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As I’ve already said elsewhere, I’ve been working on this entry and trying to give a precise definition based on my hunches of what guys like Steenrod really meant by “a convenient category of topological spaces”. (I must immediately admit that I’ve never read his paper with that title. Of course, he meant specifically compactly generated Hausdorff spaces, but nowadays I think we can argue more generally.)
I also said elsewhere that my proposed axiom on closed and open subspaces might be up for discussion. The other axioms maybe not so much: dropping any of them would seem to be a deal-breaker for what an algebraic topologist might consider “convenient”. Or so I think.
I’ve just taken a quick look through Steenrod’s paper, which is very nice. He confirms my hunch that closure under certain subspace inclusions should be one of the desiderata, but perhaps I spoke too soon when I asserted that all known examples are closed under all open subspace inclusions. (That’s okay; I can live with that.) Otherwise I am beginning to feel that convenient category of topological spaces is on fairly solid historical ground.
I hadn’t heard of these $\aleph_0$-spaces of E. Michael.
some student sends me an email telling me that after looking at some $n$Lab entries it has not become clear whether or not the category of locally compact topological spaces is cartesian closed. Maybe somebody feels like expanding on that
Of course, he meant specifically compactly generated Hausdorff spaces, but nowadays I think we can argue more generally.
On the contrary, Steenrod meant any category of topological spaces which is closed with respect to a number of constructions (a little bit more than what we call closed cartesian category) AND contains most of the standard examples. He actually lists abstract requirements on a convenient category before exhibiting the prime example, the category of compactly generated Hausdorff spaces.
A point of history. Steenrod (1967) was not the originator of the term. Ronnie used the term in that same sense in his thesis (1961) and in two published papers. I am not sure if Ronnie was the earliest to use the term in that way. It would seem that Steenrod had heard Ronnie talk on the subject, had seen the papers and had found the term useful. It is unfortunate that the term has become ’hallowed’ when it looks just like a convenient term to use. In any case, Todd, you can ask Ronnie for his version or look at his Topology and Groupoids book page 199 where he tells the story in some detail. (If someone wants to include something along these lines please don’t use this but quote Ronnie or get him to write a short historical note expressly for the Lab.) It is perhaps a variant of Baez’s Law.
@Urs #3: well, I guess everyone knows “no it isn’t”, but I’d be happy to expand on that.
@Zoran #4: I don’t believe I wrote that (doesn’t sound like me! (-:), but thanks for the information. I’ll edit later.
@Tim #5: thanks for mentioning this! I’ll send Ronnie some email if his book isn’t available online.
I see that Mike did react to my request from #3. Thanks!!
@Zoran: whoops, it seems I did write that after all, but that was evidently before I had looked at Steenrod’s paper. It doesn’t matter, since nothing of that sort was said in the nLab article. The article seems consistent with your information.
I guess everyone knows “no it isn’t”,
As I said, there was a student who emailed me who did not know it, looked at the entry, and still did not get that information. So the entry needed to be improved.
but I’d be happy to expand on that.
Mike already did.
Last I looked, Mike didn’t give an explicit example that Callot is seeking. But an example might be $\mathbb{N}^\mathbb{N}$ where $\mathbb{N}$ has the usual discrete topology. Discrete spaces are of course locally compact, and the compact-open topology is the usual product topology. But this exponential is not locally compact, because there is an explicit homeomorphism between this space and the space of irrationals in the interval $(0, 1)$ with its usual subspace topology. This homeomorphism is given by the regular continued fraction expansion.
Sorry, Urs, I didn’t mean to sound snotty back in #6. By “everyone” I actually meant the core band of contributors here; sorry for the confusion. And I had already gotten the message that Mike addressed this.
No problem, Todd. Let’s just all try to keep in mind that the nLab is used not just by us contributors.
Yes. I apologize. I had been in a hurry.
Thank you Todd, but I still don t know whether this is enough. Isn t it, that actually one has to proove that there is no topology $\X$ on $\mathbb{N}^{\mathbb{N}}$ such that $(\mathbb{N}^\mathbb{N},\X)$ is locally compact and such that X is describing continous convergence - to get a real counterexample with this. I knew the argument that $\mathbb{N}^{\mathbb{N}}$ endowed with the product topology resp. the compact-open-topology is not locally compact before. It ll be great if you could help me with this!
@Callot: fair enough. I didn’t mean to seem dismissive in throwing out that example. Let me try again.
I thought the question is whether the category of locally compact Hausdorff spaces is cartesian closed. Certainly “cartesian” is not an issue: the cartesian product of two locally compact Hausdorff spaces is their usual topological product.
Next, we examine “closure”. An exponential $\mathbb{N}^\mathbb{N}$ in $LCH$ would be, by definition, a representing object for the functor
$\hom_{LCH}(- \times \mathbb{N}, \mathbb{N}): LCH^{op} \to Set$and this determines the ’right’ topology on $\mathbb{N}^\mathbb{N}$, never mind whether or not this happens to be the compact-open topology. If you agree with this, then the course is fixed and the topology on $\mathbb{N}^\mathbb{N}$ has to be the product topology, for we have
$\hom_{LCH}(X \times \mathbb{N}, \mathbb{N}) \cong \hom_{LCH}(\sum_{\mathbb{N}} X, \mathbb{N}) \cong \prod_{\mathbb{N}} \hom(X, \mathbb{N}) \cong \hom(X, \prod_{\mathbb{N}} \mathbb{N})$(all isomorphisms natural in $X$; the first isomorphism uses discreteness of $\mathbb{N}$). And then the argument I gave above applies. Is this satisfactory?
There is only one notion of cartesian closedness, so we are certain to agree. I only have limited page views of that Google book, but a category is cartesian closed if it has finite products and if for any two objects $X$, $Y$, there is an object $Y^X$ (thought of as a “space of maps from $X$ to $Y$”) such that for any object $Z$, there is a bijection between the set of maps $Z \to Y^X$ and the set of maps $Z \times X \to Y$, and this bijection is natural in $Z$.
I have looked at the nLab page on cartesian closed categories, and it is rather impoverished. I intend to add some expositional details, but I also encourage you to begin learning category theory from an established text, such as Categories for the Working Mathematician. You will be amply repaid for your efforts.
I am guessing your native language is German. My first name probably does look a little unusual to native German speakers (my in-laws are German by birth). Sometimes it’s spelled T-o-d, but mine has two d’s. It’s an old English name with several meanings, including “fox” – traces of that meaning can be seen in English surnames like “Todhunter”. Of course “Tod” has a meaning in German, but it’s not one of my favorites. :-) I am named Todd because there is a (somewhat distant) family connection to Abraham Lincoln, whose wife’s name was Mary Todd Lincoln.
It seems to me that my last reply to Callot in #16 is still not quite a proof; the issue is not whether $\mathbb{N}^\mathbb{N}$ under the usual topological product topology is locally compact (it is of course not), but whether infinite products exist in the category of locally compact Hausdorff spaces. They do not, but we should still give an honest proof.
To settle this for good, I’ll change the example to $\mathbb{R}^\mathbb{N}$. If such a countable product existed in $L C H A b$, then we could easily give it a structure of Hausdorff TVS over the real numbers. But it is well-known that a locally compact Hausdorff TVS is finite-dimensional. So $LCHAb$ is not complete. It follows that $LCH$ cannot be complete either (it it were, then the category of abelian group objects $LCHAb$ would also be complete).
I have added some words to this effect at locally compact space.
I have fixed the code for the numbered Definitions/Theorems at conventient category of topological spaces.
Notice that with the new CSS, if you use wrong code such as
.num_thm
(which is not recognized)
instead of
.num_theorem
(which is), then the formatting comes out really bad now.
I have tried to bring the list of references here more into chronological order.
This list is lacking more comments on what it is that the given items contributed to the idea of convenient categories.
I gather that the three articles by R. Brown (now here) are claimed to be precursors to Steenrod’s, and so I have moved them out of the list of other approaches (quasi-topological and what not) and related to Steenrod’s article more closely.
I have added pointer to:
and
and I have grouped this (here), together with Escardo, Lawson & Simpson 2004, under:
“Discussion in the generality that subsumes compactly generated topological spaces and Delta-generated topological spaces and all cases of subcategory-generated spaces in between:”
I have expanded the Examples-section a fair bit:
In a new first subsection “Categories of colimits of generating spaces” (here) I have recorded more of the original insights from Vogt 1971, already much along the lines of the later development by Escardo, Lawson & Simpson 2004 (which was the only reference that the prvious version of the entry had mentioned here).
Then in a second new subsection (here) I have recorded the observation that once one has gone to the “really convenient” (J. Smith) category of Delta-generated space, the door opens up yet ever more convenience by embedding into the quasi-topos of diffeological spaces, then the cohesive topos of smooth sets, and finally the cohesive $\infty$-topos of smooth $\infty$-groupoids – whose shape modality still sees the correct homotopy types of all topological spaces, under this embedding.
I have also added a diagram showing this state of affairs.
[ typo in diagram now fixed ]
added this pointer:
added pointer to:
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