[Parse it as (locally compact)ly generated.]

I stumbled across this one whilst supervising an undergraduate thesis. Convenient categories for homotopy theory (e.g. CGWH) have been discussed here before. As an alternative to CGWH, Rainer Vogt proposed the category of locally compactly generated spaces; see also the recent German-language point-set topology textbook Grundkurs Topologie by Gerd Laures and Markus Szymik.

If (as is now usual) one means (compact Hausdorff)ly generated when one says compactly generated, then the category of compactly generated spaces is a full subcategory of the category of locally compactly generated spaces. Back in 1971, Vogt asked whether this inclusion is strict or not. Do we know the answer yet?

  • $\begingroup$ In case anyone has the same confusion I had, let me point out that (locally compact Hausdorff)ly generated spaces are the same as (compact Hausdorff)ly generated spaces, as observed by Vogt. And it seems to me that similarly, (locally compact)ly generated spaces should be the same as (compact)ly generated spaces, so the question is really about dropping the Hausdorff assumption rather than "going local". $\endgroup$ – Tim Campion Dec 6 '15 at 18:42

The paper "A distinguishing example in k-spaces" by John Isbell constructs an example of a locally compact space $X$ which is not compact-Hausdorffly generated.

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  • $\begingroup$ This is a straightforward answer, but I'm just curious about the ramifications. Are there interesting things one can do is this larger category that couldn't be done in the smaller one? $\endgroup$ – Greg Friedman Sep 11 '15 at 4:37
  • $\begingroup$ @Greg: I'm no expert, but 1) aesthetics: the cartesian product in the locally compactly generated case is closer to the usual one; 2) economy: if you were rebuilding the theory from scratch then the locally compactly generated approach involves less work. $\endgroup$ – David J. Green Sep 11 '15 at 7:02
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    $\begingroup$ On the other hand, this paper appears to show that the larger category is not cartesian closed, which was one of the main motivations for this whole business in the first place. Also, I find it striking reading Isbell's introduction that he seems to have wanted a sober example but couldn't find one. For example, this means that he wasn't able to find a ring R for which he could show that Spec R is not (compact Hausdorff)ly generated. I wonder whether this is known... $\endgroup$ – Tim Campion Dec 6 '15 at 18:48

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