Twisted complexes and simplicial homotopies

Abstract

We consider the dg-category of twisted complexes over simplicial ringed spaces. It is clear that a simplicial map $f:({\mathscr {U}},{\mathscr {R}})\rightarrow ({\mathscr {V}}, {\mathscr {S}})$ between simplicial ringed spaces induces a dg-functor $f^*:\mathrm{Tw}({\mathscr {V}}, {\mathscr {S}})\rightarrow \mathrm{Tw}({\mathscr {U}}, {\mathscr {R}})$ where $\mathrm{Tw}({\mathscr {U}}, {\mathscr {R}})$ denotes the dg-category of twisted complexes on $({\mathscr {U}},{\mathscr {R}})$. 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 0th 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 $({\mathscr {U}},{\mathscr {R}})$ satisfies some additional conditions.