *:sSetcoChS^* \colon sSet \to coCh

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

Main property

Theorem 2.15 in McClure and Smith Cochain shows that SeqSeq is an E-infinity operad in cochain complexes.

By definition, the operad SeqSeq 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 nE_n-algebras

One can easily identify a suboperad Seq nSeq_n of the operad SeqSeq 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 nn-cubes operad (Theorem 3.5 in Cochain).

References

Cochain?

James E. McClure, Jeffrey H. Smith,

Multivariable cochain operations and little nn-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.