Smooth singular complexes and diffeological principal bundles
Let \(\mathcal{D}\) denote the category of diffeological spaces. The author [\textit{H. Kihara}, J. Homotopy Relat. Struct. 14, No. 1, 51--90 (2019; Zbl 1459.58001)] constructed diffeologies on\N\[\N\Delta^{p}=\left\{ (x_{0},\ldots,x_{p})\in\mathbb{R}^{p++1}\mid\Sigma x_{i}=1,x_{i}\geq0\text{ for any }i\right\} \,(p\geq0)\N\]\Nto define the singular complex \(\mathcal{S}^{\mathcal{D}}(X)\) of a diffeological space \(X\). \N\textit{G. Hector} [in: Analysis and geometry in foliated manifolds. Proceedings of the VII international colloquium on differential geometry, Santiago de Compostela, Spain, July 26--30, 1994. Singapore: World Scientific. 55--80 (1995; Zbl 0993.58500)] used the sets \(\Delta^{p}\) rigged with the subdiffeology of \(\mathbb{R}^{p++1}\) \((p\geq0)\) to define the singular complex \(\mathcal{S}_{\mathrm{sub}}^{\mathcal{D}}(X)\) of a diffeological space \(X\). \N\textit{J. D. Christensen} and \textit{E. Wu} [New York J. Math. 20, 1269--1303 (2014; Zbl 1325.57013)] also used the affine spaces\N\[\N\mathbb{A}^{p}=\left\{ (x_{0},\ldots,x_{p})\in\mathbb{R}^{p++1}\mid\Sigma x_{i}=1\right\}\N\]\Nrigged with the subdiffeology of \(\mathbb{R}^{p++1}\) \((p\geq0)\) to define the singular complex \(\mathcal{S}_{\mathrm{aff}}^{\mathcal{D}}(X)\) to construct a model structure on \(\mathcal{D}\).\N\NThe main result of this paper is the following theorem.\N\NTheorem. The natural morphisms of simplicial sets\N\[\N\mathcal{S}_{\mathrm{aff}}^{\mathcal{D}}(X)\rightarrow \mathcal{S}_{\mathrm{sub}}^{\mathcal{D}}(X)\rightarrow \mathcal{S}^{\mathcal{D}}(X)\N\]\Nare weak equivalences.\N\NThe author characterizes diffeological principal bundles,\ extending the characteristic classes for \(\mathcal{D}\)-numerable principal bundles to those for diffeological principal bundles.
scientific article; zbMATH DE number 7904846
scientific article