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\begin{document}

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\section*{definable set}

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{ind_and_prodefinable_sets}{Ind- and pro-definable sets}\dotfill \pageref*{ind_and_prodefinable_sets} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A [[definable set]] of an $\mathcal{L}$-theory $T$ in a [[first-order language]] is an [[equivalence class]] of [[formulas]] which evaluate the same way in every model of $T$. (In the case that $T$ is empty, $\mathbf{Mod}(T)$ is just the class of $\mathcal{L}$-structures.)

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

Let $L$ be a first-order language and $T$ a theory in $L$. Any formula $\phi(x_1,\ldots,x_n)$, whose only free variables could be among $x_1,\ldots,x_n$, together with a [[model]] $M$ for $T$ evaluates to a truth value on each $a = (a_1,\ldots,a_n)\in  M^n$, hence it determines the subset $\phi(M)\subset M^n$ of all $a$ such that $\phi(a)$. Two formulas $\phi,\psi$ with the same number of free variables are equivalent if $\phi(M) = \psi(M)$ for every model $M$ of $T$. A \textbf{definable set} $X$ in $T$ is an equivalence class of formulas in $L$ under this relation. Consider the category $\mathcal{M}_{el}(L)$ of $L$-[[structure in model theory|structures]] and [[elementary embedding|elementary embeddings]] and its full subcategory $\mathcal{M}_{el}(T)$ whose objects are the models of $T$. We can also view a \textbf{definable set} for a theory $T$ as a [[functor]] $\mathcal{M}_{el}(T)\to Set$ which is of the form $M\mapsto\phi(M)$ for some $L$-formula $\phi$.

By $X(M)$ denote the set of all $a\in M$ such that $X(a)$ holds, i.e. $\phi(a)$ is true for any choice of representative $\phi\in X$.

Given two definable sets $X,Y$ in $T$ a \textbf{definable function} $f: X\to Y$ is a definable subset of the product sort $X\times Y$ such that $f_M\in X(M)\times Y(M)$ is a function (we also write $f_M : X(M)\to Y(M)$) for every model $M$ of $T$. Analogously a \textbf{definable equivalence relation} is a definable subset $R\subset X\times X$ such that $R(M)$ is an equivalence relation for every model $M$ of $T$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\textbf{Proposition.} Given a small category $\mathcal{M}$ and two functors $F,G:\mathcal{M}\to Set$ the natural transformations $\alpha : F\to G$ are in 1-1 correspondence with functors $\tilde\alpha : \mathcal{M}\to Set$ such that $\tilde\alpha(A) \subset  F(A)\times G(A)$ and such that $\tilde\alpha(A)$ is a functional relation.

Proof. If $\alpha:F\to G$ is an $Ob(\mathcal{M})$-indexed family with components $\alpha_A: F(A)\to G(A)$ viewed as functional relations $\tilde\alpha_A\subset F(A)\times G(A)$, then the extension to a functor means\newline that there is a (automatically unique) dotted arrow $\tilde\alpha_f$

\begin{displaymath}
\itexarray{
\tilde\alpha_A & \stackrel{\tilde\alpha_f}\hookrightarrow & \tilde\alpha_B\\
\downarrow && \downarrow \\
F(A)\times G(A)&\stackrel{F(f)\times G(f)}\longrightarrow & F(B)\times G(B)
}
\end{displaymath}
making the diagram commutative (once the existence of $\tilde\alpha_f$ is known, the functoriality of $\tilde\alpha_{f\circ g} = \tilde\alpha_f\circ\tilde\alpha_g$ comes by uniqueness of $\tilde\alpha_f$ and the functoriality of $F\times G$ which it restricts from). That means that for every $x\in F(A)$ $(x,\alpha_A(x))$ should be sent to $(F(f)(x), G(f)(\alpha_A(x)))$ by $\tilde\alpha_f$, what is a restriction of $F(f)\times G(f)$, but by $\alpha_B$ being \emph{functional} relation this must be the same as $(F(f)(x), \alpha_B(F(f)(x)))$, i.e. that $G(f)(\alpha_A(x))=\alpha_B(F(f)(x))$. Q.E.D.

In other words, functoriality of $\tilde\alpha$ is the same as $\alpha$ being natural.

\textbf{Corollary.} Definable functions for $L$ (for $T$) are in 1-1 correspondence with natural transformations of \textbf{definable sets} as functors $\mathcal{M}_{el}(L)\to Set$ (resp. $\mathcal{M}_{el}(T)\to Set$).

For a fixed (multi-sorted, first-order) theory $T$, definable sets and definable functions form a category $Def(T)$. This category (or any equivalent category) is the [[syntactic category]] of the theory. Models of $T$ can be recovered as left exact functors $Def(T)\to Set$. Notice that definable sets are functors from models to sets, and models are functors definable sets to sets; just the latter are the ``evaluation functionals'' among all functors.

See also [[definable groupoid]].

An $\infty$-definable set is an intersection of definable sets.

We can also consider a \emph{relative} version of a definable set (definability with parameters).

Given definable sets $X,Y$, a \emph{fixed} model $M$ and a \emph{fixed} set $A\subset X(M)$, we say that an element $b\in Y(M)$ is definable over $A$ if there is $(a_1,\ldots,a_n)\in A^n$ and a $T$-definable function $f:X^n\to Y$ such that $f(a_1,\ldots,a_n)=b$. All elements $b$ definable over $A$ form the subset $Y(A)\subset Y(M)$. A subset $B\subset M$ is \textbf{definably closed} if it is closed under definable functions. Given $A\subset M$, there is the minimal definably closed subset $B$ such that $A\subset B\subset M$, the \textbf{definable closure} $B = dcl(A)$ of $A$.

If we extend the language by the constants from $A$ we get a new language $L_A$. The definable sets in language $L$ over $A$ are the definable sets in language $L_A$ over $\emptyset$.

\hypertarget{ind_and_prodefinable_sets}{}\subsection*{{Ind- and pro-definable sets}}\label{ind_and_prodefinable_sets}

One can also study [[ind-object]]s and [[pro-object]]s in the category of definable sets and definable functions.

\begin{defn}
\label{type-definable-set}\hypertarget{type-definable-set}{}
An important special case of a pro-definable set that shows up in model theory are \textbf{type-definable sets}, which are just infinite conjunctions of definable sets. (There is no restriction that the formulas representing these definable sets all sit inside a finite product of [[type|sorts]].)

\end{defn}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[constructible set]]


\item [[recursive subset]]


\item [[computable set]]


\item [[definable groupoid]]



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

\begin{itemize}%
\item Moshe Kamensky, \emph{Ind- and Pro- definable sets}, \href{http://arxiv.org/abs/math/0608163}{math.LO/0608163}


\item Jan Holly, \emph{Definable operations on sets and elimination of imaginaries}, Proc. Amer. Math. Soc. \textbf{117} (1993), no. 4, 1149--1157, \href{http://www.ams.org/mathscinet-getitem?mr=1116261}{MR93e:03052}, \href{http://dx.doi.org/10.2307/2159546}{doi}, \href{http://www.ams.org/journals/proc/1993-117-04/S0002-9939-1993-1116261-6/S0002-9939-1993-1116261-6.pdf}{pdf}


\item [[David Kazhdan]], Lecture notes in model theory, \href{http://www.ma.huji.ac.il/~kazhdan/Notes/motivic/b.pdf}{pdf}


\item D. Haskell, E. Hrushovski, H.D.Macpherson, \emph{Definable sets in algebraically closed valued fields: elimination of imaginaries}, J. reine und angewandte Mathematik \textbf{597} (2006), \href{http://www.ams.org/mathscinet-getitem?mr=2264318}{MR2007i:03040}, \href{http://dx.doi.org/10.1515}{doi}/CRELLE.2006.066)



\end{itemize}
[[!redirects definable sets]] [[!redirects pro-definable set]] [[!redirects pro-definable sets]] [[!redirects ind-definable set]] [[!redirects ind-definable sets]] [[!redirects type-definable set]] [[!redirects type-definable sets]] [[!redirects definable subset]] [[!redirects definable subsets]]



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