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Morgan Rogers (he/him) (Dec 08 2020 at 16:31):I spent many hours today writing my answer to a stackexchange question about cocompactness, or more specifically the strong cocompactness which is the dual of *finite presentability*. It was interesting to discover that the strongly cocompact topological spaces coincide with the finite discrete spaces.
I find that stackexchange questions like this are a good temperature check of the status of a categorical concept. If a question gets comments like "Cocompactness is not as useful as compactness", it's a good sign that anyone who has come across this concept has encountered more resistance than they had motivation for looking into it further. So why did I take an interest in cocompactness? That's a topic for another stream!
Nathanael Arkor (Dec 08 2020 at 16:48):Taking topological spaces morally to be locales, we have that Loc° is the category of frames, which is essentially algebraic, and thus locally presentable. The compact objects here are the finitely presentable frames. The cocompact objects in Loc then arise by duality. In this sense cocompactness is no less natural than compactness if you are studying geometrical concepts, which are arising from algebraic concepts via Stone dualities.
Nathanael Arkor (Dec 08 2020 at 16:50):(I'm not claiming this is particularly insightful, just that I was surprised not to see this perspective mentioned in the question or comments there.)
Nathanael Arkor (Dec 08 2020 at 16:51):Just a note that "local strong presentability" is already used for the analogue with sifted colimits, rather than filtered colimits, so "strong cocompactness" may be a red herring in that regard.
Morgan Rogers (he/him) (Dec 08 2020 at 16:52):Nathanael Arkor said:
(I'm not claiming this is particularly insightful, just that I was surprised not to see this perspective mentioned in the question or comments there.)
If I'd thought of that, since ruling out non- spaces was easy, a different strategy involving ruling out non-sober spaces might have been more efficient, so thanks for the comment!
Morgan Rogers (he/him) (Dec 08 2020 at 16:55):Nathanael Arkor said:
Just a note that "local strong presentable" is already used for the analogue with sifted colimits, rather than filtered colimits, so "strong cocompactness" may be a red herring in that regard.
Strong compactness is already used plenty in the topos theory literature (compactness corresponding to the weaker condition of preserving directed unions of subobjects). Hopefully it's not too confusing, but good to know also.
Jens Hemelaer (Dec 08 2020 at 17:01):I wonder what the cocompact objects are in the category of locally compact abelian groups? Because Pontryagin duality is an equivalence of this category with its opposite, the "cocompact" objects are the Pontryagin duals of the finitely presentable objects.
In this way, you find as "cocompact" objects the finite abelian groups and the circle group and finite products of these (maybe there are others).
This is also a situation where "compact" and "cocompact" conflict with existing notions.
Morgan Rogers (he/him) (Dec 09 2020 at 11:55):[Mod] Morgan Rogers said:
I spent many hours today writing my answer to a stackexchange question about cocompactness, or more specifically the strong cocompactness which is the dual of *finite presentability*. It was interesting to discover that the strongly cocompact topological spaces coincide with the finite discrete spaces.
It was perhaps more interesting to discover (thanks to @Reid Barton pointing out an erroneous claim which I hadn't proved) that in fact the only strongly cocompact space is the singleton space!