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moving the following old discussion from out of the entry to here, just for the record (it concerns a bygone version of the entry):
+– {: .query} Tim: As I read the entry on nice topological spaces, it really refers to ’nice categories’ rather than ’nice spaces’! I have always thought of spaces such as CW-complexes and polyhedra as being ’locally nice’, but the corresponding categories are certainly not ’nice’ in the sense of nice topological space. Perhaps we need to adjust that other entry in some way.
Toby: You're right, I think I've been linking that page wrongly. (I just now did it again on homotopy type!) Perhaps we should write locally nice space or locally nice topological space (you pick), and I'll fix all of the links tomorrow.
Tim:I suggest locally nice space. (For some time I worked in Shape Theory where local singularities were allowed so the spaces were not locally nice!) There would need to be an entry on locally nice. I suggets various meanings are discussed briefly, e.g. locally contractible, locally Euclidean, … and so on, but each with a minimum on it as the real stuff is in CW-complex etc and these are the ’ideas’.
Mike: Why not change the page nice topological space to be about CW-complexes and so on, and move the existing material there to something like convenient category of spaces, which is also a historically valid term? I am probably to blame for the current misleading content of nice topological space and I’d be happy to have this changed.
Toby: I thought that nice topological space was supposed to be about special kinds of spaces, such as locally compact Hausdorff spaces, whose full subcategories of $\Sp$ are also nice. (Sort of a counterpoint to the dichotomy between nice objects and nice categories, whose theme is better fit by the example of locally Euclidean spaces). CW-complexes also apply —if you're interested in the homotopy categories.
Mike: Well, that’s not what I thought. (-: I don’t really know any type of space that is nice and whose corresponding subcategory of Top is also nice. The category of locally compact Hausdorff spaces, for instance, is not really all that nice. In fact, I can’t think of anything particularly good about it. I don’t even see any reason for it to be complete or cocomplete!
I think it would be better, and less confusing, to have separate pages for “nice spaces” and “nice categories of spaces,” or whatever we call them. And, as I said, I don’t see any need to invent a new term like “locally nice.”
When algebraic topologists (and, by extension, people talking about $\infty$-groupoids) say “nice space” they usually mean either (1) an object of some convenient category of spaces, or (2) a CW-complex-like space, between which weak homotopy equivalences are homotopy equivalences. Actually, there is a precise term for the latter sort: an m-cofibrant space, aka a space of the (non-weak) homotopy type of a CW complex.
Toby: I thought the full subcategory of locally compact Hausdorff spaces was cartesian closed? Maybe not, and it's not mentioned above.
But you can see that most of the examples above list nice properties of their full subcategories. And the page begins by talking about what a lousy category $\Top$ is. So it seems clearly wrong that you can't make $\Top$ a nicer category by taking a full subcategory of nice spaces. (Not all of the examples are subcategories, of course.)
Mike: It’s true that locally compact Hausdorff spaces are exponentiable in $Top$. However, I don’t think there’s any reason why the exponential should again be locally compact Hausdorff.
I guess you are right that one could argue that compactly generated spaces themselves are “nice,” although I think the main reason they are important is that the category of compactly generated spaces is nice. I propose the following:
Toby: I believe that the compact Hausdorff reflection (the Stone–Čech compactification) of $Y^X$ is an exponential object.
Anyway, your plan sounds fine, although nice category of spaces might be another title. (I guess that it's up to whoever gets around to writing it first.) Although I'm not sure that people really mean m-cofibrant spaces when they speak of nice topological spaces when doing homotopy theory; how do we know that they aren't referring to CW-complexes? (which is what I always assumed that I meant).
Mike: I guess nice category of spaces would fit better with the existing cumbersomely-named dichotomy between nice objects and nice categories. I should have said that when people say “nice topological space” as a means of not having to worry about weak homotopy equivalences, they might as well mean (or maybe even “should” mean) m-cofibrant space. If people do mean CW-complex for some more precise reason (such as wanting to induct up the cells), then they can say “CW complex” instead.
Re: exponentials, the Stone-Čech compactification of $Y^X$ will (as long as $Y^X$ isn’t already compact) have more points than $Y^X$; but by the isomorphism $Hom(1,Y^X)\cong Hom(X,Y)$, points of an exponential space have to be in bijection with continuous maps $X\to Y$.
Toby: OK, I'll have to check how exactly they use the category of locally compact Hausdorff spaces. (One way is to get compactly generated spaces, of course, but I thought that there was more to it than that.) But anyway, I'm happy with your plan and will help you carry it out.
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