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.
The synopsis of the paper goes as follows.

§ 2

recalls the definition of a diffeological space, summarizing the equivalence between the category \(\mathsf{Set}^{\Delta^{\mathrm{op}}}\)of simplicial sets and [H. Kihara, Smooth homotopy of infinite-dimensional \(C^{\infty}\)-manifolds. Providence, RI: American Mathematical Society (AMS) (2023; Zbl 1542.58001)].

§ 3

recalls crucial results on local systems [A. Gómez-Tato et al., Trans. Am. Math. Soc. 352, No. 4, 1493–1525 (2000; Zbl 0939.55010)], establishing the following theorem.
Theorem. 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 \[ \mathsf{Ho}\left( \mathcal{M}_{\mathbb{Q}}\right) \overset{\simeq }{\longrightarrow}\mathsf{fib}\mathbb{Q}\text{-}\mathsf{Ho}\left( \mathsf{Diff}_{\ast}\right) \] where \(\mathsf{Ho}\left( \mathcal{M}_{\mathbb{Q}}\right) \)is the homotopy category of minimal local systems.

§ 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 \(\left\vert {}\right\vert _{D}\).

§ 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.

Appendix A

establishes that the functor \(k\)from the category Stfdof stratifolds [M. Kreck, Differential algebraic topology. From stratifolds to exotic spheres. Providence, RI: American Mathematical Society (AMS) (2010; Zbl 1420.57002)] to \(\mathsf{Diff}\)[T. Aoki and 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.

Appendix B

constructs a commutative algebraic model for the unreduced suspension of a connected closed manifold, which was used in §5.