Homotopy Colimits of DG Categories and Fukaya Categories

TL;DR

This paper introduces a new method for computing homotopy colimits of dg categories, enabling practical calculations of wrapped Fukaya categories for Weinstein manifolds, with applications to cotangent bundles of lens spaces.

Contribution

It develops a cylinder object for semifree dg categories and provides a practical formula for homotopy colimits, advancing computations in Fukaya categories.

Findings

Computed wrapped Fukaya categories of cotangent bundles of lens spaces.

Showed the endomorphism algebra of the cotangent fibre as a homotopy invariant.

Demonstrated applications to cotangent bundles and plumbing spaces.

Abstract

We construct a new cylinder object for semifree differential graded (dg) categories in the category of dg categories. Using this, we give a practical formula computing homotopy colimits of semifree dg categories. Combining it with the result of Ganatra, Pardon, and Shende, we get a formula computing wrapped Fukaya categories of Weinstein manifolds using their sectorial coverings. This formula has lots of applications including a practical computation of the wrapped Fukaya category of any cotangent bundle or plumbing space. In this paper, we compute wrapped Fukaya categories of cotangent bundles of lens spaces using their Heegaard decomposition. From the computation, we show that the endomorphism algebra of the cotangent fibre is a full invariant of the homotopy type of lens spaces.