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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeJul 2nd 2020

Added a reference to diffeological spaces.

• CommentRowNumber2.
• CommentAuthorDmitri Pavlov
• CommentTimeJul 2nd 2020

The 1977 definition of Chen spaces appears to be equivalent to diffeological spaces, or am I missing something?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJul 2nd 2020

You are saying this as if it is in contradiction to something?

I don’t feel responsible for the entry “Chen space”, but over in the references of diffeological space it does attribute the definition to Chen, rediscovered, apparently independently, by Souriau.

• CommentRowNumber4.
• CommentAuthorDmitri Pavlov
• CommentTimeJul 2nd 2020
• (edited Jul 2nd 2020)

Baez and Hoffnung at the bottom of page 8 of their paper https://arxiv.org/abs/0807.1704v4 write the following:

Stacey has given a more general comparison of Chen spaces versus diffeological spaces [40]. To briefly summarize this, let us write ChenSpace for the category of Chen spaces, and DiffeologicalSpace for the category of diffeological spaces. Stacey has shown that these categories are not equivalent. However, he has constructed some useful functors relating them. These take advantage of the fact that every open subset of R n becomes a Chen space with its subspace smooth structure, and conversely, every convex subset of R n becomes a diffeological space.

And at the bottom of page 9 they write:

With a little work, it follows that both Ch ♯ and Ch ♭ embed DiffeologicalSpace isomorphically as a full subcategory of ChenSpace: a ‘reflective’ subcategory in the first case, and a ‘coreflective’ one in the second. The embedding Ch ♭ is a bit strange: as shown by Stacey, even the ordinary closed interval fails to lie in its image!

What am I missing?

Andrew Stacey writes at the bottom of page 32 in https://arxiv.org/abs/0802.2225:

Chen spaces. Let us now compare Chen spaces and Souriau spaces. We can show that they are different by exhibiting a Souriau space whose images under the two extension functors are different.

This Souriau space is [0, 1] with its usual diffeology. The Chen space Ch ♯ ([0, 1]) is easily seen to be the standard Chen space structure on [0, 1]. In particular, it contains the identity on [0, 1]. On the other hand, every plot in Ch ♭ ([0, 1]) factors through a C ∞ -map U → [0, 1] from an open subset of some Euclidean space into [0, 1]. The identity on [0, 1] does not have this property: any factorisation of the inclusion [0, 1] → R via some open set U must extend outside [0, 1]. Hence Ch ♯ ([0, 1]) , Ch ♭ ([0, 1]).

One can extend this example to see that the main difference between Chen spaces and Souriau spaces is the ability to “approach boundary points at speed”. In the Chen realm, one can approach a point at speed and stop. In the Souriau realm, one must always be able to go a little further. Now suppose that one wishes to declare that a certain point cannot be approached from certain directions. One therefore wishes to consider one-sided derivatives at those points. In the Chen realm, this presents no difficulties: one approaches at a steady speed along those directions that one is allowed to approach along. In the Souriau realm, this is more problematical. One has to approach along an allowed direction and then “bounce back”, so ones speed has to approach zero as one nears the point of interest.

Looking at the original paper by Chen https://doi.org/10.1090/s0002-9904-1977-14320-6, Definition 1.2.1 basically talks about sheaves of sets on the site of convex subsets of R^n.

However, it does not require these subsets to be open or closed, which means that we get something quite different from Souriau’s diffeological spaces.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJul 3rd 2020

Oh, I see. Thanks for highlighting. So I have adjusted the discussion of the references over at diffeological space, accordingly (here).