Twisted complexes and simplicial homotopies

Abstract

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 $\text{Tw}(\mathcal{U}, \mathcal{R})$ denotes the dg-category of twisted complexes on $(\mathcal{U},\mathcal{R})$. In this paper we prove that for simplicial homotopic maps $f$ and $g$, there exists an $A_{\infty}$-natural transformation $\Phi: f^*\Rightarrow g^*$ between induced dg-functors. Moreover the $0$th component of $\Phi$ is an objectwise weak equivalence. If we restrict ourselves to the full dg-subcategory of twisted perfect complexes, then we prove that $\Phi$ admits an $A_{\infty}$-quasi-inverse when $(\mathcal{U},\mathcal{R})$ satisfies some additional conditions.