TY - JOUR

T1 - On the notion of a semi-abelian category in the sense of Palamodov

AU - Kopylov, Yaroslav

AU - Wegner, Sven Ake

N1 - Funding Information:
Acknowledgements Both authors would like to thank the referee for several useful comments which helped to improve this article. The first-named author was partially supported by the Russian Foundation for Basic Research (Grant 09-01-00142-a), the State Maintenance Program for the Leading Scientific Schools and Junior Scientists of the Russian Federation (NSh-6613.2010.1), and the Integration Project “Quasiconformal Analysis and Geometric Aspects of Operator Theory” of the Siberian Branch and the Far East Branch of the Russian Academy of Sciences.

PY - 2012/10

Y1 - 2012/10

N2 - In the sense of Palamodov, a preabelian category is semi-abelian if for every morphism the natural morphism between the cokernel of its kernel and the kernel of its cokernel is simultaneously a monomorphism and an epimorphism. In this article we present several conditions which are all equivalent to semi-abelianity. First we consider left and right semi-abelian categories in the sense of Rump and establish characterizations of these notions via six equivalent properties. Then we use these properties to deduce the characterization of semi-abelianity. Finally, we investigate two examples arising in functional analysis which illustrate that the notions of right and left semi-abelian categories are distinct and in particular that such categories occur in nature.

AB - In the sense of Palamodov, a preabelian category is semi-abelian if for every morphism the natural morphism between the cokernel of its kernel and the kernel of its cokernel is simultaneously a monomorphism and an epimorphism. In this article we present several conditions which are all equivalent to semi-abelianity. First we consider left and right semi-abelian categories in the sense of Rump and establish characterizations of these notions via six equivalent properties. Then we use these properties to deduce the characterization of semi-abelianity. Finally, we investigate two examples arising in functional analysis which illustrate that the notions of right and left semi-abelian categories are distinct and in particular that such categories occur in nature.

KW - Category of bornological spaces

KW - Preabelian category

KW - Quasi-abelian category

KW - Semi-abelian category

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U2 - 10.1007/s10485-011-9249-0

DO - 10.1007/s10485-011-9249-0

M3 - Article

AN - SCOPUS:84865150012

VL - 20

SP - 531

EP - 541

JO - Applied Categorical Structures

JF - Applied Categorical Structures

SN - 0927-2852

IS - 5

ER -