A Homological View of Categorical Algebra

We approach categorical algebra with the goal of enabling homological and homotopical methods. From the ground up, we develop what is needed to work effectively with chain complexes and their homology. Thus we start by assuming only the existence of a zero-object, kernels and cokernels. For effective computations with chain complexes, we only require one additional structural axiom, namely the di-exactness condition. By assuming that any composite of a normal monomorphism followed by a normal epimorphism can be written as a normal epimorphism followed by a normal monomorphism, we arrive at the concept of a di-exact category. Here the Snake Lemma, a primordial version of the Short Five Lemma, and border cases of the 3 3-Lemma hold. In the course of this development, we also incorporate familiar structures such as normal categories, homological categories, and semiabelian categories. Remarkable is, that a finitely cocomplete homological category is semiabelian if and only if it satisfies the di-exactness condition.