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? 



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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:


1. Extend the sheaf $\Phi$ to closed subsets in our manifolds
2. 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:



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> 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.
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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)? 



Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?
I recommend a very nice paper of Sander Kupers, "Three applications of delooping to $h$-principles". In an appendix he discusses and compares various convenient categories of spaces when working with continuous sheaves. However Kupers does not work with quasitopological spaces in the body of the paper. His preferred convenient category of spaces is instead the category of sheaves of sets on the site of manifolds and open embeddings (smooth, PL or topological). In the smooth case these are often called "differentiable spaces". 


One reason this is a convenient category to work with is that in many applications one considers continuous sheaves $U \mapsto F(U)$ where $F(U)$ is a "moduli space" parametrizing some sort of "objects over $U$". In algebraic geometry it is a well established principle that moduli spaces are fruitfully described by specifying the functor they represent, i.e. as a sheaf of sets on some site. For example, the Grassmannian $G(k,n)$ represents the functor assigning to a space $S$ the set of all rank $k$ subbundles of the trivial rank $n$ vector bundle over $S$, and this is a more useful description of $G(k,n)$ than a specification of a topology on the set of $k$-dimensional subspaces of affine $n$-space. Moreover, in situations where the moduli functor is not representable, it is often more useful to work with the moduli functor itself rather than some coarse approximation of it. Importantly, for a continuous sheaf described in this way, the various notions of flexibility, microflexibility, softness etc get a very concrete meaning in terms of the moduli functor.


As a specific example, Kupers considers the case that $F(U)$ is the "space of foliations on $U$". In this case $F(U)$ can be defined in a very convenient way: it is taken to be the functor taking a manifold $M$ to the set of foliations on $M \times U$ of codimension $\dim(U)$ which are transverse to the projection onto $M$. 


However one could consider equally well sheaves of sets on other geometrically natural sites, and have many of the same advantages. One could consider sheaves on topological spaces with respect to the open cover topology, as in the theory of topological stacks. Or, indeed, one could work with quasi-topological spaces: a quasi-topological space is a particular type of sheaf of sets on the category of compact Hausdorff spaces, with respect to the topology defined by finite families of jointly surjective maps. The reason it is natural to impose descent for such a fine topology is that any surjective map of compact Hausdorff spaces is a quotient map. I think this explains at least morally why quasi-topological spaces should arise naturally when studying continuous sheaves: we want $F(U)$ to be a moduli space, so it is natural to describe it as a sheaf of sets, and by retaining control of the moduli functor we can force also the value of the sheaf on a codimension zero submanifold with boundary, say, to have a modular interpretation. 




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Now let me also say that I believe that quasi-topological spaces are simply not very natural objects to work with and are probably never really the right tool for the job. 


I want to compare them to a notion of ringed space introduced by T.A. Springer, in his book "Linear algebraic groups". In the first part of this book Springer introduces a "baby version" of scheme theory, adequate for describing the theory of varieties over a field. One way in which the theory is simplified is by the nonstandard definition of *ringed space* that he works with. He fixes a ground field $k$, and declares that a ringed space is a topological space $X$, together with a subsheaf of the sheaf of $k$-algebras $U \mapsto \mathrm{Funct}(U,k)$ on $X$. This simplifies life in some ways. For example, he can then define a morphism of ringed spaces to be a morphism of underlying topological spaces, satisfying the *property* of being compatible with the structure sheaves, rather than including the extra data of a map of sheaves. And he completely avoids talking about "locally" ringed spaces in this way, which would otherwise be necessary to have an equivalence of categories between $k$-algebras and affine schemes over $k$. 


But in the end this definition is obviously a kludge - it is simply much more natural to work with an arbitrary sheaf of rings, without imposing this concreteness condition, and it is somehow an "accident" that it produces the "correct" theory when the scheme is of finite type over a field. 


Now the definition of a quasi-topological space is strikingly similar. Recall that a quasi-topological space consists of a set $W$, and a subsheaf of the sheaf of sets $S \mapsto \mathrm{Funct}(S,W)$ on the category of compact Hausdorff spaces. Morphisms are natural transformations. It seems very natural to expect the category of *all* sheaves of sets on compact Hausdorff spaces to be a more flexible and convenient category than this one. When this question was asked, there was no established name for a sheaf of sets on compact Hausdorff spaces, but now we have one: this is precisely the notion of a **condensed set** of Clausen-Scholze. I would place a bet that any occurence of quasi-topological space in the literature is more conveniently described as a condensed set.


A final remark is that the algebro-geometric study of moduli spaces also teaches us that we should allow the parametrized objects to have automorphisms, which leads to replacing sheaves of sets with sheaves of groupoids and the theory of stacks. Once one has passed to 1-stacks it is natural to proceed further to higher stacks. I would therefore expect differentiable higher stacks or condensed anima to be useful categories of spaces for studying $h$-principles in general.  


