Lemma 24.25.12. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let (\mathcal{A}, \text{d}) be a sheaf of differential graded algebras on (\mathcal{C}, \mathcal{O}). Let R be a set and for each r \in R let an injective map \mathcal{M}_ r \to \mathcal{M}'_ r of acyclic differential graded \mathcal{A}-modules be given. There exists a functor M : \textit{Mod}(\mathcal{A}, \text{d}) \to \textit{Mod}(\mathcal{A}, \text{d}) and a natural transformation j : \text{id} \to M such that
j_\mathcal {M} : \mathcal{M} \to M(\mathcal{M}) is injective and a quasi-isomorphism,
for every solid diagram
\xymatrix{ \mathcal{M}_ r \ar[r] \ar[d] & \mathcal{M} \ar[d]^{j_\mathcal {M}} \\ \mathcal{M}'_ r \ar@{..>}[r] & M(\mathcal{M}) }
a dotted arrow exists in \textit{Mod}(\mathcal{A}, \text{d}) making the diagram commute.
Proof.
We define M(\mathcal{M}) as the pushout in the following diagram
\xymatrix{ \bigoplus _{(r, \varphi )} \mathcal{M}_ r \ar[r] \ar[d] & \mathcal{M} \ar[d] \\ \bigoplus _{(r, \varphi )} \mathcal{M}'_ r \ar[r] & M(\mathcal{M}) }
where the direct sum is over all pairs (r, \varphi ) with r \in R and \varphi \in \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}(\mathcal{A}, \text{d})}(\mathcal{M}_ r, \mathcal{M}). Since the pushout of an injective map is injective, we see that \mathcal{M} \to M(\mathcal{M}) is injective. Since the cokernel of the left vertical arrow is acyclic, we see that the (isomorphic) cokernel of \mathcal{M} \to M(\mathcal{M}) is acyclic, hence \mathcal{M} \to M(\mathcal{M}) is a quasi-isomorphism. Property (2) holds by construction. We omit the verification that this procedure can be turned into a functor.
\square
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