Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying to understand explicitly why we need to use these?

Background

Very roughly, the idea is that one has a sheaf $\Phi$ of topological spaces on the site of smooth manifolds. A typical examples will be the sheaf of immersions. Often there is a related sheaf of spaces $\Psi$ which has a homotopy theoretic nature. For example the sheaf of sections of the jet bundles corresponding to jets of immersions. Now $\Psi$ is something which is easy to compute, and there is a comparison map $$ \Phi \to \Psi$$ The h-principle will be satisfied if this is a weak equivalence for each manifold. By construction we usually know that this holds for $\mathbb{R}^n$, and we also know that $\Psi$ is not only a sheaf but a "homotopy sheaf". For example the value of $\Psi(U \cup V)$ will be equivalent to the homotopy fiber product $$ \Psi(U) \times^h_{\Psi(U \cap V)} \Psi(V)$$ (it is isomorphic to the ordinary fiber product by the sheaf condition).

So the h-principle would follow if we could show that $\Phi$ was also a "homotopy sheaf". One way to prove this is to show that the restriction maps $\Phi(U) \to \Phi(U_0)$ are Serre fibrations.

This is usually too strong for open manifolds $U_0$. For example for $\Psi$ the sheaf of smooth functions, this restriction map is not surjective and it is easy to show it is not a Serre fibration. This is related to the fact that the sheaf of smooth functions is not flabby.

However the sheaf of smooth functions is soft, and we might hope to prove something similar for these sheaves of spaces. Namely we might be able to show that the restriction map to closed submanifolds (with boundary) is a Serre fibration.

So at this stage we must do two things:

- Extend the sheaf $\Phi$ to closed subsets in our manifolds
- Show that $\Phi$ satisfies this a softness condition (namely that restriction to closed subsets is a Serre fibration).

After this the general argument goes on to show that indeed $\Phi$ satisfies the h-principle.

Quasitopological spaces enter the picture in step 1. The idea is to define the value of $\Phi$ on closed subsets in the usual way, as a colimit of the spaces assigned to open subsets containing the closed subset. In Gromov's text on the h-principle he has the following cryptic remark:

There is no useful natural topology on this space; however there is a weaker structure, called a

quasi-topology, which nicely behaves under direct limits.

I have seen similar things mentioned in various papers and texts where the h-principle comes up, but none I have come across have given an example of what goes wrong and why we actually need to use these more exotic spaces. Indeed in some other texts the issue is either ignored or skirted somehow, but I am left wondering when and why we would need to use these quasi-topological spaces.

**The Question**: Coming from algebraic topology, I know we now have several convenient categories of topological spaces which are cartesian closed and fairly well behaved categorically, for example the category of compactly generated Hausdorf spaces. Why can't we just take the direct limit in one of these categories? What goes wrong and how badly?

Do I really need to worry about this when $\Phi$ is, say, the space of smooth functions (with one of the Whitney $C^\infty$-topologies or some variant thereof)?