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nLab > nLab General Discussions: syntactic category

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2014
    • (edited Nov 24th 2014)

    On g+ here is a question about the nnLab entry syntactic category asking for references. (“Who originally proved the universal property given there?”)

    I wish people would know to take such questions to the nnForum. Certainly I would hope that “steering committee” members would do so! (hint :-)

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 25th 2014

    Ok, I was just curious if anyone on G+ who doesn’t frequent the forum might know. Next suggestion would be the categories mailing list, or possibly HoTT list, as there are a bunch of people there with relevant history.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 7th 2016
    • (edited May 7th 2016)

    Is that result in syntactic+category#group proved somewhere online? I mean the one that claims that the category of contexts for the theory of group is the opposite of the category of finitely presented groups (disguised in rot13)?

    The page links to Lawvere theory, but that just relates the Lawvere theory of groups to the opposite of the category of free groups.

    I was looking there really to find out what is known about the structure of (slices of) the category of contexts for a Martin-Löf type theory, pursing the modal thoughts here, where morphisms aren’t just context extensions but any morphisms of contexts.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 7th 2016

    It should be free groups on finite sets, since a Lawvere theory only uses finite products. I learned this from Jim Dolan, so I don’t know a reference. It probably goes back to Lawvere…

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 7th 2016

    Is there not a difference? The objects of the category of contexts are contexts, such as Δ=a:G,b:G,(ab) 2=a 2b 2\Delta \;=\; a\colon G,\; b\colon G,\; (a b)^2 = a^2 b^2, while the objects of the Lawvere theory of groups are just the powers of the single generating object.

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 7th 2016
    • (edited May 7th 2016)

    I mean the Lawvere theory should be finitely presented free groups, I’m unsure of the relation to the category of contexts.

    Edit: now I check Lawvere theory I see that it is using free groups on finite sets, but just doesn’t specify so after the first slightly ambiguous mention.

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 7th 2016

    Clearly Lawvere theory pertains to the wrong doctrine in this example, since the doctrine one has in mind here is that of finite limits, not finite products. I’m not sure what is the best page to link to instead of Lawvere theory. Finite limit theories are closely related to Gabriel-Ulmer duality though.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeMay 7th 2016

    One place the general result can be found is Part D of the Elephant (the category of contexts for a finite-limit theory is the opposite of the category of finitely-presented models in Set).

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