Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Kopylov Y. A. On Some Diagram Assertions in Preabelian and P-Semi-Abelian Categories. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 4, pp. 434-443. DOI: 10.18500/1816-9791-2020-20-4-434-443, EDN: RVAYDK

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
30.11.2020
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Language: 
English
Heading: 
Mathematics
Article type: 
Article
UDC: 
512.66:517.982.2
DOI: 
10.18500/1816-9791-2020-20-4-434-443
EDN: 
RVAYDK

On Some Diagram Assertions in Preabelian and P-Semi-Abelian Categories

Autors: 
Kopylov Yaroslav A., Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences (Sobolev Institute of Mathematics)
Abstract: 

As is well known, many important additive categories in functional analysis and algebra are not abelian. Many classical diagram assertions valid in abelian categories fail in more general additive categories without additional assumptions concerning the properties of the morphisms of the diagrams under consideration. This in particular applies to the so-called Snake Lemma, or the KerCoker-sequence. We obtain a theorem about a diagram generalizing the classical situation of the Snake Lemma in the context of categories semi-abelian in the sense of Palamodov. It is also known that, already in P-semi-abelian categories, not all kernels (respectively, cokernels) are semi-stable, that is, stable under pushouts (respectively, pullbacks). We prove a proposition showing how non-semi-stable kernels and cokernels can arise in general preabelian categories.

Key words: 
P-semi-abelian category
strict morphism
semi-stable kernels and cokernels
Snake Lemma
Ker-sequence
Coker-sequence
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Received: 
15.12.2019
Accepted: 
23.03.2020
Published: 
30.11.2020
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