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Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology. (English) Zbl 1397.16008

Mem. Am. Math. Soc. 1151, vi, 149 p. (2016).
Summary: In this monograph, we extend S. Schwede’s exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology [J. Reine Angew. Math. 498, 153–172 (1998; Zbl 0923.16007)] to certain exact and monoidal categories. Therefore we establish an explicit description of an isomorphism by A. Neeman and V. Retakh [Compos. Math. 102, No. 2, 203–242 (1996; Zbl 0852.18011)], which links 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, we show that our construction behaves well with respect to structure preserving functors between exact monoidal categories. We use our main result to conclude, that the graded Lie bracket in Hochschild cohomology is an invariant under Morita equivalence. For quasi-triangular bialgebras, we further determine a significant part of the Lie bracket’s kernel, and thereby prove a conjecture by L. Menichi [J. Algebra 331, No. 1, 311–337 (2011; Zbl 1256.16010)]. Along the way, we introduce \(n\)-extension closed and entirely extension closed subcategories of abelian categories, and study some of their properties.

MSC:

16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
14F35 Homotopy theory and fundamental groups in algebraic geometry
16T05 Hopf algebras and their applications
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
18E10 Abelian categories, Grothendieck categories
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
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References:

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