Hochschild and Simplicial Cohomology

TL;DR

This paper explores the relationship between Hochschild and simplicial cohomology, introducing autopoietic cochains for broader algebra classes and connecting algebraic operations with topological cohomology.

Contribution

It introduces autopoietic cochains extending coherent cochains to more general algebras and links algebraic products with topological cohomology operations.

Findings

Identifies a subalgebra of Hochschild cochains detecting simplicial cohomology.

Shows the correspondence between algebraic and topological cohomology products.

Extends the concept of coherent cochains to broader algebra classes.

Abstract

We study a naturally occurring $E_{\infty}$-subalgebra of the full $E_2$-Hochschild cochain complex arising from coherent cochains. For group rings and certain category algebras, these cochains detect $H^*(B {\cal{C}})$, the simplicial cohomology of the classifying space of the underlying group or category, $\cal{C}$. In this setting the simplicial cup product of cochains on $B{\cal{C}}$ agrees with the Gerstenhaber product and Steenrod's cup-one product of cochains agrees with the pre-Lie product. We extend the idea of coherent cochains to algebras more general than category algebras and dub the resulting cochains autopoietic. Coefficients are from a commutative ring $k$ with unit, not necessarily a field.