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Abelian categories and definable additive categories  [PDF]
Mike Prest
Mathematics , 2012,
Abstract: We consider three (2-)categories and their (anti-)equivalence. They are the category of small abelian categories and exact functors, the category of definable additive categories and interpretation functors, the category of locally coherent abelian categories and coherent morphisms. These categories link algebra, model theory and "geometry".
Imaginaries and definable types in algebraically closed valued fields  [PDF]
Ehud Hrushovski
Mathematics , 2014,
Abstract: The text is based on notes from a class entitled {\em Model Theory of Berkovich Spaces}, given at the Hebrew University in the fall term of 2009, and retains the flavor of class notes. It includes an exposition of material from \cite{hhmcrelle}, \cite{hhm} and \cite{HL}, regarding definable types in the model completion of the theory of valued fields, and the classification of imaginary sorts. The latter is given a new proof, based on definable types rather than invariant types, and on the notion of {\em generic reparametrization}. I also try to bring out the relation to the geometry of \cite{HL} - stably dominated definable types as the model theoretic incarnation of a Berkovich point.
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.
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  [PDF]
Dennis Sieg,Sven-Ake Wegner
Mathematics , 2014, DOI: 10.1002/mana.200910154
Abstract: We show that every additive category with kernels and cokernels admits a maximal exact structure. Moreover, we discuss two examples of categories of the latter type arising from functional analysis.
Some remarks on multicategories and additive categories  [PDF]
Claudio Pisani
Mathematics , 2013,
Abstract: Categories are coreflectively embedded in multicategories via the "discrete cocone" construction, the right adjoint being given by the monoid construction. Furthermore, the adjunction lifts to the "cartesian level": preadditive categories are coreflectively embedded (as theories for many-sorted modules) in cartesian multicategories (general algebraic theories). In particular, one gets a direct link between two ways of considering modules over a rig, namely as additive functors valued in commutative monoids or as models of the theory generated by the rig itself.
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.
Cohomology of exact categories and (non-)additive sheaves  [PDF]
Dmitry Kaledin,Wendy Lowen
Mathematics , 2011,
Abstract: We use (non-)additive sheaves to introduce an (absolute) notion of Hochschild cohomology for exact categories as Ext's in a suitable bisheaf category. We compare our approach to various definitions present in the literature.
Interpretable groups are definable  [PDF]
Janak Ramakrishnan,Ya'acov Peterzil,Pantelis Eleftheriou
Mathematics , 2011,
Abstract: We prove that in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. We also prove that every definable group lives in a cartesian product of one-dimensional definable group-intervals (or one-dimensional definable groups). We discuss the general open question of elimination of imaginaries in an o-minimal structure.
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