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Local systems in diffeology

Katsuhiko Kuribayashi

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.

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.

Paper Structure

This paper contains 18 sections, 32 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. An overview of rational homotopy theory for diffeological spaces
  3. A spectral sequence for a diffeological adjunction space
  4. Remarks and future perspectives
  5. The organization of the paper
  6. The category $\mathsf{Diff}$ of diffeological spaces and de Rham complexes
  7. Diffeological spaces
  8. A model structure on $\mathsf{Diff}$
  9. De Rham complexes
  10. A candidate for a localized diffeological space
  11. Local systems
  12. The fibrewise localization of a diffeological space
  13. Examples of localizations associated with local systems
  14. A local system for an adjunction space in $\mathsf{Diff}$ and $\mathsf{Top}$
  15. An application of Theorem \ref{['thm:SS']} to a stratifold
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\text{\em fN${\mathbb Q}$-Ho}(\mathsf{Diff})$ be the full subcategory of the homotopy category $\text{\em Ho}(\mathsf{Diff})$ consisting of connected nilpotent rational diffeological spaces which correspond to connected nilpotent rational Kan complexes of finite ${\mathbb Q}$-type via the singu

Theorems & Definitions (85)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7
  • Proposition 2.8
  • ...and 75 more