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\section*{Grothendieck&#39;s Galois theory}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{preliminaries}{Preliminaries}\dotfill \pageref*{preliminaries} \linebreak
\noindent\hyperlink{grothendiecks_axioms}{Grothendieck's axioms}\dotfill \pageref*{grothendiecks_axioms} \linebreak
\noindent\hyperlink{the_classical_case_of_fields}{The classical case of fields}\dotfill \pageref*{the_classical_case_of_fields} \linebreak
\noindent\hyperlink{galois_theorem_for_locales_and_topoi}{Galois theorem for locales and topoi}\dotfill \pageref*{galois_theorem_for_locales_and_topoi} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

For a sufficiently nice [[topological space]], the [[fundamental group]] at a point can be reconstructed as a group of [[deck transformation]]s of the [[universal covering space]], which is the same as the [[automorphism]]s of the [[fiber]] over that point of the projection map. The deck transformations are [[monodromies]] induced by loops at the base point. The functor which assigns to a point the fiber functor over it, generalizes to fiber functors in the [[Tannaka duality|Tannakian formalism]] of Grothendieck which defines in more general setups the [[fundamental groupoid]] as the group of automorphisms of the appropriate fiber functor. See also [[fundamental group of a topos]].

\begin{description}

\item[Grothendieck's [[Galois theory]] was constructed in order to define for [[scheme]]s an analogue of the familiar correspondence] [[covering space]]s of $X$ : $\pi_1(X)$[[action|-sets]]

 
\end{description}
for a [[locally path-connected space|locally path connected]], [[semi-locally simply connected space|semilocally simply connected]] [[topological space]] $X$.

The objects on the left are not difficult to define for schemes (at least naively -- one really needs trivialisations over \'e{}tale [[coverage|covers]]), but it may not be entirely immediate what the [[fundamental group]] defined in terms of loops should be.

The reason Galois's name is attached to this theory is that in the case of the scheme $Spec(k)$, the objects corresponding to the covering spaces are simply [[field extension]]s $Spec(k')$. The fundamental group of schemes defined in this way is the [[algebraic fundamental group]], and is a [[profinite group]].

\hypertarget{generalizations}{}\subsubsection*{{Generalizations}}\label{generalizations}

The basic idea of Grothendieck's Galois theory may be extended to objects in an [[topos]] -- leading to a notion of [[fundamental group of a topos]] -- and then further to objects in any [[(∞,1)-topos]]. For more on this see [[homotopy group of an ∞-stack]].

\hypertarget{details}{}\subsection*{{Details}}\label{details}

\hypertarget{preliminaries}{}\subsubsection*{{Preliminaries}}\label{preliminaries}

\begin{udefn}
Given an arrow $f:x \to y$ in a category $C$ the \emph{category of arrows compatible with $f$}, denoted $Comp(f)$ is the full [[subcategory]] of the [[undercategory]] $x \downarrow C$ on the arrows that [[coequalizer|coequalize]] the same pairs $g,h:w\rightrightarrows x$ that $f$ does.

\end{udefn}
\begin{udefn}
An arrow $f:x\to y$ in a category $C$ is a \emph{[[strict epimorphism]]} if it is [[initial object|initial]] in $Comp(f)$.

\end{udefn}
It is not obvious, but a strict epimorphism is an epimorphism.

\hypertarget{grothendiecks_axioms}{}\subsubsection*{{Grothendieck's axioms}}\label{grothendiecks_axioms}

In what follows, Let $C$ be a [[category]] and $F:C \to Set$ a [[functor]]. The axioms presented here are as in

J. P. Murre, \emph{Lectures on an introduction to Grothendieck's theory of the fundamental group}, Tata Inst. of Fund. Res. Lectures on Mathematics 40, Bombay, 1967. iv+176+iv pp.

and copied also in

\begin{itemize}%
\item [[Eduardo Dubuc]], C. S. de la Vega \emph{On the Galois theory of Grothendieck}, Bol. Acad. Nac. Cienc. (Cordoba) \textbf{65} (2000) 111--136. \href{http://arxiv.org/abs/math.CT/0009145}{arXiv}

\end{itemize}
Some terminology: $X\in C$ is called \emph{finite} if $F(X)$ is a finite [[set]]. Let $\int_F C$ denote the [[category of elements]] of $F$, in which an object $(X,a)$ is called finite if $X$ is finite.

\begin{itemize}%
\item \textbf{G0)} The full subcategory of $\int_F C$ on the finite objects is [[cofinal subcategory|cofinal]].


\item \textbf{G1)} $C$ has all finite [[limit]]s


\item \textbf{G2)} $C$ has an [[initial object]], finite [[coproduct]]s and [[quotient object|quotients]] by finite [[group]]s


\item \textbf{G3)} Given $f:x\to z$ in $C$ there is a factorisation $x \stackrel{e}{\to} y \stackrel{i}{\to} z$ where $e$ is a strict epimorphism and $i$ is a mono. Also, $y$ is assumed to be a [[direct sum]]mand of $z$.


\item \textbf{G4)} $F$ preserves finite limits


\item \textbf{G5)} $F$ preserves initial object, finite sums, quotients by finite group actions and sends strict epimorphisms to surjections


\item \textbf{G6)} $F$ reflects isomorphisms



\end{itemize}

\vspace{.5em} \hrule \vspace{.5em}
The functor $F$ is called the \emph{fibre functor}, and the pair $(C,F)$ is sometimes called a \emph{Galois category}.

It follows from the axioms that $F$ is a [[pro-representable functor]]. The [[automorphism group]] of the [[pro-object]] $P$ representing $F$ is (should be. I'm not familiar enough with pro-objects) a [[profinite group]] $\pi$. This acts on $F(X) = [P,-]$ by precomposition (talking out of my depth here -- it's getting a bit vague) and so $F$ lifts to a functor to $\pi-Set$, and Grothendieck's result is that this functor is an [[equivalence of categories]].

There are several modifications one can make the above. In the case that $C$ is the category of covering spaces of a nice enough space, the functor $F$ is [[representable functor|representable]] by the [[universal covering space]], and so there is a `representable' version of the above, not needing to utilise profinite groups. One can also consider just the connected-objects version, and end up with an equivalence to the category of \emph{[[transitive action|transitive]]} $\pi$-sets.

\hypertarget{the_classical_case_of_fields}{}\subsection*{{The classical case of fields}}\label{the_classical_case_of_fields}

Even for the classical case of the inclusion of [[field]]s, Grothendieck's Galois theorem gives more general statement than the previously known. This is the Grothendieck's version of the Galois correspondence theorem for fields:

Let $K \subset L$ be a finite dimensional [[Galois extension]] of fields. Let $Gal[L : K]$ denote the group of $K$-automorphisms of $L$ and $Gal [L : K]-finSet$ the category of finite $Gal[L : K]$-sets. By $SplitfinAlg_K(L)$ denote the finite dimensional $K$-algebras which are split over $L$; here $L$ itself is an object. Consider the [[representable functor]] $h_L = Hom_K(-,L):SplitfinAlg_K(L)\to Set$. It takes values in the [[subcategory]] of finite sets and it comes with a canonical $Gal[L : K]$-action. In other words, this functor factors through $Gal [L : K]-finSet$. Moreover, the corresponding functor

\begin{displaymath}
SplitfinAlg_K(L)\to Gal [L : K]-finSet
\end{displaymath}
is an [[equivalence of categories]].

There is an infinitary version as well, generalizing the classical Galois theorem on infinitary Galois extensions.

Thus let $K\subset L$ be an arbitrary Galois extension. Now $Gal[L:K]$ denotes the profinite Galois group and $Gal[L:K]-profinSpace$ the category or profinite $Gal[L:K]$-spaces. $SplitAlg_K(L)$ denotes the category of $K$-algebras split over $L$ (possible infinite-dimensional). Then there is a canonical anti-equivalence of categories

\begin{displaymath}
SplitAlg_K(L)\to Gal [L : K]-profinSpace
\end{displaymath}
(factorizing a profinite-space version of the representable functor $Hom_K(-,L)$).

A special case of this is the following: the category of [[étale scheme|étale k-schemes]] reps. [[étale group scheme|étale group schemes]] for a field $k$ is equivalent to the category of sets equipped with an [[action]] of the [[absolute Galois group]] reps. to the category of [[Galois module|Galois modules]] of the absolute Galois group.

\hypertarget{galois_theorem_for_locales_and_topoi}{}\subsection*{{Galois theorem for locales and topoi}}\label{galois_theorem_for_locales_and_topoi}

Let $E$ be a Grothendieck topos. Then there exist an open [[localic groupoid]] $G$ such that $E$ is equivalent to the category of \'e{}tale presheaves over $G$. One of the classical references is

\begin{itemize}%
\item J. P. Murre, \emph{Lectures on an introduction to Grothendieck's theory of the fundamental group}, Tata Inst. of Fund. Res. Lectures on Mathematics \emph{40}, Bombay, 1967. iv+176+iv pp.

\end{itemize}
This is a variant of the theorem in the setting of [[locales]] from

\begin{itemize}%
\item [[Andre Joyal]], M. Tierney, \emph{An extension of the Galois theory of Grothendieck}, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp.

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

The original development of the theory by [[Grothendieck]] is in [[SGA1]].

\begin{itemize}%
\item [[Alexander Grothendieck]], (1971), S.G.A.1 - Revetements \'e{}tales et groupe fondamental, Lecture Notes in Maths. 224, Springer-Verlag.

\end{itemize}
A more recent treatment can be found in

\begin{itemize}%
\item [[Eduardo Dubuc]], C. S. de la Vega, \emph{On the Galois theory of Grothendieck}, Bol. Acad. Nac. Cienc. (Cordoba) \textbf{65} (2000) 111--136. \href{http://arxiv.org/abs/math.CT/0009145}{arXiv:math.CT/0009145}

\end{itemize}
and more related categorical and topos theoretic aspects in

\begin{itemize}%
\item [[Eduardo Dubuc]], \emph{Localic Galois theory}, Adv. Math. \textbf{175}:1 (2003), 144--167 ; \emph{On the representation theory of Galois and atomic topoi}, JPAA \textbf{186}:3 (2004) 233--275 

\end{itemize}
A very approachable account is given in

\begin{itemize}%
\item Marco Ant\`o{}nio Delgado Robalo, \emph{Galois theory towards dessins d'enfants}, masters thesis, Lisboa 2009, \href{https://dspace.ist.utl.pt/bitstream/2295/575330/1/dissertacao.pdf}{pdf}

\end{itemize}
(This has the advantage of looking towards Grothendieck's [[children's drawing|dessins d'enfants]].)

Basic intuition is explained in

\begin{itemize}%
\item Pierre Cartier, \emph{A mad day's work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry}, Bull. Amer. Math. Soc. \textbf{38} (2001), 389-408, \href{http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf}{pdf}

\end{itemize}
The construction for general [[topos]]es is described in section 8.4 of

\begin{itemize}%
\item [[Peter Johnstone]], \emph{Topos theory} , Academic Press (1977)

\end{itemize}
and, a current state of the art description is in

\begin{itemize}%
\item [[Marta Bunge]], \emph{Galois groupoids and covering morphisms in topos theory}, Galois theory, Hopf algebras, and semiabelian categories, 131--161, Fields Inst. Commun. \textbf{43}, Amer. Math. Soc., Providence, RI, 2004, \href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.15.7071}{links}.

\end{itemize}
A modern approach from classical via Grothendieck up to [[categorical Galois theory]] based on precategories and adjunctions is in

\begin{itemize}%
\item [[Francis Borceux]], [[George Janelidze]], \emph{[[Galois theories]]}, Cambridge Studies in Adv. Math. \textbf{72}, 2001. xiv+341 pp.

\end{itemize}
The application of a general Tannakian theorem of Saavaedra Rivano, as corrected by Deligne, to the ``[[differential Galois theory]]'' for differential instead of algebraic equations is in the last chapter of Deligne's [[Catégories Tannakiennes]].

\begin{itemize}%
\item [[George Janelidze]], Dietmar Schumacher, [[Ross Street]], \emph{Galois theory in variable categories}, Applied Categorical Structures \textbf{1}, Number 1, 103-110, DOI:
\item Federico G. Lastaria, \emph{On separable algebras in Grothendieck Galois theory}, Le Matematiche 51:3, 1996, \href{http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/464/0}{link}

\end{itemize}
[[!redirects Grothendieck Galois theory]] [[!redirects Grothendieck's Galois theory]] [[!redirects Grothendieck-Galois theory]] [[!redirects Grothendieck?Galois theory]] [[!redirects Grothendieck--Galois theory]] [[!redirects Galois category]]

category: Galois theory



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