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\section*{ultraproduct}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{model_theory}{}\paragraph*{{Model theory}}\label{model_theory}

[[!include model theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{settheoretic}{Set-theoretic}\dotfill \pageref*{settheoretic} \linebreak
\noindent\hyperlink{ultraproducts_of_structures}{Ultraproducts of structures}\dotfill \pageref*{ultraproducts_of_structures} \linebreak
\noindent\hyperlink{ultraproducts_of_models}{Ultraproducts of models}\dotfill \pageref*{ultraproducts_of_models} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{in_functional_analysis}{In functional analysis}\dotfill \pageref*{in_functional_analysis} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \textbf{ultraproduct} construction is an important tool in [[model theory]] that permits to produce a new structure from an infinite family of structures. The construction has a decidedly `algebraic' flavor and hence occurs naturally in applications of model theory to [[algebra]].

It is closely related via the [[?o? ultraproduct theorem]] to the [[compactness theorem]]: suppose one is given a set of formulas $S$ in some first-order language and a family of structures such that any finite subset of formulas is modeled by all but a finite number of structures. Then an ultraproduct of those structures may be used to model the entire set $S$.\footnote{Due to the outstanding importance of the compactness theorem, it is possible to prove `almost all' results in model theory by the use of ultraproducts. This approach to model theory is pursued in \hyperlink{BellSlomson}{Bell-Slomson (1969)}. For contrast, compare with the more sober view of \hyperlink{Hodges93}{Hodges (1993)}.} 

In slightly greater detail, given a family of [[structures]] of the same signature in the sense of [[model theory]] (or more specially, [[universal algebra]]), one can (assuming the [[ultrafilter principle]], a weak form of the [[axiom of choice]]) use [[ultrafilters]] to form a certain [[congruence]] on the [[direct product]] and construct a [[quotient object]], a new structure of the same signature, called an \textbf{ultraproduct}. As long as the ultrafilter is [[free ultrafilter|free]] (contains the filter of cofinite subsets), the last sentence of the preceding paragraph is validated.

An ultraproduct of some number of copies of the \emph{same} structure is called an \textbf{ultrapower}.

Another important facet, somewhat reminiscent of [[Morita equivalence]], is the [[Keisler-Shelah theorem]]: Two structures are elementary equivalent $\mathfrak{A}\equiv\mathfrak{B}$ iff they have isomorphic ultrapowers: $\mathfrak{A}^I / \mathcal{U} \cong \mathfrak{B}^I / \mathcal{U}$ for some set $I$ and ultrafilter $\mathcal{U}$. This deep result says that the partially syntactic concept of \emph{elementary equivalence} can be characterized in purely semantic form with the help of ultrapowers.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

First we present the bare-bones set-theoretic construction; then we discuss structures and models.

\hypertarget{settheoretic}{}\subsubsection*{{Set-theoretic}}\label{settheoretic}

\begin{defn}
\label{}\hypertarget{}{}
Let $X$ be a set, and let $U$ be an [[ultrafilter]] on $X$, which may be regarded as an element $U: 1 \to \beta X$ of the [[Stone-Cech compactification]] $\beta X = \hom_{Bool}(2^X, 2)$ with its usual topology. The \textbf{ultrapower functor} over $U$ is the functor

\begin{displaymath}
Set/X \simeq Sh(X_{disc}) \stackrel{i_\ast}{\to} Sh(\beta X) \stackrel{U^\ast}{\to} Set
\end{displaymath}
where $i_\ast$ is the [[direct image functor]] between categories of [[sheaves]] induced from the [[inclusion map|inclusion]] $i: X \to \beta X$ of $X$ as a [[discrete space|discrete]] [[subspace]], and the [[inverse image functor]] $U^\ast$ is also known as the ``taking the [[stalk]]'' functor at the point $U \in \beta X$.

\end{defn}
Let us extract a more concrete description. Let $\{Y_x: x \in X\}$ be an $X$-indexed family of sets, which we can view as an object $Y \to X$ of $\Set/X$. We may view this object as a sheaf over $X$ as discrete space; as a presheaf, it takes an open set (an arbitrary subset $A \subseteq X$) to the set of sections over $A$ which is $\prod_{x \in A} Y_x$. The direct image functor $i_\ast$ takes this presheaf to the presheaf which sends an arbitrary open set $\mathcal{V} \subseteq \beta X$ to $\prod_{x \in X \cap \mathcal{V}} Y_x$, and then the stalk functor $U^\ast$ sends this to the [[filtered colimit]]

\begin{displaymath}
colim_{\mathcal{V} \ni U} \prod_{x \in X \cap \mathcal{V}} Y_x
\end{displaymath}
and since the basic opens $\mathcal{V}_A = \{U' \in \beta X: A \in U'\}$ with $A \in U$ are [[cofinal functor|cofinal]] in the system of open neighborhoods of $U$ (ordered by reverse inclusion), the colimit may be written more simply as

\begin{displaymath}
colim_{A \in U} \prod_{x \in A} Y_x
\end{displaymath}
and this is the \emph{ultraproduct}, often written as $\prod_{x \in X} Y_x/U$.\footnote{The quotient notation is traditional but (ever so slightly) misleading. If all the $Y_x$ are [[inhabited sets]], then all the maps $\prod_{x \in A} Y_x \to \prod_{x \in B} Y_x$ ($B \subseteq A$) are quotient maps and the ultraproduct is a quotient of the full product $\prod_{x \in X} Y_x$; there the notation is apt. If one of the $Y_x$ is empty, then the full product is empty and thus the ultraproduct is not such a quotient. But it is useful to allow for empty models (when they exist)! The correct definition which works for all cases is as we have it: a filtered colimit over restricted products. See \hyperlink{Makkai1980}{p. 186} and \hyperlink{MathOverflow2012}{this MO Discussion} for a discussion of this topic.}  When all the fibers $Y_x$ are the same set $Z$, this is written as $Z^X/U$ and called an \emph{ultrapower} of $Z$.

\hypertarget{ultraproducts_of_structures}{}\subsubsection*{{Ultraproducts of structures}}\label{ultraproducts_of_structures}

Now suppose each $Y_x$ carries a structure specified by a [[signature (in logic)|signature]]; in other words, for each function symbol $f$ and predicate symbol $R$ of the signature there are specified operations and subsets

\begin{displaymath}
\omega_f: Y_x^{ar(f)} \to Y_x, \qquad \rho_R \rightarrowtail Y_x^{ar(R)}
\end{displaymath}
where $ar$ denotes arity (constants being considered function symbols of arity zero). Then of course the products $\prod_{a \in A} Y_x$ carry structures canonically induced from those on the $Y_x$. As all of the function symbols $f$ and predicate symbols $R$ have finite arity, and since the taking of filtered colimits commutes with finite limits in $Set$, we get a canonically induced structure on the ultraproduct $\prod_{x \in X} Y_x/U$. More briefly: since $U^\ast i_\ast$ preserves finite limits, it takes an $X$-indexed finitary structure $Y_x$ in $Set/X$ to a finitary structure of the same type in $Set$.

Intuitively (and adopting language used by Lawvere), what is going on is that we have a variable structure varying over some domain $X$, and we consider a point $U$ of its compactification $\beta X$ as some kind of ``ideal point at infinity'', and then we ``freeze'' (or localize) the variation by passing to germs of the variation at that point $U$.

Notice that if we replace the word ``ultrafilter'' by the word ``filter'', the general structural facts mentioned here would remain true. That is to say: if $F$ is a filter consisting of subsets of $X$, then the colimit $colim_{A \in F} \prod_{x \in A}$ (with such $A$ ordered by reverse inclusion) is still directed or filtered, and we would still get a structure of given finitary type by passing to the colimit. Here, instead of ultrapowers and ultraproducts, one speaks of \emph{reduced powers} and \emph{reduced products}.

\hypertarget{ultraproducts_of_models}{}\subsubsection*{{Ultraproducts of models}}\label{ultraproducts_of_models}

To be written.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The [[hyperreal number]]s (\href{http://en.wikipedia.org/wiki/Hyperreal_numbers}{wikipedia}) and nonstandard integers in [[nonstandard analysis]] are obtained as [[denumerable set|countable]] ultrapowers with help of [[free ultrafilters]] on $\mathbb{N}$. Such ultrafilters contain all [[cofinite subset|cofinite]] subsets of integers, but not only them. See \href{http://en.wikipedia.org/wiki/Ultraproduct}{wikipedia:ultraproduct}.

From Michael Barr's \href{http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1986__27_2/CTGDC_1986__27_2_93_0/CTGDC_1986__27_2_93_0.pdf}{Models of Sketches}

\begin{quote}%
Unlike [[limits]] and [[colimits]], an ultraproduct is not defined by any [[universal mapping property]]. Of course, if the category has limits and ([[filtered colimit|filtered]]) colimits, then it has ultraproducts constructed as colimits of [[products]] \ldots{} But usually the category of models of a [[coherent theory]] (such as the theory of [[discrete field|fields]]) lacks products and hence does not have categorical ultraproducts.


\end{quote}
\hypertarget{in_functional_analysis}{}\subsubsection*{{In functional analysis}}\label{in_functional_analysis}

(Appropriately-defined) ultrapowers of [[Banach space|Banach spaces]] allow one to embed an infinite-dimensional space $X \hookrightarrow X^{\mathcal{U}}$ inside a larger Banach space such that for every bounded family $(x_i)_{i \in I}$ and every ultrafilter $\mathcal{U}$ on $I$, the $\mathcal{U}$-limit $\lim_{i, \mathcal{U}}$ exists (just take the [[germ]] $[(x_i)]$).

There are various ways to formalize this: Henson's [[positive bounded logic]], [[continuous logic]], or ad hoc variants (e.g. ``Banach space structures'', ``normed space structures'', etc.)

As one might expect, things which are only ``approximately true'' inside $X$ become ``exactly true'' in $X^{\mathcal{U}$. For example, if $\lambda$ is an [[approximate eigenvalue]] (e.g. on the boundary of the [[spectrum of an operator|spectrum]]) for a bounded operator $T$, then the germ of the sequence of approximate eigenvectors of $\lambda$ becomes a \emph{bona fide} [[eigenvector]] with eigenvalue $\lambda$ for $T^{\mathcal{U}}$ on $X^{\mathcal{U}}$.

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[model theory]]


\item [[ultrafilter monad]]


\item [[large cardinal]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Standard references in model theory for ultraproducts are

\begin{itemize}%
\item [[John Bell|J. L. Bell]], A. B. Slomson, \emph{Models and Ultraproducts: An Introduction} , North-Holland Amsterdam 1969. (Dover reprint)


\item P. C. Eklof, \emph{Ultraproducts for Algebraists} , pp.105-137 in Barwise (ed.), \emph{Handbook of Mathematical Logic} , Elsevier Amsterdam 1977.



\end{itemize}
For a more recent textbook treatment see

\begin{itemize}%
\item [[Wilfrid Hodges]], \emph{Model Theory} , Cambridge University Press 1993. (sec. 9.5)

\end{itemize}
The categorical perspective on ultraproducts goes back to

\begin{itemize}%
\item T. Okhuma, \emph{Ultrapowers in categories} , Yokohama Math. J. \textbf{14} (1966) pp.17-37.


\item S. Fakir, L. Haddad, \emph{Objets coh\'e{}rents et ultraproduits dans les cat\'e{}gories} , Journal of Algebra \textbf{21} (1972) pp.410--421.



\end{itemize}
For the vacuum engineering point of view of `freezing variation' \footnote{This perspective was apparently inspired by joined work with [[Miles Tierney]] on the use of [[forcing]] in the context of the [[continuum hypothesis]] in early topos theory.}  in the context of non-standard analysis see

\begin{itemize}%
\item [[F. William Lawvere]], \emph{Variable sets etendu and variable structure in topoi} , Lecture notes University of Chicago 1975.

\end{itemize}
The following papers are relevant for understanding the ultraproduct construction via [[codensity monad|codensity monads]]:

\begin{itemize}%
\item D. P. Ellerman, \emph{Sheaves of structures and generalized ultraproducts} , Annals of Mathematical Logic \textbf{7} (1974) pp.163--195.


\item J. F. Kennison, \emph{Triples and compact sheaf representation} , JPAA \textbf{20} (1981) pp.13-38.


\item [[Tom Leinster]], \emph{Codensity and the Ultrafilter Monad} , TAC \textbf{28} no. 13 (2013) pp.332-370. (\href{http://www.tac.mta.ca/tac/volumes/28/13/28-13abs.html}{tac})



\end{itemize}
For a fine point concerning the definition of ultraproducts:

\begin{itemize}%
\item \href{https://mathoverflow.net/q/105397}{MathOverflow2012}


\item [[M. Makkai]], \emph{On full embeddings} , JPAA, 1980



\end{itemize}
See also

\begin{itemize}%
\item [[M. Makkai]], \emph{Ultraproducts and categorical logic} , Lectures Notes in Math. \textbf{1130}, Springer 1985, pp. 222--309.

\end{itemize}
For ultraproducts in functional analysis see e.g.

\begin{itemize}%
\item Jose Iovino, \emph{Applications of model theory to functional analysis}

\end{itemize}
[[!redirects ultraproduct]] [[!redirects ultraproducts]] [[!redirects ultrapower]] [[!redirects ultrapowers]] [[!redirects reduced product]] [[!redirects reduced power]] [[!redirects reduced products]] [[!redirects reduced powers]]



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