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Labbified an MO discussion at coproduct-preserving representable.
Interesting, thanks! I added a comment about finiteness/infiniteness. Isn’t coproduct-preservation also used as a notion of “finiteness” in triangulated categories?
Thanks for the addition! It’s true that this article could use a little more ’philosophy’ or intuition on the significance of coproduct-preservation, which depending on the application would appear to pull in different directions (connectedness in the case of extensive categories, compactness/finiteness/smallness for say modules over a Noetherian ring, etc.).
I haven’t thought about the triangulated categories example. It might help me to consider some examples, like what sort of behaviors can one expect by looking at derived categories of various abelian categories. Can you say something about this?
I have added the floating toc compact object - contents
We are about at the limit of my knowledge about triangulated categories, but I think the phrase you want to search for is “compactly generated triangulated category”. Triangulated-category theorists call these “compact objects”.
What is the connection between these compact objects and ideas such as pseudocompact rings, linear compact ones ditto modules, etc. (I should know this and in asking am being lazy. :-()
There was an old Lecture Notes in Maths, that looked at linear compactness from a logical POV linking it with ultraproducts and things like that…. but I forget the name. (I have a copy but not here at home.)
Tim, your guess is as good as mine. What is annoying to me is how ill-chosen the terminology ’compact’ seems to be (say in the additive category sense), when a straightforward rewriting of the definition to $Top$ yields the notion not of compact space, but of connected space! Moreover, it refers to coproduct-preserving representables for some people, and to filtered-cocontinuous representables for others. None of these gives you compact spaces when applied to $Top$. It’s a terminological mess!
Maybe, just maybe, one can characterize compact spaces by some such condition – and maybe we should try working this out here. For example, I’m wondering whether some variation along the following lines could work: “in the category of Hausdorff spaces and closed continuous maps, a space $X$ is compact if $\hom(X, -)$ preserves filtered colimits.” It may take some tweaking to get it right.
Of course, if an additive functor on an additive category preserves filtered colimits, then it preserves coproducts, since any coproduct is a filtered colimit of finite coproducts, and the latter are absolute for additive functors. I would guess that the use of “compact” in triangulated categories came from this observation, together with the fact that triangulated categories, being homotopy categories, admit very few actual colimits aside from coproducts, so this is in some sense the strongest “shadow” of preserving filtered colimits that can be “seen” from the triangulated world. Probably you know all that. Is “compact” meaning “coproduct-preserving representable” used outside of triangulated categories? That seems to me much harder to justify.
I used to get up in arms about the use of “compact object” to mean “filtered-cocontinuous representable” since, as you say, compact spaces are not the finitely presentable objects in $Top$. But I’ve sort of given up worrying about it. A space is compact, of course, just when the top element is a filtered-cocontinuous representable in its open-set lattice, or more generally in the category of topological spaces and open inclusions. So if we think of “compactness” in this sense as primarily a general category-theoretic notion, but one which applies primarily to “algebraic” sorts of categories, then we can argue that since compact objects per se in a category like $Top$ are ill-behaved, by “compact space” we can mean a space that is compact in some other related category. It’s definitely a bit kludgy, but there are other terminological battles I’d rather spend my time fighting (and actually, as time goes by I find I have less and less desire to spend any time fighting terminological battles…).
Thanks for your observations, Mike. I thought people did use ’compact’ to mean coproduct-preserving representable, even without a whisper of triangulated categories. I’d have to check, but I think it might be in some of those old books like Freyd’s Abelian Categories, for instance.
I bring up this particular battle because Tim was querying about the relation to other notions that seem to have closer contact with the traditional topological notion of compactness (specifically, linearly compact rings and such), and my thinking was that if only that traditional notion could be brought under the umbrella we are discussing in some proper (non-kludgy) way, we would be better able to grasp the possible relations in an intuitive way. Of course, all this might be a bit chimerical (and I don’t expect anyone but myself to spend time on it).
I thought people did use ’compact’ to mean coproduct-preserving representable, even without a whisper of triangulated categories.
I haven’t encountered it, I don’t think, but you could very well be right. In any case I think I would stand by my statement that such a use is “much harder to justify”. (-: (Not that I’m especially thrilled with the usage in triangulated categories either, but that seems to me to be quite common and unlikely to go away any time soon.)
If you can find a nice way to regard compact spaces as directed-cocontinuous representables in a $Top$-like category, I think that would be very interesting (and satisfying) — I didn’t mean to sound dismissive or discouraging. You have a good point that it might actually be helpful in comparing things. I think I worked out once something along those lines for sequentially compact subsequential spaces, but I don’t remember how exact the correspondence was.
but you could very well be right
I could just as easily be wrong. :-) I misinterpreted some remarks on this at compact object; Freyd actually uses the word “small”, which I like better (although the $R$-module $K$ mentioned in the “negative results” section of coproduct-preserving representable doesn’t actually scream ’small’ to my ears (-: ). And the MO participants of that discussion might very well have been thinking of triangulated categories.
I didn’t mean to sound dismissive or discouraging
You didn’t.
Getting back to my question, I was noting that duals of Grothendieck categories (in the sense of abelian categories) involve compactness in the linear sense. see U. Oberst, Duality theory for Grothendieck categories, Bull. Amer. Math. Soc. Volume 75, Number 6 (1969), 1401-1407.
see also this MO question
I once got a ‘certain distance’ into an investigation on the characterisation of those commutative rings for which the derived functors of the limit functor vanished on systems of fg-modules from some point $n$ onwards, so the holim was an n-type or something like that. Chris Jensen had done the case n=1 in his lecture note on the $lim^i$ and it had involved compactness of the ring. This raises the question as to when although the ’coproducts etc’ are preserved less strictly. i.e. in the linearly compact case, the limit functor is exact, but in the next case the $lim^k$ are zero from $k=2$ onwards (on the relevant type of modules). This would correspond to a categorification of compactness, perhaps.
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