Is there a notion of smoothness for maps of real closed spaces in the sense of Schwartz? [1]
Ideally it would have the following properties:
- Every smooth map of real closed spaces is locally of the form $(0,1)^n\times U\to U$;
- For every smooth map of schemes $S\to T$, the induced map of real closed spaces $S_r\to T_r$ is smooth.
(note that all maps $S_r\to T_r$ induced by a smooth map satisfy the first condition, since étale maps induce local homeomorphisms and every smooth map is étale locally a projection)
[1] Schwartz, Niels, Real closed spaces, Rocky Mt. J. Math. 14, 971-972 (1984). ZBL0576.14023.
