On the Ker-Coker-sequence in a semiabelian category

The Ker-Coker-sequence and its generalization in some classes of additive categories

“…We now demonstrate a version of Theorem 1 of [10] for P-semi-abelian categories: Theorem 1. The following hold:…”

“…It was demonstrated how some assumption about the properties of one of the morphisms α, β, or γ influences the exactness of (I) and the properties of the morphisms constituting the sequence. In this article we consider two generalizations of [10].…”

“…in a semi-abelian category in the sense of Palamodov (in [7,[10][11][12][13] these categories were called preabelian). The difference from the case of a quasi-abelian category is that in a P-semi-abelian category, for the validity of many of the arguments of [10], we have to impose the condition of stability under pushouts (pullbacks) on the kernels (cokernels) of some morphisms in the diagram. For example, the construction of the connecting morphism δ for the Ker-Coker-sequence is possible if coker ψ 0 is a stable cokernel or ker ϕ 1 is a stable kernel in the above sense.…”

The five- and nine-lemmas in P-semi-abelian categories

“…Throughout the sequel, the ambient category A is assumed P-semi-abelian. We will need the following lemma from [4], whose proof coincides almost verbatim with that of Lemma 6 in [2].…”

“…The proofs of N. V. Glotko are based on the theorems about the Ker-Coker-sequence in a quasi-abelian category [2,3].…”