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# ON THE HOMOLOGY THEORY OF MODULES

Journal of the Faculty of Science. Section I 01/1954;
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##### Article: Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology
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ABSTRACT: In this thesis, we extend S. Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore we establish 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 so called 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 quasitriangular Hopf algebras, we further determine a significant part of the Lie bracket's kernel. Along the way, we introduce $n$-extension closed and entirely extension closed subcategories of abelian categories, and study some of their properties.
03/2014;
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##### Article: An Introduction to Topos Theory
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ABSTRACT: The purpose of this text is to equip the reader with an intuitive but precise understanding of elementary structures of category and topos theory. In order to achieve this goal, we provide a guided tour through category theory, leading to the definition of an elementary (Lawvere–Tierney) topos. Then we turn to the investigation of consequences of this definition. In particular, we analyse in detail the topos Set 2 op , the internal structure of its subobject classifier and its variation over stages. Next we turn to the discussion of the interpretation of a logic and language in topos, viewed as a model of higher order intuitionistic multisort type theory, as well as the geometric perspective on a topos, viewed as a category of set-valued sheaves over base category equipped with a Grothendieck topology. This text is designed as an elementary introduction, written in a self-contained way, with no previous knowledge required.
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##### Article: A note on homology in categories
Annals of Mathematics 01/1959; · 2.82 Impact Factor