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Are all CGWH (compactly generated weak Hausdorff) spaces m-cofibrant? If not, is there a convenient category of topological spaces where all of the spaces are m-cofibrant? Can we have our cake and eat it too, or do we have to make a choice between these two nice properties (cartesian-closedness and the Whitehead property)?
No. (-:
How cryptic... Unfortunately, I asked questions that imply contradictory results, so could you at least specify which question(s) have negative answers?
Although an explanation of the reasoning seems like it could be interesting as well.
Well, I was in a hurry. (-:
Definitely, not all CGWH spaces are m-cofibrant. I’m pretty sure the topologist’s sine curve is CGWH, but it is not m-cofibrant or else it would be contractible, since it is weakly contractible.
It’s harder to give a definite counterexample to the second question, but I’m almost positive the answer is no, and certainly no known convenient category has that property. Things like the topologist’s sine curve are pretty easy to construct, so I expect that they will exist in any topological category containing, say, the real numbers, and having some limits and colimits.
There’s an amazing fact that if X is m-cofibrant and Y is compact, then $X^Y$ is again m-cofibrant, so that in particular the loop space of an m-cofibrant space is again m-cofibrant. But the amazingness of this is because it’s not true without such an assumption on Y.
Your third and fourth questions, which are the ones that contradict each other, seem to be basically summarizing the first two. I think the answer is “no, we can’t have our cake and eat it too.”
It's harder to give a definite counterexample to the second question, but I'm almost positive the answer is no, and certainly no known convenient category has that property. Things like the topologist's sine curve are pretty easy to construct, so I expect that they will exist in any topological category containing, say, the real numbers, and having some limits and colimits.
I guess I'll ask on MO.
Is it intended (and if so, is it wise?) that we have separate entries called
It would seem to me that these entries should be merged, and one made a redirect to the other. No?
I see your point. However, I’m currently playing Bourbaki over at convenient category of topological spaces, focusing on what I thought were the original desiderata of people like Steenrod. I’d actually like to keep a technical distinction between these phrases, with “convenient” meaning what I’m saying it means (we can discuss this of course), and with “nice” having a wider and vaguer meaning.
A good way of explaining what I mean is that all convenient categories are “nice”, but compact Hausdorff spaces are also nice, and so are sequential spaces. There also also cartesian closed categories of spaces which arise in domain theory that may be nice for logicians and domain theorists, but not very convenient for most working topologists. Ditto for categories of locales. So “nice” for me would mean nice for a general mathematical purpose at hand, whereas “convenient” is technical and has mostly to do with the needs of algebraic topologists.
And although they are not categories of topological spaces, I would also place various toposes and quasitoposes of “spaces” under the general umbrella of “nice”, including for example Johnstone’s topological topos and the quasitopos of subsequential spaces. Possibly “nice” (but this to me is debatable) is Spanier’s quasitopological spaces.
Edit: the entry nice category of topological spaces does not currently exist, but nice category of spaces does. I think this is very much in keeping with what I said above.
Psh, if you were playing Bourbaki, you’d come up with a new name for it as well, like a “magma”, a “filter”, or a “uniform space”, or “semigroup”…
Bad joke? Don’t care!
I agree with Todd #7.
I find Spaniers quasitopological spaces intriguing, largely since they are one of the few categories I know of in which anyone has seriously considered doing mathematics which is not well-powered or well-copowered and whose forgetful functor to Set has large fibers. But perhaps those are some of the reasons their niceness is debatable, and they certainly don’t seem to ever have caught on.
whose forgetful functor to Set has large fibers
Yes! Even the two-point set has a proper class of quasitopologies on it. This alone probably makes them not very “nice” for a lot of people.
Okay, so then we should keep the two entries separate but maybe put in the kind of discussion along the lines Todd just gave in #7 to make clear what’s going on.
Okay, I put in such a discussion at convenient category of topological spaces. I also added other content.
Thanks, Todd. I think this is very useful.
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