  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.
