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The page Chu construction claims that
Pontryagin duality is fully embedded in the larger duality which obtains on $Chu(Top, S^1)$, where $Top$ is a nice category of spaces.
It’s not entirely clear to me how this happens: where does the abelian group structure come from? Is the claim that an abelian group structure on a sufficiently nice space $G$ is completely determined by the topology of $G$, the topology of its dual group $\widehat{G}$, and the evaluation map $\widehat{G}\times G\to S^1$? Should that be obvious?
The paper of Barr cited at Pontryagin dual instead embeds Pontryagin duality in a “separated” form of the Chu construction $Chu(Ab,S^1)$.
I probably wrote that. I’m pretty sure I must have had Barr’s paper in mind. Since the sentence is so short, I think it can be regarded as essentially a typo, which should be fixed.
That was my first thought too. But then I looked again at the preceding examples where, for instance, Stone duality embeds in $Chu(Set,2)$, so that for instance the algebraic structure of a Boolean algebra is captured by the “2-valued characters”, and wondered whether something similar might be going on here. However, if you can’t think of a way in which that would work, I’ll go ahead and change the page.
…although I’m not sure exactly what it should say either. Barr’s paper doesn’t use the full Chu construction $Chu(Ab,S^1)$, only a subcategory of it where the pairing separates points on both sides, and that’s something that hasn’t been discussed on this page. Would Pontryagin duality also be embedded in something larger but easier to describe, like $Chu(TopAb,S^1)$?
I’d have to think of it some more. Somewhere in some of Pratt’s notes (for his 1999 Summer Workshop at Comimbra) he mentions vector spaces over F_2 as embedding into Chu(Set, 2) [pages 16-17], and so a fleeting thought might be that something similar could work for Chu(Top, S^1), but that’s only a half-assed thought. It could be, as you say, that something along the lines of $Chu(TopAb, S^1)$ would work.
Is $TopAb$ symmetric monoidal closed? And if so, is Pontryagin duality its actual internal-hom into $S^1$?
I was thinking of a sm closed variant of Top to work with, like compactly generated, then passing to the algebras of a commutative monad on Top to get the answer ’yes’. Since locally compact Hausdorff spaces are compactly generated, I think we get the answer ’yes’ to your second question, using compact-open topologies as usual. But I just got out of a nap, and maybe I’m not fully alert yet.
That does seem to me as though it should work. According to our page compact-open topology, the compact-open topology agrees with the internal-hom in compactly generated spaces when the domain is compactly generated Hausdorff. Maybe I’ll put that version on the page, along with a pointer to Barr’s paper and maybe a ponderment about $Chu(Top,S^1)$.
Cool; thanks Mike!
There is a discussion on MathOverflow of this claim:
Armed with just this much knowledge, and knowledge of how star-autonomous categories behave (as categorified versions of Boolean algebras, or perhaps better Boolean rigs)
It was added to the article in Revision 2, way back in 2009.
Deleted the false claims about *-autonomous categories categorifying Boolean algebras or rigs. Mentioned linear logic instead.
added the original reference:
Previously the only reference given in the Idea-section was to Pratt, with the words
The construction has been extensively developed by Pratt 1999
No mentioning was (and is) made of Barr’s development of the theory. Is this intentional? I am not an expert on the subject, but just looking at the number of early articles that Barr has on the subject, it seems odd.
added pointer to
and reworded the first couple of paragraph of the Idea-section.
moving this old query-box discussion out of the entry:
+–{.query} Hi Toby; could I get you to explain the aside about Boolean rigs above? I’m thinking Boolean algebras is appropriate, as we have $I \to x^* \wp x$, $x \otimes x^* \to D$ [where $\wp$ denotes Girard’s “par” and $D$ denotes the dualizer], together with appropriate triangular equations, categorifying the inequalities $1 \leq (\neg x) \vee v$ and $x \wedge (\neg x) \leq 0$ in a Boolean algebra. —Todd
Now that I go to write Boolean rig, I'm not so sure. I just know that $Chu(P X,\empty)$ at measurable space is not (even classically) a Boolean algebra. I'll get back to you in a day or less. —Toby
Right, I agree. The Chu construction applied to a complete Heyting algebra is merely a $*$-autonomous quantale, not a $*$-autonomous frame (which would be a complete Boolean algebra), as you noted at measurable space. —Todd =–
Hmm, I think then that the aside should also be removed. Or can someone explain it?
Edit: Never mind; I see Dmitri removed the bit about Boolean rig last year.
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