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Patrick Nicodemus (Aug 23 2021 at 00:09):Question:
Can the category of compactly generated spaces be characterized in any nice categorical way as a cocompletion of the category of compact hausdorff spaces? Suppose I don't know what a topological space is, the category of CG spaces presumably is some full subcategory of the free cocompletion of CompHaus,
https://ncatlab.org/nlab/show/free+cocompletion#free_cocompletion_of_large_categories
How can we characterize this subcategory?
David Michael Roberts (Aug 23 2021 at 02:15):Not an answer, but one could perhaps see it as the concrete (?small) sheaves on CompHaus, taking a suitable Grothendieck (pre)topology on CompHaus. I wouldn't call it the free cocompletion, since finite coproducts of compact Hausdorff spaces are still compact Hausdorff. So perhaps the coherent topology is needed, generated by the (finitely) extensive topology and suitable quotient maps of Hausdorff spaces.
Mike Shulman (Aug 23 2021 at 04:10):Concrete sheaves on CompHaus, for the topology of finite coverings, are quasi-topological spaces.
Mike Shulman (Aug 23 2021 at 04:12):The definition of k-space suggests that they should have some universal property relative to the way thath CompHaus sits inside the concrete category Top.
Ivan Di Liberti (Aug 23 2021 at 06:32):Mike Shulman said:
The definition of k-space suggests that they should have some universal property relative to the way thath CompHaus sits inside the concrete category Top.
Aren't they precisely their colimit closure in Top?
Roman Kniazev (Aug 23 2021 at 08:09):It seems that this article of Escardó, Lawson and Simpson might be useful. In my understanding, in this context, it is somewhat important to distinguish between compactly generated Hausdorff spaces and compactly generated spaces without separation axiom. For the latter, the paper says (by Lemma 3.2) that they are exactly the colimit closure of compact Hausdorff spaces in Top.
Roman Kniazev (Aug 23 2021 at 08:09):It doesn't answer the original question, though, as we are still in the ambient category of topological spaces.
Patrick Nicodemus (Aug 23 2021 at 12:26):Ok! Thanks. To give some context, I was thinking about how the CH spaces are monadic over Sets so i was thinking about how it would look to do topology by starting with CH spaces as 'convergence algebras' and then building the more general CG spaces by gluing.
Oscar Cunningham (Aug 23 2021 at 13:29):I believe this is the idea behind condensed mathematics.
Reid Barton (Aug 23 2021 at 14:06):Are you familiar with the relational presentation of general topological spaces? Seems like there might be a nice way to cut out the CG spaces from that, but I don't know.
Reid Barton (Aug 23 2021 at 14:08):I guess the question is whether you want to produce exactly the category of CG spaces, or just something similar. I think the CG spaces have some essential algebraic/relational character, e.g., they don't quite form a quasitopos.
David Michael Roberts (Aug 24 2021 at 10:37):I do wonder how one can recognise compactly generated (?weak) Hausdorff spaces among condensed sets.
David Michael Roberts (Aug 24 2021 at 10:42):@Mike Shulman I was more thinking small concrete sheaves, rather than concrete sheaves of small sets (as the nLab puts it).
Jens Hemelaer (Aug 24 2021 at 11:08):There is an inclusion of -topological spaces into condensed sets for which all maps from points are quasicompact, and this inclusion functor has a left adjoint. The left adjoint lands into the compactly generated topological spaces (this is in "Lectures on Condensed Mathematics", Proposition 2.15 and below). So I think that the compactly generated Hausdorff spaces are precisely the condensed sets such that the unit of this adjunction is an isomorphism?
Weakly Hausdorff spaces are not necessarily , so then the usual way of associating a condensed set to it fails.
Mike Shulman (Aug 24 2021 at 13:04):David Michael Roberts said:
Mike Shulman I was more thinking small concrete sheaves, rather than concrete sheaves of small sets (as the nLab puts it).
Meaning what? Small colimits of representables in the category of large sheaves that happen to also be concrete? Those two requirements seem to me somewhat in tension; it's not obvious to me that the concrete sheaf represented by a compactly generated space is small in that sense.
David Michael Roberts (Aug 25 2021 at 01:46):@Mike Shulman
I wrote in haste. All I meant was that surely there are quasitopological spaces that aren't small sheaves, right? And a compactly-generated space should be the colimit of a small diagram of compact Hausdorff spaces, right? I'm not sure what you mean by 'large sheaves' (non-small sheaves? Sheaves of sets that are possibly large relative to the given universe?)
Mike Shulman (Aug 25 2021 at 01:48):What do you mean by "small sheaf"?
David Michael Roberts (Aug 25 2021 at 06:40):Hmm, is it the difference between being a sheaf that is a small colimit of representables (what I assumed is meant), and some other notion? Is there a worry about the Grothendieck topology not being subcanonical? I admit I haven't gone back to your paper on the matter to check definitions and discussion...
Mike Shulman (Aug 25 2021 at 07:45):You were the first one to use the phrase "small sheaves", so I was just trying to figure out what you meant. I don't think it's standard yet, is it?
By "large sheaves" I meant sheaves of large sets, that being the only ambient place I know of where you could talk about things with the property of being a small colimit of representable sheaves -- the category of small-set-valued sheaves is, as far as I know, not closed under small colimits.
Mike Shulman (Aug 25 2021 at 07:50):It seems plausible that a compactly generated space would be a colimit in Top of a small diagram of compact Hausdorff spaces, although it's not immediately obvious to me how to prove it, since the definition of compact generation refers to all compact Hausdorff spaces. But I also don't see how to conclude that the corresponding representable presheaf on compact Hausdorff spaces would be analogous colimit of representables. (In particular, it seems troubling that the relevant topology on CptHaus used for quasi-topological spaces consists only of finite covers.)
Jens Hemelaer (Aug 25 2021 at 08:59):In this book by Warner it is stated, on page 21 of the pdf, that a Hausdorff compactly generated space is the filtered colimit of its compact Hausdorff subspaces.
In the category of condensed sets every object is a (small) colimit of compact Hausdorff spaces, so by going through condensed sets you can show that every compactly generated space is a (small) colimit of compact Hausdorff spaces. But probably there is a more direct way to show this as well.