Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology
Author(s)
Bibliographic Information
Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology
（Memoirs of the American Mathematical Society, no. 1151）
American Mathematical Society, c2016
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Note
Includes bibliographical references
"Volume 243, number 1151 (fourth of 4 numbers), September 2016"
Description and Table of Contents
Description
In this monograph, the author extends S. Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore the author establishes an explicit description of an isomorphism by A. Neeman and V. Retakh, which links $\mathrm{Ext}$-groups with fundamental groups of categories of extensions and relies on expressing the fundamental group of a (small) category by means of the associated Quillen groupoid.
As a main result, the author shows that his construction behaves well with respect to structure preserving functors between exact monoidal categories. The author uses his main result to conclude, that the graded Lie bracket in Hochschild cohomology is an invariant under Morita equivalence. For quasi-triangular bialgebras, he further determines a significant part of the Lie bracket's kernel, and thereby proves a conjecture by L. Menichi. Along the way, the author introduces $n$-extension closed and entirely extension closed subcategories of abelian categories, and studies some of their properties.
Table of Contents
Introduction
Prerequisites
Extension categories
The Retakh isomorphism
Hochschild cohomology
A bracket for monoidal categories
Application I: The kernel of the Gerstenhaber bracket
Application II: The $\mathbf{Ext}$-algebra of the identity functor
Appendix A. Basics
Bibliography.
by "Nielsen BookData"