bases are finite in each dimension).

For example, if a simplicial set $X$ admits an algorithm generating a (finite) list of its non-degenerate distinguished simplices in any given dimension (we call it *effective* in Section \[sec:polynomiality\]), then its normalized chain complex $C_*X$ is effective.

Reductions, strong equivalences
-------------------------------

\[s:effReduction\]

We recall that a *reduction* (also called *contraction* or *strong deformation retraction*) $C_*{\xRightarrow{\ \ }}C'_*$ between two chain complexes is a triple $(\alpha,\beta,\eta)$ such that $\alpha{\colon}C_*{\rightarrow}C'_*$ and $\beta{\colon}C'_*{\rightarrow}C_*$ are equivariant chain maps with $\alpha\beta={\mathop\mathrm{id}\nolimits}$ and $\eta$ is an equivariant chain homotopy on $C_*$ with $\partial\eta+\eta\partial={\mathop\mathrm{id}\nolimits}-\beta\alpha$; moreover, we require that $\eta\beta=0$, $\alpha\eta=0$ and $\eta\eta=0$. The following diagram illustrates this definition:

$$(\alpha,\beta,\eta){\colon}C_*{\xRightarrow{\ \ }}C'_*\quad\equiv\quad\xymatrix@C=30pt{
C_* \ar@(ul,dl)[]_{\eta} \ar@/^/[r]^\alpha & C'_* \ar@/^/[l]^{\beta}
}$$

Reductions are used to solve homological problems in $C_*$ by translating them to $C_*'$ and vice versa, see [@SergerGenova]; a particular example is seen at the end of the proof of Lemma \[l:lift\_ext\_one\_stage\]. While for this principle to work, chain homotopy equivalences would be enough, they are not sufficient for the so-called perturbation lemmas (we will introduce them later), where the real strength of reductions lies.

For the following definition, we consider pairs $(C_*,D_*)$, where $C_*$ is a chain complex and $D_*$ 