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