A Homological View of Categorical Algebra

Abstract

We provide a foundation for working with homological and homotopical methods in categorical algebra. This involves two mutually complementary components, namely (a) the strategic selection of suitable axiomatic frameworks, some well known and some new, and (b) the development of categorical tools for effective reasoning and computing within those frameworks. The selection of axiomatic frameworks begins `from the ground up' with z-exact categories. These are pointed categories in which every morphism has a kernel and cokernel. Then we progress all the way to abelian categories, en route meeting contexts such as Borceux--Bourn homological categories and Janelidze--M\'arki--Tholen semiabelian categories. We clarify the relationship between these axiomatic frameworks by direct comparison, but also by explaining how concrete examples fit into the selection. The outcome is a fine-grained set of criteria by which one can map varieties of algebras (in the sense of Universal Algebra) and topological models of algebraic theories into the various frameworks. The categorical tools for effective computation deal mostly with situations involving universal factorizations of morphisms, with exact sequences, and with the homology of chain complexes. Further, the categorical tools for computation include the `basic diagram lemmas' of homological algebra, that is the (Short) $5$-Lemma, the $(3\times 3)$-Lemma, the Snake Lemma, and applications thereof. We find that these tools are even available in some surprisingly weak categorical environments, such as the category of pointed sets. In discovering such features, we make systematic use of what we call the self-dual axis of a category.