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nLab > Latest Changes: Chu construction

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  1.  
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeJun 27th 2018

    Added link to Vaughan Pratt’s Linear Process Algebra.

    diff, v32, current

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeOct 13th 2019

    Added the polycategorical viewpoint.

    diff, v33, current

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeOct 14th 2019

    Added the universal property, generalized from Pavlovic to polycategories.

    diff, v34, current

  4. Fix minor typo in diagram

    Anthony Hart

    diff, v35, current

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeApr 18th 2020

    The page Chu construction claims that

    Pontryagin duality is fully embedded in the larger duality which obtains on Chu(Top,S 1)Chu(Top, S^1), where TopTop 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 GG is completely determined by the topology of GG, the topology of its dual group G^\widehat{G}, and the evaluation map G^×G→S 1\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)Chu(Ab,S^1).

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 18th 2020

    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.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeApr 18th 2020

    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)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.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeApr 18th 2020

    …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)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)Chu(TopAb,S^1)?

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 18th 2020

    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)Chu(TopAb, S^1) would work.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeApr 18th 2020

    Is TopAbTopAb symmetric monoidal closed? And if so, is Pontryagin duality its actual internal-hom into S 1S^1?

    • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 18th 2020

    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.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeApr 18th 2020

    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)Chu(Top,S^1).

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeApr 19th 2020

    Fixed up the discussion of Pontryagin duality.

    diff, v36, current

    • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 19th 2020

    Cool; thanks Mike!

    • CommentRowNumber15.
    • CommentAuthorvaleriadepaiva
    • CommentTimeFeb 22nd 2021

    minor typo

    diff, v37, current

    • CommentRowNumber16.
    • CommentAuthorvaleriadepaiva
    • CommentTimeFeb 22nd 2021

    small typo

    diff, v37, current

    • CommentRowNumber17.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 12th 2022
    • (edited Jun 12th 2022)

    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.

    • CommentRowNumber18.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 13th 2022

    Deleted the false claims about *-autonomous categories categorifying Boolean algebras or rigs. Mentioned linear logic instead.

    diff, v38, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2023

    added publication data to the references and brought them into chronological order

    diff, v39, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2023
    • (edited Jul 11th 2023)

    while I was at it, I have brushed-up some of the typesetting, such as of the commuting diagrams

    diff, v39, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2023
    • (edited Oct 27th 2023)

    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.

    diff, v41, current

    • CommentRowNumber22.
    • CommentAuthorTim_Porter
    • CommentTimeOct 27th 2023

    Typo fixed

    diff, v42, current

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeOct 28th 2023

    added pointer to

    and reworded the first couple of paragraph of the Idea-section.

    diff, v43, current

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeOct 28th 2023

    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→x *℘x I \to x^* \wp x , x⊗x *→D x \otimes x^* \to D [where ℘\wp denotes Girard’s “par” and DD denotes the dualizer], together with appropriate triangular equations, categorifying the inequalities 1≤(¬x)∨v1 \leq (\neg x) \vee v and x∧(¬x)≤0x \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(PX,∅)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 =–

    diff, v43, current

    • CommentRowNumber25.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 4th 2023
    • (edited Nov 4th 2023)

    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.

1 to 25 of 25

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