Let $X$ be a (say, topological) space and $i: Z\hookrightarrow X$ be a closed subspace. Let $Sh(X)$ and $Sh(Z)$ denote the categories of sheaves of abelian groups on $X$ and $Z$ respectively. Similarly let $Ch(X)$ and $Ch(Z)$ denote the categories of complexes of sheaves of abelian groups on $X$ and $Z$ respectively.

Let $\mathcal{F}^{\cdot}\in Ch(X)$ and $G^{\cdot}\in Ch(Z)$. Suppose we have a morphism $u: i^*\mathcal{F}^{\cdot} \to \mathcal{G}^{\cdot}$ which is invertible up to chain homotopy. Then can we extend $\mathcal{G}^{\cdot}$ and $u$ to $X$? In other words, can we find a complex $\mathcal{H}^{\cdot}\in Ch(X)$ and a morphism $v:\mathcal{F}^{\cdot}\to\mathcal{H}^{\cdot}$ such that

1. $v$ is also invertible up to chain homotopy.
2. $i^*\mathcal{H}^{\cdot}\cong \mathcal{G}^{\cdot}$ degreewisely (not only up to homotopy).
3. The pullback of the morphism, $i^*v$, is homotopical equivalent to $u$?

Let $X$ be a (say, topological) space and $i: Z\hookrightarrow X$ be a closed subspace. Let $Sh(X)$ and $Sh(Z)$ denote the categories of sheaves of abelian groups on $X$ and $Z$ respectively. Similarly let $Ch(X)$ and $Ch(Z)$ denote the categories of complexes of sheaves of abelian groups on $X$ and $Z$ respectively.

Let $\mathcal{F}^{\cdot}\in Ch(X)$ and $G^{\cdot}\in Ch(Z)$. Suppose we have a morphism $u: i^*\mathcal{F}^{\cdot} \to \mathcal{G}^{\cdot}$ which is invertible up to chain homotopy. Then can we extend $\mathcal{G}^{\cdot}$ and $u$ to $X$? In other words, can we find a complex $\mathcal{H}^{\cdot}\in Ch(X)$ and a morphism $v:\mathcal{F}^{\cdot}\to\mathcal{H}^{\cdot}$ such that

1. $v$ is also invertible up to chain homotopy.
2. $i^*\mathcal{H}^{\cdot}\cong \mathcal{G}^{\cdot}$ degreewisely (not only up to homotopy).
3. The pullback of the morphism, $i^*v$, is homotopical equivalent to $u$?

We can put some restrictions, say bounded, on the complexes.