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Is there a definition of a "$C^k$-analytic object with singularities" for $k > 0$ as a locally ringed space? I know orbifolds, but in algebraic geometry, one can have more general schemes/varieties.

I don't see how to define e.g. a $C^k$-function for $k > 0$ on $X := V(xy) \subset \mathbf{R}^2$ to $\mathbf{R}$.

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