Twisted complexes and simplicial homotopies

In this paper we consider the dg-category of twisted complexes over simplicial ringed spaces. It is clear that a simplicial map $f: ($\mathcal{U}$,$\mathcal{R}$)\to ($\mathcal{V}$, $\mathcal{S}$)$ between simplicial ringed spaces induces a dg-functor $f^*: $\text{Tw}$($\mathcal{V}$, $\mathcal{S}$)\to $\text{Tw}$($\mathcal{U}$, $\mathcal{R}$)$where$$\text{Tw}$($\mathcal{U}$, $\mathcal{R}$)$denotesthedg-categoryoftwistedcomplexeson$($\mathcal{U}$,$\mathcal{R}$)$. In this paper we prove that for simplicial homotopic maps $f$ and $g$, there exists an $A_{\mathrm{\infty }}$-natural transformation $\mathrm{\Phi }:f^{\ast }\Rightarrow g^{\ast }$ between induced dg-functors. Moreover the $0$th component of $\mathrm{\Phi }$ is an objectwise weak equivalence. If we restrict ourselves to the full dg-subcategory of twisted perfect complexes, then we prove that $\mathrm{\Phi }$ admits an $A_{\mathrm{\infty }}$-quasi-inverse when $(\mathcal{U},\mathcal{R})$ satisfies some additional conditions.