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 -