Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology / Reiner Hermann.

Format:

Book

Author/Creator:

Hermann, Reiner, 1986- author.

Series:

Memoirs of the American Mathematical Society ; v. 243, no. 1151.

Memoirs of the American Mathematical Society, 0065-9266 ; volume 243, number 1151

Language:

English

Subjects (All):

Homotopy theory.

Geometry, Algebraic.

Associative rings.

Rings (Algebra).

Physical Description:

1 online resource (158 pages) : illustrations.

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2016]

Summary:

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.

Contents:

Cover

Title page

Introduction

Background

Main results

Outline

Conventions

Chapter 1. Prerequisites

1.1. Exact categories

1.2. Monoidal categories

1.3. Examples: Exact and monoidal categories

Chapter 2. Extension categories

2.1. Definition and properties

2.2. Homotopy groups

2.3. Lower homotopy groups of extension categories

2.4. -Extension closed subcategories

Chapter 3. The Retakh isomorphism

3.1. An explicit description

3.2. Compatibility results

3.3. Extension categories for monoidal categories

Chapter 4. Hochschild cohomology

4.1. Basic definitions

4.2. Gerstenhaber algebras

Chapter 5. A bracket for monoidal categories

5.1. The Yoneda product

5.2. The bracket and its properties

5.3. The module case -Schwede's original construction

5.4. Morita equivalence

5.5. The monoidal category of bimodules

Chapter 6. Application I: The kernel of the Gerstenhaber bracket

6.1. Introduction and motivation

6.2. Bialgebroids

6.3. A monoidal functor

6.4. Comparison to Linckelmann's result

Chapter 7. Application II: The \Ext-algebra of the identity functor

7.1. The evaluation functor

7.2. Exact endofunctors

7.3. \Ext-algebras and adjoint functors

7.4. Hochschild cohomology for abelian categories

Acknowledgements

Appendix A. Basics

A.1. Homological lemmas

A.2. Algebras, coalgebras, bialgebras and Hopf algebras

A.3. Examples: Hopf algebras

Bibliography

Main references

Supplemental references

Back Cover.

Notes:

"Volume 243, number 1151 (fourth of 4 numbers), September 2016."

Includes bibliographical references.

Description based on print version record.