Rational homotopy theory and differential graded category

TL;DR

This paper extends Sullivan's rational homotopy theory to non-simply connected spaces using differential graded categories, establishing a model category structure and an equivalence for spaces with finite fundamental groups.

Contribution

It introduces a new framework generalizing Sullivan's theory to non-simply connected spaces via closed tensor dg-categories and proves an equivalence with certain homotopy categories.

Findings

Established a model category structure on small closed tensor dg-categories.

Proved an equivalence between homotopy categories of certain spaces and dg-categories.

Extended rational homotopy theory to include non-simply connected spaces.

Abstract

We propose a generalization of Sullivan's de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan's theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions.