  Grandis's non-abelian homological algebra generalizes standard homological
algebra in abelian categories to \textit{homological categories}, which are a
broader class of categories including for example the category of lattices and
Galois connections. Here, we prove that if $\mathsf{C}$ is any category with an
ideal of null morphisms with respect to which (co)kernels exist, then the arrow
category of $\mathsf{C}$ is a homological category. This broadens the
applicability of Grandis's framework substantially. In particular, one can form
the homology of chain complexes in $\mathsf{C}$ by taking the homology objects
to be morphisms of $\mathsf{C}$, which one may think of as maps from an object
of cycles to an object of chains modulo boundaries.
  One situation to which Grandis's original framework does not apply is
\textit{$\varepsilon$-curved homological algebra}. This refers to chain
complexes of normed spaces whose differential squares to zero only
approximately, in the sense that $\|d^2\| \leq \varepsilon$ for some
$\varepsilon > 0$. This is relevant for example in the theory of approximate
representations of groups, where Kazhdan has successfully employed
$\varepsilon$-curved homological techniques in an ad-hoc manner. We develop
some basics of $\varepsilon$-curved homological algebra and note that our
result on arrow categories facilitates the application of Grandis's theory.
