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nLab > Latest Changes: Łoś ultraproduct theorem

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    • CommentRowNumber1.
    • CommentAuthorjesse
    • CommentTimeJan 4th 2017

    To fix a grey link, I started a page for the Łoś theorem.

    On a side note, should the pages ultrapower and ultraproduct maybe be merged?

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 4th 2017

    I support the merge of ultrapower and ultraproduct. Thanks for your work!

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJan 5th 2017

    Thanks! I don’t understand the proof of the theorem given though: what is a “point of a definable set” and how does it relate to the truth of a first-order formula? Is excluded middle being silently used somewhere to make our pretopos morphism into a Heyting category morphism?

    • CommentRowNumber4.
    • CommentAuthorjesse
    • CommentTimeJan 5th 2017
    • (edited Jan 5th 2017)

    @Mike: In the proof, ’taking points of X inside a model M’ means ’evaluating the functor M at X’. (The terminology’s motivated by thinking of X as an algebraic variety and M as a field.)

    The proof as written is a little confusing. The only thing that needs to be done is to read off how the subobject *M(X)^*M(X) of the filtered colimit *M^*M is computed: it’s the colimit of subobjects ∏ j∈SM j(X)\prod_{j \in S} M_j(X) of ∏ j∈JM j\prod_{j \in J} M_j (for S∈𝒰S \in \mathcal{U}), so for a sequence (x i) i∈I(x_i)_{i \in I}, the germ [(x i)][(x_i)] is in *M(X)^*M(X) if and only if (x i) i∈I(x_i)_{i \in I} is sent by one the transition maps of the colimit (which is just truncation to a subset of indices inside 𝒰\mathcal{U}) to one of the ∏ j∈SM j(X)\prod_{j \in S} M_j(X) (for some S∈𝒰S \in \mathcal{U}).

    • CommentRowNumber5.
    • CommentAuthorjesse
    • CommentTimeJan 5th 2017
    • (edited Jan 5th 2017)

    I guess you were also asking how one gets the statement of the Los’ theorem for sentences. I think one gets around this by identifying sentences with their sort (type), i.e. a true sentence ψ\psi is equivalent in the syntactic category to the formula ψ∧(x=x)\psi \wedge (x = x), so some object Y(ψ)Y(\psi) in Def(T)\mathbf{Def}(T).

    So we test the truth of ψ\psi in *M^*M by seeing if the quotient (whence pretopos, and writing *Y^*Y for *M(Y(ψ))^*M(Y(\psi)) and Y iY_i for M i(Y(ψ))M_i(Y(\psi)))

    *Y(ψ)/ *Y(ψ)^* Y(\psi) / ^*Y(\psi)

    is nonempty, which corresponds to the sequence

    ((Y i(ψ)/Y i(ψ)) i∈I\left((Y_i(\psi) / Y_i(\psi)\right)_{i \in I}

    being supported on a subset of indices that are in 𝒰\mathcal{U}.

    • CommentRowNumber6.
    • CommentAuthorThomas Holder
    • CommentTimeDec 9th 2020

    Added a refernce to

    • Hisashi Aratake, Sheaves of Structures, Heyting-Valued Structures, and a Generalization of Łoś’s Theorem , arXiv:2012.04317 (2020). (abstract)

    diff, v7, current

    • CommentRowNumber7.
    • CommentAuthorJem Lord
    • CommentTimeMay 17th 2022

    Noted where choice is used, and fixed the colimit part of the proof (“colimits commute with colimits” only works when the indexing category of the inner colimit doesn’t depend on the object used in the outer).

    diff, v8, current

    • CommentRowNumber8.
    • CommentAuthorShamrock
    • CommentTimeJul 11th 2024
    • (edited Jul 11th 2024)
    Does anyone know if the genralization of Proposition 2.2 to ultraproducts in the infinity category of spaces holds? I think the same argument works for preservation of finite limits, the initial object, and coproducts, but I'm a little concerned about the quotients of equivalence relations (which should turn into the statement that geometric realizations of groupoid objects are preserved, I guess?)

1 to 8 of 8

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