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\section*{model-theoretic Galois theory}

\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{the_case_of_finite_extensions}{The case of finite extensions}\dotfill \pageref*{the_case_of_finite_extensions} \linebreak
\noindent\hyperlink{the_case_for_infinite_extensions}{The case for infinite extensions}\dotfill \pageref*{the_case_for_infinite_extensions} \linebreak
\noindent\hyperlink{the_modeltheoretic_absolute_galois_group}{The model-theoretic absolute Galois group}\dotfill \pageref*{the_modeltheoretic_absolute_galois_group} \linebreak
\noindent\hyperlink{the_fundamental_theorem_of_modeltheoretic_galois_theory}{The fundamental theorem of model-theoretic Galois theory}\dotfill \pageref*{the_fundamental_theorem_of_modeltheoretic_galois_theory} \linebreak
\noindent\hyperlink{model_theory_and_the_tannakian_formalism}{Model theory and the Tannakian formalism}\dotfill \pageref*{model_theory_and_the_tannakian_formalism} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{discussion}{Discussion}\dotfill \pageref*{discussion} \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}

Classical [[Galois theory]] is about the classification of intermediate field extensions in a (field-theoretic) [[algebraic closure]] of a ground field according to the structure of a [[profinite group|profinite automorphism group]].

This can be adapted to the setting of a [[monster model|universal domain]] of any [[first-order theory]] which [[elimination of imaginaries|eliminates imaginaries]]: we can analogously classify [[definable closure|definably closed]] subsets of a [[definable closure|model-theoretic algebraic closure]] of some parameter set according to the structure of a profinite automorphism group.

In the language of [[Grothendieck's Galois theory]], we can sketch the next two sections as: the category of finite $A$-definable sets in a [[monster model]] $\mathbb{M}$ equipped with the forgetful functor to [[Set]] is a [[Galois category]].

\hypertarget{the_case_of_finite_extensions}{}\subsection*{{The case of finite extensions}}\label{the_case_of_finite_extensions}

It will be instructive to look at the case of finite extensions first, as the case of infinite extensions will be a version of the same argument but souped-up with formalities about profinite groups.

Recall the fundamental theorem of Galois theory for finite extensions of fields:

\textbf{Theorem.} For $L/K$ a finite, [[separable field extension|separable]], [[normal field extension]], there is an order-reversing bijective correspondence

\begin{displaymath}
\operatorname{Sub}_{\mathbf{Grp}} \left( \operatorname{Aut}(L/K)\right) \leftrightarrows \operatorname{Int} \left(L/K\right)
\end{displaymath}
between the subgroups of the group of field automorphisms of $L$ fixing $K$ pointwise, and the intermediate field extensions between $K$ and $L$. The correspondence is given by sending a subgroup to its field of fixed points, and an intermediate extension to its stabilizer subgroup.

This can be translated to general model-theoretic language as follows: we work in a monster model $\mathbb{M} \models T$ for a complete first-order theory which [[elimination of imaginaries|eliminates imaginaries]]. Let $A$ be a small parameter set in $M$. We have the following

\textbf{Dictionary}, between our general $T$ eliminating imaginaries and the special case $T = [[ACF]]$ the theory of an algebraically closed field.

\begin{itemize}%
\item $A \supseteq K$ corresponds to ``$A$ is an extension of $K$''.
\item If $A$ is an extension of $K$, the [[definable closure]] of $A$ corresponds to the [[perfect field|perfect hull]] of the field $K(A)$ generated by adjoining $A$.
\item If $A$ is an extension of $K$, the [[algebraic closure]] of $A$ corresponds to the separable closure of the field $K(A)$ generated by adjoining $A$.
\item If $A$ is a definably closed extension of $K$ with $A = \operatorname{dcl}(\gamma \cup K)$ for $\gamma$ finite, this corresponds to $K(A)$ being finitely-generated.
\item The [[orbit]] of an element $\ell \in \mathbb{M}$ under $\Aut(\mathbb{M}/K)$ corresponds to conjugates of $\ell$ in $K(\ell)$ under $\operatorname{Aut}(K(\ell)/K)$.
\item The size of this orbit corresponds to the degree of the field extensions $K(\ell)/K$.
\item If this orbit is finite, $\ell$ is said to be \emph{algebraic} over $K$.
\item The condition ``$\forall \ell \in L \supseteq K$, $\operatorname{Orb}_{\operatorname{Aut}(\mathbb{M}/L)}(\ell) \subseteq L$'' corresponds to ``$L$ is a normal extension of $K$''.

\end{itemize}
When $L$ is a normal extension of $K$, $L$ splits into $\operatorname{Aut}(\mathbb{M}/K)$-orbits, and so the latter group acts via restriction on $L/K$. We call the image of the induced group homomorphism $\operatorname{Aut}(\mathbb{M}/K) \to \Sym(L/K)$ $\operatorname{Aut}(L/K)$.

With this in place, we can show:

\textbf{Theorem.} Let $K$ be a definably closed parameter set. Let $A$ be a normal extension of $K$ generated by the finite algebraic tuple $\gamma$. Then there is an order-reversing bijective correspondence between the subgroups of $\operatorname{Aut}(A/K)$ and the definably closed intermediate extensions of $A/K$. The correspondence is given by maps $\mathsf{Fix}$ sending a subgroup to its fixed points and $\mathsf{Stab}$ sending an intermediate definably closed extension to its stabilizer subgroup.

\emph{Proof.} By saturation, $\mathsf{Fix}$ is well-defined, and $\mathsf{Stab}$ is clearly well-defined.

$\mathsf{Fix}$ is left-[[inverse]] to $\mathsf{Stab}$: by saturation in the [[monster model|monster]], any fixed points of $\mathsf{Stab}(B)$ for $K \subseteq B \subseteq A$ must be in the definable closure of $B$, so whenever $B$ is definably closed, $\mathsf{Fix} \left( \mathsf{Stab} (B) \right) \subseteq B$, with the reverse inclusion immediate.

$\mathsf{Stab}$ is left-inverse to $\mathsf{Fix}$: for $H$ a subgroup of $\operatorname{Aut}(A/K)$, note that for any $c$ a [[elimination of imaginaries|code]] for the $H$-orbit of $\gamma$ and for any $\sigma \in \operatorname{Aut}(\mathbb{M}/K)$, $\sigma$ fixes $c$ if and only if the restriction $\sigma \restriction A$ permutes $H.\gamma$. Since $\gamma$ generates $A$, any automorphism in $\operatorname{Aut}(A/K)$ is determined by where it sends $\gamma$, so $\sigma \restriction A \operatorname{.} \gamma \in H.\gamma \iff \sigma \restriction A \in H$.

In particular, since $H.\gamma$ is finite, $c$ is actually $H.\gamma$-definable, hence $A$-definable. Since $A$ is definably closed, $c \in A$, and so $c \in \mathsf{Fix}(H)$. If $g \in \mathsf{Stab}(\mathsf{Fix}(H))$, $g$ in particular fixes $c$, hence $g \in H$, and it's clear that $H \subseteq \mathsf{Stab}(\mathsf{Fix}(H))$. $\square$

\hypertarget{the_case_for_infinite_extensions}{}\subsection*{{The case for infinite extensions}}\label{the_case_for_infinite_extensions}

We'll get the case for infinite extensions by just classifying all subextensions of $\operatorname{acl}(A)/\operatorname{dcl}(A)$ at once.

\hypertarget{the_modeltheoretic_absolute_galois_group}{}\subsubsection*{{The model-theoretic absolute Galois group}}\label{the_modeltheoretic_absolute_galois_group}

Let $\mathbb{M} \models T$ be a [[monster model]]. Let $A$ be a small parameter set. $\operatorname{acl}(A)$ is a normal extension of $A$, because every finite $A$-definable set splits into $\operatorname{Aut}(\mathbb{M}/A)$-orbits. The \emph{absolute Galois group} $\operatorname{Gal}(A)$ of $A$ is $\operatorname{Aut}(\operatorname{acl}(A)/\operatorname{dcl}(A)$.

(For example, in $\mathsf{ACF}$, this recovers the usual absolute Galois group.)

Now, $\operatorname{acl}(A)$ is the colimit of the diagram of finite $A$-definable sets. From [[commutativity of limits and colimits]], we know that whenever $F : \mathbf{C} \to G \text{-} \mathbf{Set}$ is a cofiltered diagram of $G$-sets, then taking orbits of $\underset{\longleftarrow}{\lim}F$ is the same as taking a limit of the orbits. Dually, if we take automorphism groups, we get:

\begin{displaymath}
\operatorname{Aut}\left(\underset{\longrightarrow}{\lim} F\right) \simeq \underset{\longleftarrow}{\lim} \left(F(c) \right).
\end{displaymath}
So $\operatorname{Gal}(A)$ is profinite.

\hypertarget{the_fundamental_theorem_of_modeltheoretic_galois_theory}{}\subsubsection*{{The fundamental theorem of model-theoretic Galois theory}}\label{the_fundamental_theorem_of_modeltheoretic_galois_theory}

\textbf{Theorem.} Let $T$ be a first-order theory which [[elimination of imaginaries|eliminates imaginaries]], and let $\mathbb{M} \models T$ be a [[monster model|monster]]. Let $A \subseteq \mathbb{M}$ be a small parameter set. Then there is a bijective order-reversing correspondence

\begin{displaymath}
\mathsf{Fix} : \operatorname{Sub}_{\text{Krull-closed}}\left(\operatorname{Gal}(A)\right) \leftrightarrows \operatorname{Sub}_{\text{dcl-closed}}\left(\operatorname{acl}(A)/\operatorname{dcl}(A)\right) : \mathsf{Stab}
\end{displaymath}
given by taking a subgroup closed in the [[profinite group|profinite topology]] of $\operatorname{Gal}(A)$ to its fixed points and by taking a definably-closed intermediate extension of $\operatorname{acl}(A)/\operatorname{dcl}(A)$ to its stabilizer.

\emph{Proof.} That $\mathsf{Fix}$ is left-inverse to $\mathsf{Stab}$ again follows from being in a monster.

On the other hand, let $H$ be a closed subgroup. $H$ is an intersection of basic open subgroups $H_i$ which are preimages of $\operatorname{Aut}(B_i/A)$, for $B_i$ finite and $A$-definable. Fixing an ordering on the $B_i$ and treating them as tuples, obtain codes $c_i$ for the orbit of each $B_i$ under $\operatorname{Aut}(B_i/A)$. Since each $B_i$ is finite $A-definable$, $c_i$ is $A$-algebraic. The stabilizer of each $c_i$ is precisely $H_i$, so $H = \bigcap_{i \in I} \mathsf{Stab}(c_i)$. Since each $c_i$ is fixed by $H$, whenever $g \in \mathsf{Stab} \left(\mathsf{Fix}(H) \right)$, then in fact $g \in H$. $\square$

\hypertarget{model_theory_and_the_tannakian_formalism}{}\subsection*{{Model theory and the Tannakian formalism}}\label{model_theory_and_the_tannakian_formalism}

In the motivating examples (see below) it turns out that Galois (i.e. relative automorphism) groups are themselves definable (i.e. arise as interpretations in the model of internal groups in $\mathbf{Def}(T)$.) In this case what you're taking the Galois group of must formally resemble an [[internal diagram]] on this [[group object|internal group]]; in model theory these are studied as what model theorists call \emph{internal covers}. A structure theorem of Hrushovski makes this correspondence explicit: internal covers are torsors of definable groupoids and vice-versa; see \href{http://arxiv.org/abs/math/0603413}{here}.

All internal groups in $\mathbf{Def}(\mathsf{ACF})$ are in fact algebraic groups, so this is reminiscent of reconstruction results arising from [[Tannaka duality]]. As it turns out, one can make this analogy explicit and recover a slightly-weakened version of the Tannakian formalism for algebraic groups using the theory of internal covers; see the paper by Moshe Kamensky \href{https://arxiv.org/abs/0908.0604}{here}.

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

If $T = \mathsf{ACF}$ the [[theory of algebraically closed fields]], this recovers the classical fundamental theorem of Galois theory.

If $T = \mathsf{DCF}$ the [[theory of differentially closed fields]], this recovers [[differential Galois theory]]. (In fact, Kolchin's work was what inspired Poizat to introduce imaginaries and work out classical Galois theory in a model-theoretic setting.)

Any theory $T$ can be conservatively interpreted inside a theory $T^{\operatorname{eq}}$ which eliminates imaginaries and hence ``admits a Galois theory.'' This is the [[coherent logic|coherent]] special case of a result Olivia Caramello spells out in very general terms in her monograph on topological (toposic) Galois theory.

\hypertarget{discussion}{}\subsection*{{Discussion}}\label{discussion}

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

\begin{itemize}%
\item [[Galois theory]]
\item [[Grothendieck's Galois theory]]
\item [[elimination of imaginaries]]
\item [[definable closure|algebraic type]]
\item [[Tannaka duality]]

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

\begin{itemize}%
\item Bruno Poizat, \emph{Une Theorie de Galois Imaginaire}.


\item Bruno Poizat, \emph{A Course in Model Theory}.


\item Alice Medvedev and Ramin Takloo-Bighash, \href{http://arxiv.org/abs/0909.4340}{\emph{An invitation to model-theoretic Galois theory}}



\end{itemize}
[[!redirects model theoretic Galois theory]]



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