Cartan–Eilenberg resolution

Definition

Let $\mathcal{A}$ be an abelian category and let $C_\bullet$ be a chain complex in $\mathcal{A}$. A Cartan–Eilenberg resolution of $C_\bullet$ is a double complex $P_{\bullet,\bullet}$ of projective objects with $P_{p,q} = 0$ for $q < 0$. Its horizontal differential is $d^h \colon P_{p,q} \to P_{p-1,q}$ and its vertical differential is $d^v \colon P_{p,q} \to P_{p,q-1}$. It comes with a chain map $\varepsilon \colon P_{\bullet,0} \to C_\bullet$ with $\varepsilon \circ d^v = 0$ on $P_{\bullet,1}$. These data must satisfy, for every $p$:

1. if $C_p = 0$, then $P_{p,q} = 0$ for all $q$;
2. the maps induced by $\varepsilon$ on boundaries, $B_p(P_{\bullet,\bullet}, d^h) \to B_p(C)$, form a projective resolution of $B_p(C)$;
3. the maps induced by $\varepsilon$ on homology, $H_p(P_{\bullet,\bullet}, d^h) \to H_p(C)$, form a projective resolution of $H_p(C)$.

Here $B_p(P_{\bullet,\bullet}, d^h)$ and $H_p(P_{\bullet,\bullet}, d^h)$ are the complexes, with vertical differential, of horizontal boundaries and horizontal homology in column $p$.