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Aronszajn and Sikorski subcartesian differential spaces. (English) Zbl 1487.58003

In this paper, it is shown that there is a natural transformation between two categories of subcartesian differential space and Arzojan subcartesian space.
A subcartesian space, \(S\), is a Hausdorf differential space whose every point \(x\in S\) has a neighbourhood \(U_x\) diffeomorphic to a differential subspace \(V_x\) of \(\mathbb{R}^{n_x}\).
A subcartesian space of Aronszajn space is a Hausdorf space with an atlas \(\mathfrak{A}=\{\varphi:U_\varphi\rightarrow V_\varphi\}\), where \(\varphi:U_\varphi\rightarrow V_\varphi\) is a homeomorphism between \(U_\varphi\) and and open subset \(V_\varphi\subset \mathbb{R}^{n_\varphi}\), such that
1. The domains \(\{U_\varphi|\varphi\in\mathfrak{A}\}\) forms an open cover for \(S\).
2. For every \(\varphi,\psi\in\mathfrak{A}\) and every \(x\in U_\varphi\cap V_\psi\) there exists a \(C^\infty\)-mapping \(s\) extending \(\psi\circ\varphi^{-1}: \varphi(U_\varphi \cap U_\psi)\rightarrow\psi(U_\varphi\cap U_\psi)\) in a neighbourhood of \(\varphi(x)\in\mathbb{R}^{n_\varphi}\). Also, there exists a \(C^\infty\)-mapping \(t\) extending \(\varphi\circ\psi^{-1} :\psi(U_\varphi\cap U_\psi)\rightarrow \varphi(U_\varphi \cap U_\psi)\) in a neighbourhood of \(\psi(x)\in\mathbb{R}^{n_\psi}\) [see the paper].
The definition of a smooth morphism of Arzojan subcartesian spaces is given as follows. Let \((S_1,\mathfrak{A}_1)\) and \((S_2,\mathfrak{U}_2)\) be two subcartesian of Arzojan spaces. A map \(\chi:S_1\rightarrow S_2\) is smooth, if for every \(x\in S_1\), there exist \(\varphi_1\in\mathfrak{U}_1\) and \(\varphi_2\in\mathfrak{U}_2\) such that \(x\in U_{\varphi_1}\), \(\chi(x)\in U_{\varphi_2}\) and \(\varphi_2\circ \chi\circ\varphi^{-1}_1:V_{\varphi_1}\rightarrow V_{\varphi_2}\) extends to a \(C^\infty\)-map of a neighbourhood of \(\varphi_1(x)\in\mathbb{R}^{n_{\varphi_1}}\) to a neighbourhood of \(\varphi_2(\chi(x))\in\mathbb{R}^{n_{\varphi_2}}\) [see the paper].
In this case, a natural transformation is constructed between the category of subcartesian spaces with a differential structure and the category of subcartesian spaces of Arzojan.

MSC:

58A40 Differential spaces
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References:

[1] N. Aronszajn, Subcartesian and subRiemannian spaces, Notices Amer. Math. Soc. 14 (1967), 111.
[2] R. Sikorski, Abstract covariant derivative, Colloq. Math. 18 (1967), 252-272. · Zbl 0162.25101
[3] R. Sikorski, Introduction to Differential Geometry, Państwowe Wydawnictwo Naukowe, Warszawa, 1972 (in Polish). · Zbl 0255.53001
[4] J. Śniatycki, Differential Geometry of Singular Spaces and Reduction of Symmetry, Cambridge Univ. Press, Cambridge, 2013. · Zbl 1298.58002
[5] P. Walczak, A theorem on diffeomorphisms in the category of differential spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 325-329. · Zbl 0258.58001
[6] Richard Cushman, Jędrzej Śniatycki Department of Mathematics and Statistics University of Calgary Calgary, Alberta T2N 1N4, Canada E-mail: r.h.cushman@gmail.com sniatycki@gmail.com
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