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One aim of the Froelicher spaces page is to translate some topological properties into properties of Froelicher spaces. What I want is to define certain intrinsic properties of Froelicher spaces by analogy with some property of topological spaces. Although there are (at least) two functors from Froelicher spaces to topological spaces, I’m not interested in saying “A Froelicher space has property X if its underlying topological space does.” but rather in finding some close analogy with property X which means that I can define it intrinsically, and then try to prove some theorem along the lines of “If a Froelicher space has property fr-X then its underlying topological space has property top-X.”.
Often the Froelicher space stuff comes in two flavours: functional and curvaceous, depending on whether the functions or the curves are primary in its definition. Sometimes these coincide, sometimes not.
For manifolds of mapping spaces, I need some notion of compactness. What I really want is to be able to say “$X$ is dum-de-dum compact so in $\mathbb{R} \times X$, neighbourhoods of $\{0\} \times X$ look like $(-\epsilon, \epsilon) \times X$.”.
For topological spaces, this condition is equivalent to sequential compactness. It’s not so obvious how to translate sequential definitions into ones with a path-flavour so that they go over to Froelicher spaces, but I decided to introduce the notion of a “smoothly convergent sequence”. The idea is that a sequence $(x_n)$ smooth converges to $x$ (in a Froelicher space) if there is a smooth curve $c \colon \mathbb{R} \to X$ with $c(0) = x$ and $c$ interpolates the sequence $(x_n)$ as it approaches $0$ (in a “sufficiently nice” topological space then one can do something similar with sequences and continuous paths). This is not quite precise, and that’s the point of this post. Clearly, as I’m inventing this as I’m going along, I could make this definition be what I like so the temptation is strong to make it so that I have the same equivalence: $X$ is curvaceously compact if and only if neighbourhoods of $\{0\} \times X$ in $\mathbb{R} \times X$ look like $(-\epsilon,\epsilon) \times X$ (incidentally, this is not a satisfactory definition of curvaceously compact since the RHS is purely topological).
For that, I need to make the definition of “smoothly convergent sequence” as follows:
Let $(t_n) \to 0$ in $\mathbb{R}$. A sequence $(x_n)$ converges smoothly to $x$ with respect to $(t_n)$ if there is a smooth curve $c \colon \mathbb{R} \to X$, a sequence $(r_n) \to 0$ in $\mathbb{R}$, and a smooth function $s \colon \mathbb{R} \to \mathbb{R}$ such that $c(r_n) = x_n$, $c(0) = x$, and $s(r_n) = t_n$.
The point of this is that in the topological case, when one fits a path to a sequence then the choice of “control points” (i.e. the points where the curve goes through the sequence) is not all that important - one can reparametrise just about any choice to any other choice (subject to monotonicity). In the smooth case, that’s not true: not all control points are equivalent, so it seems to make sense that you should specify the control points. But then one could set up a poset (ish) of families of control points, with $(t_n) \preceq (s_n)$ if there is a smooth function taking one set to the other. That’s sort of what the above definition tries to encode.
This is probably getting a bit complicated, so let me try to summarise. I have a choice. I want to define two things (“smoothly convergent sequence” and “curvaceous compactness”) and the second definition depends on the first. To be useful, “curvaceous compactness” needs to imply that product property which imposes a certain requirement on the definition. That means that I don’t have complete freedom in my definitions, and they need to be a little more complicated that the topological versions. But on to which definition should I load the complication? Is it best to keep the elementary concept (smoothly convergent sequence) and put the weight on the derived ones, or the other way around?
Obviously, I can change things later if I wish, but it’d be nice to have some advice before setting out!
(nLab pages: Froelicher spaces and manifolds of mapping spaces)
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