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1. • CommentRowNumber1.
   • CommentAuthorjesse
   • CommentTimeJan 4th 2017
   • PermaLink
   Author: jesse
   Format: MarkdownItexTo fix a grey link, I started a page for the [[Łoś ultraproduct theorem|Łoś theorem]]. On a side note, should the pages [[ultrapower]] and [[ultraproduct]] maybe be merged?

   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?

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 4th 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI support the merge of [[ultrapower]] and [[ultraproduct]]. Thanks for your work!

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

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJan 5th 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThanks! 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?

   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?

4. • CommentRowNumber4.
   • CommentAuthorjesse
   • CommentTimeJan 5th 2017
   • (edited Jan 5th 2017)
   • PermaLink
   Author: jesse
   Format: MarkdownItex@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 $^*M(X)$ of the filtered colimit $^*M$ is computed: it's the colimit of subobjects $\prod_{j \in S} M_j(X)$ of $\prod_{j \in J} M_j$ (for $S \in \mathcal{U}$), so for a sequence $(x_i)_{i \in I}$, the germ $[(x_i)]$ is in $^*M(X)$ if and only if $(x_i)_{i \in I}$ is sent by one the transition maps of the colimit (which is just truncation to a subset of indices inside $\mathcal{U}$) to one of the $\prod_{j \in S} M_j(X)$ (for some $S \in \mathcal{U}$).

   @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 $^*M(X)$ of the filtered colimit $^*M$ is computed: it’s the colimit of subobjects $\prod_{j \in S} M_j(X)$ of $\prod_{j \in J} M_j$ (for $S \in \mathcal{U}$), so for a sequence $(x_i)_{i \in I}$, the germ $[(x_i)]$ is in $^*M(X)$ if and only if $(x_i)_{i \in I}$ is sent by one the transition maps of the colimit (which is just truncation to a subset of indices inside $\mathcal{U}$) to one of the $\prod_{j \in S} M_j(X)$ (for some $S \in \mathcal{U}$).

5. • CommentRowNumber5.
   • CommentAuthorjesse
   • CommentTimeJan 5th 2017
   • (edited Jan 5th 2017)
   • PermaLink
   Author: jesse
   Format: MarkdownItexI 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 $\psi$ is equivalent in the syntactic category to the formula $\psi \wedge (x = x)$, so some object $Y(\psi)$ in $\mathbf{Def}(T)$. So we test the truth of $\psi$ in $^*M$ by seeing if the quotient (whence pretopos, and writing $^*Y$ for $^*M(Y(\psi))$ and $Y_i$ for $M_i(Y(\psi))$) $$^* Y(\psi) / ^*Y(\psi)$$ is nonempty, which corresponds to the sequence $$\left((Y_i(\psi) / Y_i(\psi)\right)_{i \in I}$$ being supported on a subset of indices that are in $\mathcal{U}$.

   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 $\psi$ is equivalent in the syntactic category to the formula $\psi \wedge (x = x)$, so some object $Y(\psi)$ in $\mathbf{Def}(T)$.

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

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

   is nonempty, which corresponds to the sequence

   $$\left((Y_i(\psi) / Y_i(\psi)\right)_{i \in I}$$

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

6. • CommentRowNumber6.
   • CommentAuthorThomas Holder
   • CommentTimeDec 9th 2020
   • PermaLink
   Author: Thomas Holder
   Format: MarkdownItexAdded a refernce to * Hisashi Aratake, _Sheaves of Structures, Heyting-Valued Structures, and a Generalization of Łoś's Theorem_ , arXiv:2012.04317 (2020). ([abstract](https://arxiv.org/abs/2012.04317)) <a href="https://ncatlab.org/nlab/revision/diff/%C5%81o%C5%9B+ultraproduct+theorem/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/%C5%81o%C5%9B+ultraproduct+theorem/7">v7</a>, <a href="https://ncatlab.org/nlab/show/%C5%81o%C5%9B+ultraproduct+theorem">current</a>

   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

7. • CommentRowNumber7.
   • CommentAuthorJem Lord
   • CommentTimeMay 17th 2022
   • PermaLink
   Author: Jem Lord
   Format: MarkdownItexNoted 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). <a href="https://ncatlab.org/nlab/revision/diff/%C5%81o%C5%9B+ultraproduct+theorem/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/%C5%81o%C5%9B+ultraproduct+theorem/8">v8</a>, <a href="https://ncatlab.org/nlab/show/%C5%81o%C5%9B+ultraproduct+theorem">current</a>

   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