Hochschild Cohomology, Monoid Objects and Monoidal Categories

TL;DR

This paper advances the categorical understanding of Hochschild cohomology for monoid objects in monoidal categories, offering intuitive formulations and extending the cohomology theory with new perspectives and equivalences.

Contribution

It introduces a more intuitive Hochschild cochain complex and extends the cohomology definition to bimodule objects, with an alternative cosimplicial formulation.

Findings

Provides an intuitive Hochschild cochain complex formulation

Extends Hochschild cohomology to bimodule objects

Offers an equivalent cosimplicial object formulation

Abstract

This paper expands further on a category theoretical formulation of Hochschild cohomology for monoid objects in monoidal categories enriched over abelian groups, which has been studied in arXiv:1605.00842. This topic was also presented at ISCRA, Isfahan, Iran, April 2019. The present paper aims to provide a more intuitive formulation of the Hochschild cochain complex and extend the definition to Hochschild cohomology with values in a bimodule object. In addition, an equivalent formulation of the Hochschild cochain complex in terms of a cosimplicial object in the category of abelian groups is provided.