Local systems in diffeology (Q6612007)

\textit{H. Kihara} [J. Homotopy Relat. Struct. 14, No. 1, 51--90 (2019; Zbl 1459.58001); Smooth homotopy of infinite-dimensional \(C^{\infty}\)-manifolds. Providence, RI: American Mathematical Society (AMS) (2023; Zbl 1542.58001)] has succeeded in introducing a Quillen model structure on the category \(\mathsf{Diff}\) of diffeological spaces. This paper, by relating the local systems over \(K\left( \pi,1\right) \)-spaces to the model structure of \(\mathsf{Diff}\), proposes a framework of rational homotopy theory for diffeological spaces.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] recalls the definition of a diffeological space, summarizing the equivalence between the category \(\mathsf{Set}^{\Delta^{\mathrm{op}}}\)\ of simplicial sets and [\textit{H. Kihara}, Smooth homotopy of infinite-dimensional \(C^{\infty}\)-manifolds. Providence, RI: American Mathematical Society (AMS) (2023; Zbl 1542.58001)].\N\N\item[\S 3] recalls crucial results on \textit{local systems} [\textit{A. Gómez-Tato} et al., Trans. Am. Math. Soc. 352, No. 4, 1493--1525 (2000; Zbl 0939.55010)], establishing the following theorem.\N\NTheorem. Let \(\mathsf{Ho}\left( \mathsf{Diff}_{\ast}\right) \)\ be the homotopy category of pointed diffeological spaces with its full subcategory \(\mathsf{fib}\mathbb{Q}\)-\(\mathsf{Ho}\left( \mathsf{Diff}_{\ast}\right) \)\ of fiberwise rational connected diffeological spaces of finite type. Then there exists an equivalence of categories\N\[\N\mathsf{Ho}\left( \mathcal{M}_{\mathbb{Q}}\right) \overset{\simeq }{\longrightarrow}\mathsf{fib}\mathbb{Q}\text{-}\mathsf{Ho}\left( \mathsf{Diff}_{\ast}\right)\N\]\Nwhere \(\mathsf{Ho}\left( \mathcal{M}_{\mathbb{Q}}\right) \)\ is the homotopy category of minimal local systems.\N\N\item[\S 4] gives examples of local systems associated with relative Sullivan algebras, seeing that a relative Sullivan model for a fibration in \(\mathsf{Set}^{\Delta^{\mathrm{op}}}\)\ gives rise to the fiberwise localization of a diffeological space via the realization functor \(\left\vert {}\right\vert _{D}\).\N\N\item[\S 5] constructs a spectral sequence converging to the singular de Rham cohomology of a diffeological adjunction space 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.\N\N\item[Appendix A] establishes that the functor \(k\)\ from the category \textsf{Stfd}\ of stratifolds [\textit{M. Kreck}, Differential algebraic topology. From stratifolds to exotic spheres. Providence, RI: American Mathematical Society (AMS) (2010; Zbl 1420.57002)] to \(\mathsf{Diff}\)\ [\textit{T. Aoki} and \textit{K. Kuribayashi}, Cah. Topol. Géom. Différ. Catég. 58, No. 2, 131--160 (2017; Zbl 1378.18016)] assigns an adjunction space in \(\mathsf{Diff}\)\ to a \(p\)-stratifold up to smooth homotopy.\N\N\item[Appendix B] constructs a commutative algebraic model for the unreduced suspension of a connected closed manifold, which was used in \S 5.\N\end{itemize}