О некоторых диаграммных утверждениях в предабелевых и P-полуабелевых категориях

Yaroslav A Kopylov

doi:10.18500/1816-9791-2020-20-4-434-443

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.

Highlights

Many classical diagram assertions valid in abelian categories fail in more general additive and nonadditive 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 Ker-Coker-sequence
It is natural to expect that possible generalizations of the Snake Lemma in the non-abelian setting would require additional conditions on the morphisms of the diagram under consideration

Summary

INTRODUCTION

Many classical diagram assertions valid in abelian categories fail in more general additive and nonadditive 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 Ker-Coker-sequence (see, for example, [1, 2] or [3]). ❨❛✳ ❆✳ ❑♦♣2❧♦✈✳ ❖♥ ❙♦♠❡ ❉✐❛❣r❛♠ ❆ss❡rt✐♦♥s ✐♥ Pr❡❛❜❡❧✐❛♥ ❛♥❞ P ✲❙❡♠✐✲❆❜❡❧✐❛♥ ❈❛t❡❣♦r✐❡s an analog of this assertion holds for the larger class of P -semi-abelian categories. This is the main result of the present article. In Section 4., we prove the above mentioned main result on the exactness of the Ker- and Coker-sequences (Theorem 1)

PREABELIAN AND P -SEMI-ABELIAN CATEGORIES

SEMI-STABLE KERNELS AND COKERNELS IN A PREABELIAN CATEGORY

THE LEFT AND RIGHT HOMOLOGY OBJECTS

A GENERALIZATION OF THE SNAKE LEMMA