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nLab > Latest Changes: Kriegl and Michor's cartesian closed category of manifolds

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    • CommentRowNumber1.
    • CommentAuthorAndrew Stacey
    • CommentTimeSep 27th 2012

    Back in the early ’80s, Kriegl and Michor came up with a variant on the notion of “smooth manifold” that produced a cartesian closed category. Their remarks on this in A Convenient Setting … are interesting reading for putting this in context, but nonetheless I’ve been meaning to take a look at their definition for a while to see what the bones of the proposal are.

    I’ve put up a basic page with just the definition at Kriegl and Michor’s cartesian closed category of manifolds. There’s more detail at A convenient setting for differential geometry and global analysis (lspace). I wasn’t sure how to split the pages; at the moment there’s not enough detail on the nlab page but I think that the nlab page shouldn’t have details on the actual paper.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 28th 2012
    • (edited Sep 28th 2012)

    Back in the early ’80s, Kriegl and Michor came up with a variant on the notion of “smooth manifold” that produced a cartesian closed category.

    Does that category sit faithfully in diffeological spaces, do you know? That would be an important property to mention, if true.

    I’ve put up a basic page with just the definition at Kriegl and Michor’s cartesian closed category of manifolds.

    Currently this seems to be an orphaned page. (?) I’d suggest to link such pages to others from which one might expect a reader to want to arrive at them. What do you think? Probably smooth manifold and also maybe diffeological space or the like should point to this page with some helpful comment on why the reader might want to follow that link.

    • CommentRowNumber3.
    • CommentAuthorAndrew Stacey
    • CommentTimeSep 28th 2012

    I’ve added a little more to the page, in particularly drawing out the underlying category of smooth spaces, and linked it from generalized smooth space.

    There isn’t much of a reason why someone would want to look at this - even the authors don’t revisit it in A Convenient Setting! But I got curious about other definitions of “manifold” and thought I should record what I found. There are some interesting ideas there, even if the whole structure is not so, namely:

    1. What is a tangent space? Part of the definition here is of a space TMT M which plays the role of a tangent space over MM, but it is part of the given data and not constructed organically from MM, so it got me thinking: if I gave you a smooth space XX and another space YY and claimed that YY was the “tangent space” of XX, what would you expect that to mean?

    2. Transportable structures. Since they don’t start in a category of generalised smooth spaces, they have to have some extra structure to build smooth spaces (pre manifolds) out of some more basic data. They use parallel transport to do that, so one can imagine replacing local triviality by transportable in lots of contexts and seeing what happened.

    • CommentRowNumber4.
    • CommentAuthorpmichor
    • CommentTimeAug 23rd 2021
    If a manifold in this sense has Banach tangent spaces, then it is a traditional Banach manifold. This is not stated in the papers, but the the proof for finite dimensional tangent spaces carries over by using the inverse function theorem on Banach spaces instead of the finite dimensional one.
    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 24th 2021

    Changed references to Michor’s 1984 paper from cryptic MR numbers. Also minor cleanup after what I guess was Peter M’s addition.

    diff, v5, current

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