This entry may need to be merged with cochain on a simplicial set.
Given a simplicial set , the simplicial cochains on form a cochain complex. The cohomology of this cochain complex computes the cohomology of the simplicial set .
As a special case, if is the singular simplicial set of a topological space , then the simplicial cochains of are precisely the singular cochains of .
Given an abelian group , the simplicial cochains functor is a functor
It is defined as the composition of the simplicial chains functor (with integer coefficients)
with the dualization functor
The simplicial cochains of a simplicial set with coefficients in a commutative ring admit an action of the sequence operad, which turns 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 February 1, 2021 at 02:33:06. See the history of this page for a list of all contributions to it.