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I left a counter-query underneath Zoran’s query at compactly generated space. It may be time for a clean-up of this article; the query boxes have been left dangling and unanswered for quite some time. Either proofs or references to detailed proofs would be welcome.
Yes, some clean-up would be nice. I replied to the queries.
In that case, Zoran’s point stands.
Looks like the remark in question was originally added by Ronnie Brown over a year ago.
There is still left-over query box discussion at compactly generated space between Zoran, Todd, Mike and Toby . It looks to me like all issues have been clarified there. Does anyone feel like brushing up the entry?
Actually, looking at it again I see maybe an interpretation of the remark that might make it make sense. If by $k Top(X,Y)$ we mean a specific topology which is not necessarily a k-space, then cartesian closure of the category of k-spaces and continuous maps would be about a homeomorphism relating to the k-ifications of this $k Top(X,Y)$ when X and Y are k-spaces, while cartesian closure of the category of all spaces and k-continuous maps could be phrased either as saying the same thing (invoking the fact that they are equivalent) or about a k-homeomorphism relating to $k Top(X,Y)$ itself rather than its k-ification. And in the latter case saying that it is actually a homeomorphism rather than just a k-homeomorphism would be saying something stronger, relating to a characterization of $k Top(X,Y)$ as an arbitrary space rather than (its k-ification) as a k-space. Is this making sense?
Unfortunately, I am too much in a hectic mode before the long travel next week, so I can not delve into this discussion until about next Thursday…
I’ve edited the entry to clarify some, by distinguishing notationally between the category of k-spaces and continous maps, and the category of all spaces and k-continuous maps. Although they are equivalent, the question at issue seems to revolve around their non-identical-ness.
I still do not have enough time to delve into this but I was told by my student that
is extremely clear about when the weakly Hausdorff assumption is needed and when not. I quoted it in $n$Lab.
Thanks for the reference; I had a look at it and tried to clarify the entry a trifle further.
compactly generated space says that a topological space is compactly generated iff
$X$ is an identification space of a disjoint union of compact Hausdorff spaces.
It is clear to see that a quotient space of a disjoint union of compact Hausdorff spaces if compactly generated, namely every compact Hausdorff space is compactly generated, disjoint union of compactly generated Hausdorff is compactly generated Hausdorff and any quotient of a compactly generated Hausdorff space is compactly generated. Conversely, there are many ideas of covers of $X$ by compact Hausdorff spaces, one needs to choose good enough such that the definition for compactly generated (not necessarily even weakly Hausdorff) space can be tested only on the morphisms from the cover: then one does the identification by identifying points with the same image. However, any attempt which I tried in a quick attempt use the axiom of choice, most often within a proper class or something of the sort. What is the clean set-theoretic way to do it ?
What makes you think there is one? (-:
So what the entry claims ? It says that it is a characterization of a compactly generated space. Even if we use the axiom of choice is it within a set, or we entail a change of universe or what ? Something is claimed and my attempts to understand it get into set-theoretical nightmare, if I want a full generality.
Well, I don’t know the answer. I was just saying, just because something is true doesn’t necessarily mean it can be proven in a set-theoretically clean way. (-:
I felt that :)
I have reorganized the sections at compactly generated space a little (check if you agree that it is better now), added a stubby Examples-section and a reference.
By the way, there is still lots of query-box discussion there. Maybe it can be removed or else turned into definite content?
I moved the (still empty) section on “weak Hausdorffification” to weakly Hausdorff space. Maybe k-space and compactly generated space should be separate pages too?
In compactly generated topological space there is an inconclusive discussion box about local cartesian closure of the category of compactly generated spaces. I want to bring your attention to this paper I have just stumbled upon
Booth, Peter I. The exponential law of maps. II. Math. Z. 121 (1971), 311–319
where the author claims that the category of compactly generated spaces over a Hausdorff space is cartesian closed. Unfortunately, for the definition of the internal hom-object he refers to another paper of his where the definition is stated in a different language (and involves further references) and I find it difficult to put the definition and the proof together. Perhaps someone will be motivated enough to take a closer look and decide whether the proof is valid.
I have added in, here, the argument for why every CW-complex is compactly generated.
Returning to #18: I’m fairly skeptical of local cartesian closure. This paper for example says that the category of Kelley spaces (meaning compactly generated Hausdorff spaces) is not locally cartesian closed.
In the entry where it says “by Yoneda lemma arguments” I have added a pointer to the actual argument (prop.).
Regarding #5 and #6: I vote for simply giving some name such as $[X,Y]$ to the actual exponential object in compactly generated spaces and then simply saying that there is a homeomorphism $[X,[Y,Z]]\simeq [X \times Y, Z]$.
I don’t see how the present state of the entry is trying to improve on this simple fact that the entry should state.
Comment #21 should point here.
Thanks. Sorry.
I don’t have the time to re-understand what was going on in #5-6, but I agree that the simple and important facts should be stated up-front in clean notation.
There are some old queries on that page that could probably be discarded…
I just noticed that the section Local Cartesian closure contained the following query box exchange (now moved from there to here) which remains unresolved:
+– {: .query}
Mike Shulman: What precisely does “get an exponential law” mean? Do you mean that $k Top/B$ is cartesian closed if $B$ is $T_0$?
Toby: Hopefully that is explained in the new article.
Mike: Which new article? exponential law for spaces? That page doesn’t talk about fibered exponentials at all.
Toby: Seeing this later, I no longer know what article I meant.
=–
I expect that P. Booth would be an author of such an article. It might be this by Booth and Brown. I can probably firm this up soon.
@Todd here’s a better link: https://doi.org/10.1016/0016-660X(78)90049-1, title is Spaces of partial maps, fibred mapping spaces and the compact-open topology, if that rings any bells for anyone.
Thank you David.
If in doubt ask Ronnie Brown or Peter Booth. At that time (later 1970s early 80s) a lot was done on partial maps.
Removed an old query box:
Zoran Škoda: I do not understand the remark. I mean if the domain is k-space then by the characterization above continuous is the same as k-continuous. Thus if both domain and codomain are continuous then homeo is the same as k-homeo. I assume that even in noHausdorff case, the test-open topology for $X$ and $Y$ k-spaces gives a k-space and that the cartesian product has the correction for the k-spaces.
Todd Trimble: That may be just the point: that the domain is not necessarily a $k$-space. I have to admit that I haven’t worked through the details of this exposition, but one thing I tripped over is the fact that we’re dealing with all topological spaces $X$, $Y$, not just $k$-spaces.
Mike Shulman: But any topological space is isomorphic in $k\Top$ to its $k$-ification, right? So $k\Top$ might as well be defined to consist of $k$-spaces and continuous maps.
Todd Trimble: Okay, you’re right that makes sense. So in that case, it seems that Zoran definitely has a point here.
Mike Shulman: See the nForum discussion.
[Administrative note: #1 - #24 were originally from another thread with name ’compactly generated space’. Thank you to Dmitri Pavlov for pointing out that the threads should be merged; I have deleted Dmitri’s comment requesting this to avoid confusion in the future, as it now would have been placed out of sync.]
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