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\section*{examples of Frölicher spaces}

\hypertarget{examples_of_frlicher_spaces}{}\section*{{Examples of [[Frölicher Spaces]]}}\label{examples_of_frlicher_spaces}

\noindent\hyperlink{manifolds_as_frlicher_spaces}{Manifolds as Fr\"o{}licher Spaces}\dotfill \pageref*{manifolds_as_frlicher_spaces} \linebreak
\noindent\hyperlink{the_pinched_plane}{The Pinched Plane}\dotfill \pageref*{the_pinched_plane} \linebreak
\noindent\hyperlink{coequaliser_example}{Coequaliser Example}\dotfill \pageref*{coequaliser_example} \linebreak


\hypertarget{manifolds_as_frlicher_spaces}{}\subsection*{{Manifolds as Fr\"o{}licher Spaces}}\label{manifolds_as_frlicher_spaces}

Any [[smooth manifold]] defines a Fr\"o{}licher space with curves $C^\infty(\mathbb{R}, M)$ and functions $C^\infty(M, \mathbb{R})$.

\hypertarget{the_pinched_plane}{}\subsection*{{The Pinched Plane}}\label{the_pinched_plane}

Taking quotients in the category of Fr\"o{}licher spaces is straightforward: the smooth functions are those that pull-back to smooth functions on the original space.

As an example, consider the plane $\mathbb{R}^2$ quotiented out by the $x$-axis.

Let us write this as $X$. This example is closely related to taking cones and suspensions in algebraic topology. The smooth functions on $X$ are simple to describe: the set is equivalent to those smooth functions $\mathbb{R}^2 \to \mathbb{R}$ which are constant on the $x$-axis.

Now let us consider the smooth curves. Let $\alpha : \mathbb{R} \to X$ be a smooth curve. We can partition $\mathbb{R}$ into two pieces: those points that are mapped to the squashed point in $X$ and those points that aren't. Let us write $A$ for the set of points that are \textbf{not} mapped to the squashed point. Using bump functions it is easy to show that $A$ is open in $\mathbb{R}$. As the quotient map $\mathbb{R}^2 \to X$ is bijective off the $x$-axis, the restriction of $\alpha$ to $A$ has a unique lift to $\mathbb{R}^2$. Let us write $\alpha_x$ and $\alpha_y$ for the coordinate functions of this lift. Again using bump functions, it is easy to show that $\alpha_x$ and $\alpha_y$ are smooth on $A$. Furthermore, as the projection $(x,y) \mapsto y$ descends to a smooth function on $X$, $\alpha_y$ is actually the restriction to $A$ of a smooth function $\mathbb{R} \to \mathbb{R}$ which we shall also denote by $\alpha_y$. Note that $A = \alpha_y^{-1}(0)$.

The interesting part comes when looking at what happens to $\alpha_x$ at the boundary of $A$. As $A$ is open, it is a disjoint union of open intervals. The boundaries of each of these intervals forms part of the boundary of $A$ and it is simplest to start with these points. For further simplicity, let us assume that $(0,1)$ is one of the components of $A$ and we are considering the boundary point $0$. Thus we wish to consider $\lim_{t \to 0^+} \alpha_x(t)$.

The general rule is simple to state: $\alpha_y$ must go to zero faster than $\alpha_x$ (and any of its derivatives) can go to infinity.

\hypertarget{coequaliser_example}{}\subsection*{{Coequaliser Example}}\label{coequaliser_example}

Let us give an example that shows that the category of Fr\"o{}licher spaces is not [[locally cartesian closed category|locally cartesian closed]]. Consider a [[coequaliser]] diagram $\mathbb{R} \setminus \{0\} \to \mathbb{R} \amalg \mathbb{R}$ where the two maps are the inclusions into the two cofactors.

The coequaliser of this diagram is $\mathbb{R} \cup \{*\}$ where the $*$ is a doubled-point at $0$. (So this is the well-known example of a non-Hausdorff [[manifold]].) Thus any smooth function $\psi : \mathbb{R} \cup \{*\} \to \mathbb{R}$ has to satisfy $\psi(0) = \psi(*)$, which means that any smooth curve $\alpha : \mathbb{R} \to \mathbb{R} \cup \{*\}$ can choose whether to pass through $0$ or $*$ completely arbitrarily.

We consider this as a coequaliser of spaces over $\mathbb{R}$ by taking the obvious map to $\mathbb{R}$ in each case. The colimit is the same whether we work in the full category of Fr\"o{}licher spaces or just those over $\mathbb{R}$.

Now let $Y$ be any Fr\"o{}licher space and consider it as a space over $\mathbb{R}$ via the [[constant function|constant]] zero map, $\omicron : Y \to \mathbb{R}$. We take the [[fibred product]] over $\mathbb{R}$ of the coequaliser diagram. Since $\mathbb{R} \setminus \{0\}$ has no points mapping to $0$, $(\mathbb{R} \setminus \{0\}) \times_{\mathbb{R}} Y = \emptyset$. For the third space, we see that $(\mathbb{R} \amalg \mathbb{R}) \times_{\mathbb{R}} Y = Y \amalg Y$. Thus the coequaliser diagram is now $\emptyset \to Y \amalg Y$. The coequaliser is thus $Y \amalg Y$. Note that smooth curves into $Y \amalg Y$ are of the form $(i, \beta)$ where $i$ is a constant in $\{0,*\}$ that distinguishes the cofactors and $\beta : \mathbb{R} \to Y$ is a smooth curve in $Y$.

Let us consider the product over $\mathbb{R}$ of $\mathbb{R} \cup \{*\}$ with $Y$. As a set, this is just $Y \amalg Y$ again. However, as a Fr\"o{}licher space it has different functions to those on $Y \amalg Y$. The product over $\mathbb{R}$ is a subspace of $(\mathbb{R} \cup \{*\}) \times Y$ and thus a curve into it is smooth if and only if it is smooth into $(\mathbb{R} \cup \{*\}) \times Y$, whence it is smooth if and only if the projections to $\mathbb{R} \cup \{*\}$ and to $Y$ are smooth. As we are considering curves in $(\mathbb{R} \cup \{*\}) \times_\mathbb{R} Y$, the projection to $\mathbb{R} \cup \{*\}$ must have image in $\{0,*\}$. Thus \emph{any} curve is allowed by this and so the smooth curves into $(\mathbb{R} \cup \{*\}) \times_\mathbb{R} Y$ are of the form $(\alpha, \beta)$ where $\alpha$ is \emph{any} function from $\mathbb{R}$ into $\{0,*\}$ and $\beta : \mathbb{R} \to Y$ is a smooth curve in $Y$.

I notice that in some classically false versions of [[constructive mathematics]], the only functions from $\mathbb{R}$ to $\{0,*\}$ are the constant ones. It would be nice if there were a nonclassical [[dream mathematics|dream universe]] in which the category of Fr\"o{}licher spaces were locally cartesian closed! Unfortunately, the counterexample can be saved by using a continuously parametrised coproduct $\coprod_{\mathbb{R}} \mathbb{R}$ instead of $\coprod_{\{0,*\}} \mathbb{R} = \mathbb{R} \amalg \mathbb{R}$. ---Toby

[[Andrew Stacey]] I think I'd be disappointed if locally cartesian became a property of what set theory was being used! I suspect that the real reason this example works is the fact that $Y \to \mathbb{R}$ has such bad path-lifting properties and the coequaliser diagram that I chose is just a simple one that demonstrates it.

I've been pondering how one might fix this. I ought to write up Kriegl and Michor's extension of a manifold as, although they haven't done much with it, it contains some ideas that may be of interest. What makes me think of it here is that that definition of a manifold (which isn't, by the way, the one in their weighty tome) has two parts: a space and its tangent space, and relationships between them. This suggests that a smooth space, of whatever variety, should be more than just one space but some sort of diagram of spaces.

This is also suggested by trying to generalise the Frolicher ``idea'' to non-set-based theories. It's not immediately obvious how to make the saturation condition work, but the basic idea is to have objects as $(C,F,c)$ with $c : C \times F \to Hom$ and morphisms as $f : C_X \times F_Y \to Hom$. The saturation condition, whatever it is, is what is needed to make this into a category. So here it's obvious that objects are just special morphisms. Feeding this back into the definition you get a glimmer of an idea that maybe a morphism (let me go back to genuine Frolicher spaces here for clarity) $f : X \to Y$ needs some sort of saturation condition as well as a compatibility condition.

This is not the same as $Y \coprod Y$ and thus the functor $- \times_{\mathbb{R}} Y$ does not preserve colimits. It cannot, therefore, be a left adjoint and so the category of Fr\"o{}licher spaces is not locally cartesian closed.

This example works because of the structure of $\mathbb{R} \cup \{*\}$. If one were to work in an ``input only'' category, then the structure on $\mathbb{R} \cup \{*\}$ would be determined by those curves which lift to $\mathbb{R} \coprod \mathbb{R}$. Such maps could not arbitrarily swap between $0$ and $*$ because up in $\mathbb{R} \coprod \mathbb{R}$ these two points are far apart. Thus the subspace structure on $\{0,*\}$ in an ``input only'' category is \emph{discrete}. However, in the category of Fr\"o{}licher spaces the outputs control the behaviour of quotients. Functions out of $\mathbb{R} \cup \{*\}$ cannot detect the difference between $0$ and $*$. Thus curves into $\mathbb{R} \cup \{*\}$ are allowed to swap between them with aplomb. The subspace structure on $\{0,*\}$ is thus the \emph{indiscrete} structure.

Working in the category of \emph{Hausdorff} Fr\"o{}licher spaces (see \hyperlink{hausdorff}{below}) does not improve matters. Then we need to replace each coequaliser but its Hausdorffification. Now the distinction is clear since taking the product and then the coequaliser yields $Y \coprod Y$ as before but taking the coequaliser and then the product yields just $Y$.

It is also worth pointing out that with the modification of the previous paragraph, this example only involves manifolds (assuming that $Y$ is chosen to be a manifold). It therefore shows that a category extending that of smooth manifolds can \textbf{either} be locally cartesian closed \textbf{or} preserve limits and colimits from manifolds \emph{but not both}.



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