Exact couples in semiabelian categories revisited

TL;DR

This paper revisits the construction of exact couples in semiabelian categories, demonstrating how classical methods produce connected derived couples and exploring their iterative derivation under certain conditions.

Contribution

It extends the classical theory of exact couples to semiabelian categories with strict morphisms, establishing connections between left and right cohomology objects.

Findings

Derived couples are connected by a natural bimorphism.

The derivation process can be iterated under additional assumptions.

Classical constructions are valid in semiabelian categories with strict morphisms.

Abstract

Consider an exact couple in a semiabelian category in the sense of Palamodov, i.e., in an additive category in which every morphism has a kernel as well as a cokernel and the induced morphism between coimage and image is always monic and epic. Assume that the morphisms in the couple are strict, i.e., they induce even isomorphisms between their corresponding coimages and images. We show that the classical construction of Eckmann and Hilton in this case produces two derived couples which are connected by a natural bimorphism. The two couples correspond to the a priori distinct cohomology objects, the left resp. right cohomology, associated with the initial exact couple. The derivation process can be iterated under additional assumptions.