Local systems in diffeology

TL;DR

This paper develops a rational homotopy theory framework for diffeological spaces using local systems, establishing an equivalence with algebraic categories and constructing spectral sequences for cohomology computations.

Contribution

It introduces a new rational homotopy theory framework for diffeological spaces via local systems and relates it to algebraic models, extending cohomology tools.

Findings

Established an equivalence between homotopy categories of fiberwise rational diffeological spaces and algebraic local systems.

Constructed a spectral sequence converging to the singular de Rham cohomology of diffeological spaces.

Applied the spectral sequence to stratifolds and topological homotopy pushouts.

Abstract

By making use of Halperin's local systems over simplicial sets and the model structure of the category of diffeological spaces due to Kihara, we introduce a framework of rational homotopy theory for such smooth spaces with arbitrary fundamental groups. As a consequence, we have an equivalence between the homotopy categories of fibrewise rational diffeological spaces and an algebraic category of minimal local systems elaborated by G\'omez-Tato, Halperin and Tanr\'e. In the latter half of this article, a spectral sequence converging to the singular de Rham cohomology of a diffeological adjunction space is constructed with the pullback of relevant local systems. In case of a stratifold obtained by attaching manifolds, the spectral sequence converges to the Souriau--de Rham cohomology algebra of the diffeological space. By using the pullback construction, we also discuss a local system model for a topological homotopy pushout.