Non-abelian and $\epsilon$-curved homological algebra with arrow categories

Grandis’s non-abelian homological algebra generalizes standard homological algebra in abelian categories to $\mathit{\text{homological}}\mathit{\text{categories}}$, 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 $\epsilon \mathit{\text{-curved}}\mathit{\text{homological}}\mathit{\text{algebra}}$. This refers to chain complexes of normed spaces whose differential squares to zero only approximately, in the sense that $\mathrm{\parallel }{d}^2\mathrm{\parallel }\le \epsilon$ for some $\epsilon >0$. This is relevant for example in the theory of approximate representations of groups, where Kazhdan has successfully employed $\epsilon$-curved homological techniques in an ad-hoc manner. We develop some basics of $\epsilon$-curved homological algebra and note that our result on arrow categories facilitates the application of Grandis’s theory.