is a subcomplex of $C_*$. Such pairs are always understood in the *cellular* sense; i.e.  the distinguished basis of each $D_{{n}}$ is a subset of the distinguished basis of $C_{{n}}$.

A *reduction* $(C_*,D_*){\xRightarrow{\ \ }}(C'_*,D'_*)$ *of (cellular) pairs* is a reduction $C_*{\xRightarrow{\ \ }}C'_*$ that restricts to a reduction $D_*{\xRightarrow{\ \ }}D'_*$, i.e. such that $\alpha(D_*)\subseteq D'_*$, $\beta(D'_*)\subseteq D_*$, and $\eta(D_*)\subseteq D_*$.

From this reduction, we get an induced reduction $C_*/D_*{\xRightarrow{\ \ }}C'_*/D'_*$ of the quotients.

We will need to work with a notion more general than reductions, namely strong equivalences. A *strong equivalence* $C_*{\xLeftrightarrow{\ \ \ }}C'_*$ is a pair of reductions $C_*{\xLeftarrow{\ \ }}\widehat C_*{\xRightarrow{\ \ }}C'_*$, where $\widehat C_*$ is some chain complex. Similarly, a strong equivalence $(C_*,D_*){\xLeftrightarrow{\ \ \ }}(C'_*,D'_*)$ is a pair of reductions $(C_*,D_*){\xLeftarrow{\ \ }}(\widehat C_*,\widehat D_*){\xRightarrow{\ \ }}(C'_*,D'_*)$. Strong equivalences can be (algorithmically) composed: if $C_*{\xLeftrightarrow{\ \ \ }}C'_*$ and $C'_*{\xLeftrightarrow{\ \ \ }}C''_*$, then one obtains $C_*{\xLeftrightarrow{\ \ \ }}C''_*$ (see e.g. [@polypost Lemma 2.7]).

Let $C_*$ be a chain complex. We say that $C_*$ is equipped with *effective homology* if there is specified a strong equivalence $C_*{\xLeftrightarrow{\ \ \ }}C_*^{\mathrm{ef}}$ of $C_*$ with some effective chain complex $C_*^{\mathrm{ef}}$. Effective homology 