What is a good category of tame topological spaces?

Question

While the definition of a topological space nicely encapsulates the notion of points being close to each other, it doesn't seem to be a good notion of space. It has quite some problems:

1. Every definition comes with many different variations:
   • There is a zoo of separation axioms.
   • There are at least four versions of connectivity.
   • There are many distinct notions of manifolds and other spaces patched up local models (like CW-complexes).
   • Often the definitions don't even seem to be standardized (see for example my question on closed neighborhood deformation retracts or find the "right" definition of locally compact space).
2. Those variations are necessary, as there are many pathological counterexamples (like for example the topologist's sine curve, the Warsaw circle or the Hawaiian earring in the context of connectivity).
3. There are many unintuitive results (there are space filling curves, injective continuous maps are not embeddings, quotient maps are not open etc.).
4. global $\not\Rightarrow$ local. (I find it confusing at best that being path connected does not imply being locally path connected...)
5. The category $\mathsf{Top}$ is not cartesian closed. Phrased differently, the sets of maps between spaces do not carry a canonical topology in such a way that composition is continuous. (I am unsure about the set of maps having a topology being intuitive, but it is a very useful tool to have, especially when doing homotopy theory)

Arguably it is good to have such a general setting many subjects in mathematics can build upon. What bothers me is that, as soon as one wants to do something geometric / homotopical, one has to restrict to certain topological spaces, dealing with a lot of additional assumptions to avoid those pathological exceptions. Moreover often the topological notions are not really the right ones, when working in a non-topological context (a scheme seldomly is hausdorff, it may be separated though, borrowing but not using topological intuition). So my question is

What is a good category of tame topological spaces, in the sense that it

• is complete, (at least finitely) cocomplete and cartesian (or maybe monoidal) closed
• contains the prime examples of spaces like metric spaces, smooth manifolds, CW-complexes, polyhedra
• makes standard notions coincide (as much as possible) + avoids pathological counterexamples
• has a direct axiomatization (no "Hausdorff spaces, which admit a neighborhood of locally compact opens with connected fibers thingy...")

I am aware of the possibility that this question may not have a satisfying answer. The fact that genius minds like Grothendieck noted "The foundations of topology are inadequate is manifest in the form of false problems [..., which] include the existence of wild phenomena (space-filling curves, etc.) that add complications which are not essential" (On Grothendieck's Tame Topology, p.3) but not come up with a groundbreaking solution (I don't understand o-minimal structures yet, but they do not really look satisfying) leads me to think that the question might not even have one. However I feel like asking this question nonetheless will lead to insights of one kind or another...

As always: thank you for your time and considerations.

PS: The question feels vague, but I don't really know what to specify further. So any suggestions for making it more precise are very welcome...

Answer 1

Your conditions cannot happen all at the same time. The category of sequential spaces is complete, co-complete, coreflective, Cartesian closed, and it includes all of the examples you want including metric spaces. This is an example of one of Steenrod's "convenient categories" for doing algebraic topology. However, as soon as you include all metric spaces or all compactly generated spaces you're including tons of what you call "pathological" things like the Hawaiian earring. Additionally, notice that if you want your category of spaces to be complete and contain $S^1$, you're automatically going to be forced to include the Hawaiian earring and infinite dimensional torus. Both of these are connected, compact, locally-path-connected metric spaces but they are not locally contractible. How is this usually handled? The power of "tame" homotopy theory is that you are only interested in things up to weak homotopy equivalence and so you can always replace a space by a weakly homotopy equivalent CW complex. In the case of limits (like the Hawaiian earring) you can replace it with the homotopy limit in the CW-category so that you still have a CW-complex in your hand. So in algebraic topology you really want a "space" not to be so much a geometric-type space but rather an equivalence class of spaces, e.g. a "weak homotopy type." To use it maybe you have to go back and forth between a few categories like the category of compactly generated spaces and it's homotopy category and the category of CW-complexes and its homotopy category....This is really not so bad.

I'd also like to encourage you to be open to the possibility that there are many fascinating and useful theories that involve "pathological" examples. Mapping class groups of infinite type surfaces are large groups that could easily be labelled as pathological yet they are quite popular at the moment. What is considered tame/pathological very much depends on current interests and these change over time as new mathematics is developed. Many times I have found myself using mathematics that I never thought I would. If I had been completely shut off to it as inherently useless or ugly, I would have been severely restricted in what I could do. It is somewhat fashionable in certain fields to react sharply and negatively to considering notions of spaces that are anything but CW-complexes or manifolds and this self-induced bubble probably does more harm than good. I think that with experience comes an appreciation for the non-existence of mathematical objects which are the "best" or which satisfy all ideals at the same time. Rather a mathematical object like a category should only be considered "good" relative to the intended use or application.