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Categories of imaginaries for definable additive categories  [PDF]
Mike Prest
Mathematics , 2012,
Abstract: We develop some aspects of the model theory of additive structures, with particular emphasis on the abelian category of pp-imaginaries.
Additive versus abelian 2-representations of fiat 2-categories  [PDF]
Volodymyr Mazorchuk,Vanessa Miemietz
Mathematics , 2011,
Abstract: We study connections between additive and abelian 2-representations of fiat 2-categories, describe combinatorics of 2-categories in terms of multisemigroups and determine the annihilator of a cell 2-representation. We also describe, in detail, examples of fiat 2-categories associated to $\mathfrak{sl}_2$-categorification in the sense of Chuang and Rouquier, and 2-categorical analogues of Schur algebras.
On Ideals and Homology in Additive Categories  [PDF]
Lucian M. Ionescu
Mathematics , 1999,
Abstract: Ideals are used to define homological functors for additive categories. In abelian categories the ideals corresponding to the usual universal objects are principal, and the construction reduces, in a choice dependent way, to homology groups. Applications are considered: derived categories and functors.
On ideals and homology in additive categories
Lucian M. Ionescu
International Journal of Mathematics and Mathematical Sciences , 2002, DOI: 10.1155/s0161171202011675
Abstract: Ideals are used to define homological functors in additive categories. In abelian categories the ideals corresponding to the usual universal objects are principal, and the construction reduces, in a choice dependent way, to homology groups. The applications considered in this paper are: derived categories and functors.
Maximal exact structures on additive categories revisited  [PDF]
Septimiu Crivei
Mathematics , 2011,
Abstract: Sieg and Wegner showed that the stable exact sequences define a maximal exact structure (in the sense of Quillen) in any pre-abelian category. We generalize this result for weakly idempotent complete additive categories.
Recollement of additive quotient categories  [PDF]
Minxiong Wang,Zengqiang Lin
Mathematics , 2015,
Abstract: In this note, we define a recollement of additive categories, and prove that such a recollement can induce a recollement of their quotient categories. As an application, we get a recollement of quotient triangulated categories induced by mutation pairs.
Abelian categories versus abelian 2-categories  [PDF]
Teimuraz Pirashvili
Mathematics , 2008,
Abstract: Recently Dupont proved that the categories of discrete and codiscrete (or connected) objects in an abelian 2-category are equivalent abelian categories. He posses also a question whether any abelian category comes in this way. We will give a rather trivial solution of this problem in the case when a given abelian category has enough projective or injective objects.
Expansions of abelian categories  [PDF]
Xiao-Wu Chen,Henning Krause
Mathematics , 2010,
Abstract: Expansions of abelian categories are introduced. These are certain functors between abelian categories and provide a tool for induction/reduction arguments. Expansions arise naturally in the study of coherent sheaves on weighted projective lines; this is illustrated by various applications.
Comparison of Abelian categories recollements  [PDF]
Vincent Franjou,Teimuraz Pirashvili
Mathematics , 2004,
Abstract: We give a necessary and sufficient condition for a morphism between recollements of abelian categories to be an equivalence.
On Moduli Spaces for Abelian Categories  [PDF]
Vyacheslav Futorny,Marcos Jardim,Adriano Moura
Mathematics , 2007,
Abstract: We show that if A is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to several important abelian categories in representation theory, like highest weight categories.
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