A new model for dg-categories

TL;DR

This paper introduces a novel model for dg-categories using dg-Segal spaces, establishing an equivalence between their homotopy categories through new hypercover constructions.

Contribution

It defines dg-Segal spaces and complete dg-Segal spaces, and proves an equivalence with classic dg-categories under a new model structure.

Findings

Established an equivalence between dg-categories and dg-Segal spaces.

Introduced hypercover constructions for dg-categories.

Defined complete dg-Segal spaces and related them to classical Segal spaces.

Abstract

In this document, we develop a new model for the category of dg-categories. Following Rezk's example in the case of classic Segal spaces, we define dg-Segal spaces: functors between free dg-categories of finite type and simplicial spaces to which we add certain properties. We define also complete dg-Segal spaces, and make their relationship to classic Segal spaces explicit. With the help of two new hypercover constructions, and up to a certain hypothesis, we prove that there exists an equivalence between the homotopy category of dg-categories and the homotopy category of functors defined above with a model structure making the complete dg-Segal spaces into its fibrant objects.