Complements and the Krull-Schmidt theorem in arbitrary categories (Q1346408)

A new remarkable class of categories is introduced and studied. These categories (defined by the conditions: -- finitely complete and cocomplete, -- with zero object, -- certain axioms for a coimage factorization of morphisms) are the framework for the study of the direct product of objects. Direct products \(C = A \times B\) are characterized by ``inner'' properties of \(C\) and its subobjects \(A\) and \(B\). Using specific subobjects and certain endomorphisms, generalizations of the Fitting lemma and the Krull-Schmidt theorem are obtained for the above mentioned categories.