# Twisted complexes and simplicial homotopies

## TL;DR

This paper studies the dg-category of twisted complexes over simplicial ringed spaces and demonstrates that simplicial homotopic maps induce $A_{
fty}$-natural transformations between their associated dg-functors, with implications for perfect complexes.

## Contribution

It proves the existence of $A_{
abla}$-natural transformations between dg-functors induced by simplicial homotopic maps and establishes conditions for their quasi-inverses in the context of twisted perfect complexes.

## Key findings

- Existence of $A_{
abla}$-natural transformations for homotopic maps.
- Objectwise weak equivalence at the zeroth component of the transformation.
- Conditions under which the transformations admit $A_{
abla}$-quasi-inverses.

## 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.