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\begin{document}

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\section*{progroup}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{surjective_progroups_versus_localic_groups}{Surjective progroups versus localic groups}\dotfill \pageref*{surjective_progroups_versus_localic_groups} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{progroup} is a [[pro-object]] in the [[category]] [[Grp]] of [[groups]]. In other words, it is a formal [[cofiltered limit]] of groups.

\hypertarget{surjective_progroups_versus_localic_groups}{}\subsection*{{Surjective progroups versus localic groups}}\label{surjective_progroups_versus_localic_groups}

Of course, the category [[Grp]] is [[complete category|complete]], but in general a progroup represented by some cofiltered diagram of groups is not equivalent to the actual limit of that diagram in $Grp$. However, [[profinite groups]] (i.e. cofiltered systems of \emph{finite} groups) can be identified with actual limits of finite groups if we take those limits, not in $Grp$, but in the larger category $TopGrp$ of [[topological groups]]. The resulting topological groups are precisely those with [[Stone space|Stone]] topologies.

This is not true for pro-systems of non-finite groups, even if we restrict to those with surjective transition maps. The following counterexample is due to Higman and Stone, and is reproduced in (\hyperlink{Moerdijk}{Moerdijk}). Let $\omega_1$ be the set of countable ordinals, with the reverse of its usual ordering, and for $\alpha\in\omega_1$ let $S_\alpha$ be the set of strictly increasing functions $[0,\alpha]\to \mathbb{R}$. For $\alpha\lt \beta$, let $S_\beta \to S_\alpha$ be the restriction. Then each such transition map is surjective, but the inverse limit is empty. The sets $S_\alpha$ are not groups, but if we take the free [[vector space]] on each of them, we obtain a nontrivial pro-group with surjective transition maps whose limit in $Grp$, hence also in $TopGrp$, is trivial.

However, we do get an embedding on pro-groups with surjective transition maps if instead of [[Top]] we take the limit in the category [[Loc]] of [[locales]].

\begin{uprop}
The following are equivalent for a [[localic group]] $G$: 1. $G$ is a cofiltered limit of [[discrete group]]s (considered as discrete localic groups) 1. $G$ is a cofiltered limit of discrete groups with [[surjection|surjective]] transition maps. 1. The [[open subset|open]] [[normal subgroup]]s of $G$ form a [[neighborhood]] [[topological base|base]] at the identity $e\in G$.

\end{uprop}
\begin{proof}
This can be found in (\hyperlink{Moerdijk}{Moerdijk}).

\end{proof}
\begin{udefn}
A localic group with these properties is called \textbf{prodiscrete}.

\end{udefn}
We may as well assume that any surjective progroup is indexed on a directed [[poset]]. If $(G_i)_{i\in I}$ is such an inverse system, then the localic group $G=\lim_i G_i$ is presented by the following [[posite]]. The elements of the underlying poset are pairs $(x,i)$ where $x\in G_i$, with $(x,i)\le (y,j)$ when $i\le j$ and $f_{ij}(x)=y$. The coverings are given as follows: for any $j$, the element $(x,i)$ is covered by the family of all $(z,k)$ such that $k\le j$ and $(z,k)\le (x,i)$.

\begin{udefn}
A \textbf{surjective progroup}, also called a \textbf{strict progroup}, is a progroup whose cofiltered diagram consists of [[surjection]]s.

\end{udefn}
One can show that a progroup is isomorphic to a surjective one, in the category of pro-groups, if and only if it satisfies the \textbf{Mittag-Leffler condition}: for each $G_i$ the images of the functions $G_j\to G_i$ are eventually constant.

By a fundamental fact about [[locales]], if $G$ is prodiscrete and represented as the limit of a system with surjective transition maps, then the legs $G\to G_i$ of the limiting cone are also surjective (i.e. they are represented by injective [[frame]] homomorphisms). This is false for limits of topological spaces.

\begin{utheorem}
The category of prodiscrete localic groups is equivalent to the category of surjective progroups.

\end{utheorem}
\begin{proof}
In view of the \hyperlink{EquivalentCharacterizations}{above proposition} it suffices to show that for surjective progroups $(G_i)$ and $(H_j)$, with prodiscrete localic limits $G$ and $H$, we have

\begin{displaymath}
Hom_{LocGrp}(G,H) \cong \lim_j \colim_i Hom_{Grp}(G_i,H_j).
\end{displaymath}
But since $H = \lim_j H_j$, we have $Hom_{LocGrp}(G,H) \cong \lim_j Hom_{LocGrp}(G,H_j)$. Thus it suffices to show that any map from $G$ to a [[discrete group]] $K$ (such as $H_j$) factors through some essentially unique $G_i$.

But if $f\colon G\to K$ is such a map, then $ker(f)$ is an open normal subgroup of $G$. And if $p_i\colon G\to G_i$ are the projections, then the [[kernel]]s $ker(p_i)$ are a neighborhood base at $e$, so we have $ker(p_i)\subseteq ker(f)$ for some $i$, hence $f$ factors through $G/ker(p_i)$. Finally, this last is isomorphic to $G_i$, since $p_i\colon G\to G_i$ is an open surjection of locales.

\end{proof}
Any [[localic group]] $G$ has a [[classifying topos]] consisting of continuous $G$-sets, i.e. discrete locales with a $G$-action. In general, the resulting [[functor]]

\begin{displaymath}
LocGrp \to Topos
\end{displaymath}
is not an [[embedding]] into [[Topos]], but it can be shown to be so when restricted to prodiscrete localic groups. One can also characterize the toposes that are sheaves on a prodiscrete localic group as the [[Galois topos|Galois toposes]].

Most of these results have corresponding facts for pro-[[groupoids]] and prodiscrete localic groupoids. However, in full generality, the category of (even surjective) pro-groupoids does not embed into localic groupoids, since the category of [[pro-sets]] (= categorically discrete pro-groupoids) does not embed into locales (= categorically discrete localic groupoids).

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

\begin{itemize}%
\item [[Ieke Moerdijk]], \emph{Prodiscrete groups and Galois toposes}

\end{itemize}
[[!redirects progroups]] [[!redirects pro-group]] [[!redirects pro-groups]] [[!redirects prodiscrete localic group]] [[!redirects prodiscrete localic groups]]



\end{document}