Diffeological spaces and Sikorski differential spaces are each a generalisation of a smooth manifold. In their definitions, both have locality and smooth compatibility conditions. Any diffeological space is equipped with a natural topology with respect to all of its plots such that the plots are continuous, whilst any Sikorski differential space is equipped with the topology induced on a set by the family of functions, that is the weakest topology such that all functions from the family are continuous. My question is what are the similarities and differences of the two spaces.
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2$\begingroup$ tac.mta.ca/tac/volumes/25/4/25-04abs.html We compare various different definitions of "the category of smooth objects". The definitions compared are due to Chen, Frolicher, Sikorski, Smith, and Souriau. The method of comparison is to construct functors between the categories that enable us to see how the categories relate to each other. $\endgroup$– David Roberts ♦Commented Feb 12, 2019 at 11:37
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$\begingroup$ Thanks so much David $\endgroup$– JoyCommented Feb 14, 2019 at 20:24
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$\begingroup$ See also "J. Watts; Diffeologies, differential spaces, and symplectic geometry" available at arXiv:1208.3634 for a comparison. $\endgroup$– ARACommented Jul 26, 2019 at 17:10
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Diffeological spaces are certain kinds of sheaves of sets on the site of open subsets of various Euclidean (a.k.a. Cartesian) spaces.
Differential spaces are certain kinds of local $C^\infty$-ringed spaces (the sections of their structure sheaves are real-valued functions).
