nLab
simplicial cochain

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

Contents

Idea

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

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

Definition

Given an abelian group AA, the simplicial cochains functor is a functor

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

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

C:sSetsAbChC\colon sSet \to sAb \to Ch

with the dualization functor

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

Additional structures

The simplicial cochains of a simplicial set XX with coefficients in a commutative ring AA admit an action of the sequence operad, which turns C *(X,A)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. See the history of this page for a list of all contributions to it.