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    • CommentRowNumber1.
    • CommentAuthorEric
    • CommentTimeMay 5th 2010
    • (edited May 5th 2010)

    Lately I’ve been having fun learning about smooth spaces.

    Conceptually, I think of it kind of like this…

    Given a set XX, a diffeology D(X)D(X) on XX tells you which maps into XX are smooth (or not smooth).

    This is borrowed from the way I used to explain what a topology was to my colleagues (who were top researchers in their field of study but never had the need to know the formal definition of a topology):

    Given a set XX, a topology T(X)T(X) on XX tells you which subsets of XX are open (or closed).

    The two statements above are “similar”, but not quite perfect analogies (as far as I can tell).

    Is it possible to define a topology on a space in terms of maps into that space? For example,

    Given a set XX, a topology 𝒯(X)\mathcal{T}(X) on XX tells you which maps into XX are continuous (or not continuous).

    For example, if f𝒯(X)f\in\mathcal{T}(X) and g:XYg:X\to Y, we’d say gg is continuous if gf𝒯(Y)g\circ f\in\mathcal{T}(Y).

    Follow up question…

    Just as a topology can be defined in terms of open OR closed sets. Could you define a smooth space in terms of maps into XX that are NOT smooth? Kind of like the complementary definition of topology in terms of closed sets?

    PS: It seems like you can play this game all night long. For example, given a set XX, a purpleology P(X)P(X) tells you which maps into XX are purple. If fP(X)f\in P(X) and g:XYg:X\to Y, we say gg is purple if gfP(Y)g\circ f\in P(Y) :)

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 5th 2010
    • (edited May 5th 2010)

    Could you define a smooth space in terms of maps into X that are NOT smooth?

    There is one rule in this game here:

    your assignment of sets of allowed maps to a given test space must also respect the morphisms between the test spaces:

    if UU and VV are two test spaces and XX is your diffeological space or some other sheaf, and X(U)X(U) is the set of allowed ways of mapping UU into XX and X(V)X(V) correspondingly the set of ways of mapping VV into XX, then for every homomorphism of test spaces ϕ:UV\phi : U \to V the map that sends X(V)X(V) to X(U)X(U) by precomposing plots with ϕ\phi must

    1. exist (! :-)

    2. respect composition.

    Your example of non-smooth maps will violate 1. At least if I understand correctly what you mean:

    you could consider some manifold XX and declare that X(U)X(U) is the set of maps of set UXU \to X that are not smooth with respect to the smooth structure on UU and XX.

    But consider the non-smooth function H:H : \mathbb{R} \to \mathbb{R} that is constant 0 on the negative ray, constant 1 on the positive ray, and jumps at 0.

    Then let ϕ:\phi : \mathbb{R} \to \mathbb{R} be the map exp()+1\exp(-)+1. Then the composite ϕHX\mathbb{R} \stackrel{\phi}{\to} \mathbb{R} \stackrel{H}{\to} X is smooth! But for your definition you would need that the composite of every non-smooth function with a morphism of test domains is still non-smooth.

    See what I mean?

    • CommentRowNumber3.
    • CommentAuthorAndrew Stacey
    • CommentTimeMay 5th 2010

    But what you could do is to take some set of maps that you want not to be smooth and then consider the largest diffeology (or whatever) such that those maps are not smooth. It’s possible that this will be the discrete diffeology (only the constant maps smooth), and there are one or two conditions that have to be satisfied to be sure of getting anything at all (you could never have a constant map not-smooth, for example).

    On topology, you can certainly define a topology on a space by declaring a family of maps to be continuous. For example, that’s how you put a topology on a diffeological space (or Chen space or Frolicher space). Indeed, if you look at the topological sub-page of Froelicher spaces then you’ll see that I put two topologies on a Froelicher space: the curvaceous topology (strongest topology such that all smooth curves are continuous) and the functional topology (weakest topology such that all smooth functions are continuous).

    This kind of construction is also used a lot in functional analysis: the weak and weak* topologies are defined by declaring certain families of functionals continuous. It’s also used in algebraic topology: the compactly-generated topology on a space is defined in this way.