nLab simplicial cochain

This entry may need to be merged with cochain on a simplicial set.

Idea

Given a simplicial set $X$, the simplicial cochains on $X$ form a cochain complex. The cohomology of this cochain complex computes the cohomology of the simplicial set $X$.

As a special case, if $X$ is the singular simplicial set of a topological space $S$, then the simplicial cochains of $X=Sing(S)$ are precisely the singular cochains of $S$.

Definition

Given an abelian group $A$, the simplicial cochains functor is a functor

$$C^*(-,A)\colon sSet \to coCh.$$

It is defined as the composition of the simplicial chains functor (with integer coefficients)

$$C\colon sSet \to sAb \to Ch$$

with the dualization functor

$$Hom(-, A[0])\colon Ch \to coCh.$$

Additional structures

The simplicial cochains of a simplicial set $X$ with coefficients in a commutative ring $A$ admit an action of the sequence operad, which turns $C^*(X,A)$ into an E-infinity algebra.

In particular, this structure incorporates simplicial cup products of cochains, as well as Steenrod’s generalized cup products.

Related concepts

• simplicial chains

• singular cochains

• singular cohomology

References

An E-infinity algebra on simplicial cochains is constructed in

• Albrecht Dold, Über die Steenrodschen Kohomologieoperationen, Annals of Math. (2) 73 (1961), 258–294.

• Vladimir Hinich, Vadim Schechtman, On homotopy limit of homotopy algebras. K-theory, arithmetic and

  geometry (Moscow, 1984–1986), 240–264, Lecture Notes in Mathematics 1289, Springer, 1987.

• James McClure, Jeffrey Smith, Multivariable cochain operations and little $n$-cubes, Journal of the American Mathematical Society 16:3 (2003), 681-704.

  doi, arXiv.