#
nLab

simplicial cochain

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

# Contents

## 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.

Last revised on January 31, 2021 at 21:33:06.
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