Hochschild and Simplicial Cohomology

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.