The de Rham homotopy theory and differential graded category

TL;DR

This paper extends de Rham homotopy theory to non-nilpotent spaces using closed dg-categories and equivariant dg-algebras, establishing categorical equivalences and describing homotopy invariants.

Contribution

It introduces a framework connecting homotopy types with closed dg-categories, generalizing previous models and providing new tools for analyzing homotopy invariants.

Findings

Equivalence between schematic homotopy types and closed dg-categories

Description of homotopy invariants via minimal models

Examples illustrating the theory

Abstract

This paper is a generalization of arXiv:0810.0808. We develop the de Rham homotopy theory of not necessarily nilpotent spaces, using closed dg-categories and equivariant dg-algebras. We see these two algebraic objects correspond in a certain way. We prove an equivalence between the homotopy category of schematic homotopy types and a homotopy category of closed dg-categories. We give a description of homotopy invariants of spaces in terms of minimal models. The minimal model in this context behaves much like the Sullivan's minimal model. We also provide some examples. We prove an equivalence between fiberwise rationalizations and closed dg-categories with subsidiary data.