A Computational Approach to the Homotopy Theory of DG categories

TL;DR

This paper develops explicit, computable functors for homotopy theory in dg categories, enabling practical calculations and applications in symplectic topology, by constructing a cylinder functor and homotopy colimit functor for semifree dg categories.

Contribution

It introduces explicit, computable cylinder and homotopy colimit functors for semifree dg categories, establishing a cofibration category structure independently of prior work.

Findings

Constructed a cylinder functor for semifree dg categories.

Defined a homotopy colimit functor with finiteness properties.

Applied the framework to symplectic topology and provided a toy example.

Abstract

We give a specific cylinder functor for semifree dg categories. This allows us to construct a homotopy colimit functor explicitly. These two functors are "computable", specifically, the constructed cylinder functor sends a dg category of finite type, i.e., a semifree dg category having finitely many generating morphisms, to a dg category of finite type. The homotopy colimit functor has a similar property. Moreover, using the cylinder functor, we give a cofibration category of semifree dg categories and that of dg categories of finite type, independently from the work of Tabuada. All the results similarly work for semifree dg algebras. We also describe an application to symplectic topology and provide a toy example.