Homological double complex

In an abelian category, a homological double complex consists of objects $C_{p,q}$ for all $(p,q)\in {\mathbb{Z}}^2$ and morphisms $h_{p,q}:C_{p,q}⟶C_{p-1,q},v_{p,q}:C_{p,q}⟶C_{p,q-1}\mathrm{.}$ For every pair of integers they satisfy $h_{p-1,q}{h}_{p,q}=0,v_{p,q-1}{v}_{p,q}=0,h_{p,q-1}{v}_{p,q}+v_{p-1,q}{h}_{p,q}=0.$ The third equality is in $\mathrm{Hom}(C_{p,q},C_{p-1,q-1})$. The first two equalities say that every row and every column is a chain complex. They are separate axioms; anticommutation alone does not imply them.

A morphism $f:C\to D$ consists of maps $f_{p,q}:C_{p,q}\to D_{p,q}$ with $f_{p-1,q}{h}_{p,q}^C=h_{p,q}^D{f}_{p,q},f_{p,q-1}{v}_{p,q}^C=v_{p,q}^D{f}_{p,q}\mathrm{.}$ Identities and composition are componentwise. The complex is first quadrant if $C_{p,q}=0$ whenever $p<0$ or $q<0$.

Zero objects and complexes supported at one bidegree are allowed. Arrows meeting a zero object are zero, including the outgoing arrows on the axes of a first-quadrant complex. This definition requires no infinite products, coproducts, or choices of representatives.