Local systems in diffeology

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 Gómez-Tato, Halperin and Tanré. 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.

arXiv comment44 pages

References in corpus (8)

• The homotopy theory of diffeological spaces
• Homotopy theory of complete Lie algebras and Lie models of simplicial sets
• The $\mathbb{R}$-Local Homotopy Theory of Smooth Spaces
• Differential forms on diffeological spaces and diffeological gluing, I
• Rational homotopy equivalences and singular chains
• On the category of stratifolds
• Whitney Approximation for Smooth CW Complex
• Smooth Homotopy of Infinite-Dimensional $C^{\infty }$-Manifolds