Monoidal Categories and the Gerstenhaber Bracket in Hochschild Cohomology.

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.

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.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.