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\begin{document}

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\section*{localic group}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{localic_groups_versus_topological_groups}{Localic groups versus topological groups}\dotfill \pageref*{localic_groups_versus_topological_groups} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{localic_subgroups_are_closed}{Localic subgroups are closed}\dotfill \pageref*{localic_subgroups_are_closed} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The notion of [[locale]] is a ``point-less'' version of that of [[topological space]]. A \emph{localic group} is much like a [[topological group]], but there are some differences. For a groupoid generalization see [[localic groupoid]].

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

A \textbf{localic group} is a [[group object]] [[internalization|in]] the [[category]] of [[locales]].

\hypertarget{localic_groups_versus_topological_groups}{}\subsection*{{Localic groups versus topological groups}}\label{localic_groups_versus_topological_groups}

Localic groups are similar to [[topological group]]s, and many examples can be considered as either one. For instance, the real numbers $\mathbb{R}$ under addition can be considered as either a topological group or a localic group.

Since the ``space of points'' [[functor]] $Loc \to Top$ is a [[right adjoint]], it preserves [[limit]]s and hence [[group object]]s, so every localic group has an underlying topological group. However, this functor can discard information; for instance, \hyperlink{IKPR}{IKPR} constructs a nontrivial localic group with only one point.

Moreover, the ``locale of opens'' functor $Top\to Loc$ does not preserve [[product]]s, so not every topological group is a localic group---even if its underlying topological space is [[sober space|sober]] (hence is the space of points of some locale). In particular, the locale $\mathbb{Q}$ of rational numbers (with [[topology]] induced from that of $\mathbb{R}$) is not a localic group under addition, because the locale product $\mathbb{Q}\times_l \mathbb{Q}$ is ``bigger'' than the topological-space product (and in particular is not spatial), and the addition map $\mathbb{Q}\times \mathbb{Q}\to \mathbb{Q}$ cannot be extended to the locale product.

But if $G$ is a [[locally compact space|locally compact]] topological group (such as $\mathbb{R}$), then the space product $G\times G$ does agree with the locale product (using the [[ultrafilter principle]] in the proof), and hence $G$ is also a localic group.

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

\begin{itemize}%
\item Another important source of localic groups is from [[progroups]]: cofiltered limits of discrete groups.
\item The [[isotropy group of a topos]] is naturally regarded as a localic group inside the topos.

\end{itemize}
\hypertarget{localic_subgroups_are_closed}{}\subsection*{{Localic subgroups are closed}}\label{localic_subgroups_are_closed}

A remarkable fact about localic groups is the following (Corollary C5.3.2 of the [[Elephant]]; this also proves that $\mathbb{Q}$ cannot be a localic group):

\begin{utheorem}
Any [[overt space|overt]] localic subgroup of a localic group is weakly [[closed sublocale|closed]]. If the ambient localic group is in a [[Boolean topos]] then any localic subgroup is closed.

\end{utheorem}
\begin{proof}
Details can be found in C5.3.1 of the [[Elephant]], in the more general case of localic [[groupoids]]. The basic idea of the proof is to use the fact that the intersection of any two [[dense sublocale]]s is again dense (a fact which very much fails for topological spaces).

If $H\rightarrowtail G$ is a localic subgroup, we construct its closure $\bar{H}$, which is also a localic subgroup in which $H$ is dense. By [[pullback]], it follows that $H\times \bar{H} \to \bar{H} \times \bar{H}$ is [[fiberwise dense subspace|fiberwise dense]] over $\bar{H}$ via the second projection. Applying the [[automorphism]] $(g,h) \mapsto (g,g^{-1}h)$ of $G\times G$, we conclude that $H\times \bar{H} \to \bar{H} \times \bar{H}$ is also [[fiber]]wise dense over $\bar{H}$ via the ``composition'' map. Dually, $\bar{H}\times H \to \bar{H} \times \bar{H}$ is also fiberwise dense over $\bar{H}$ via the ``composition'' map, and thus (by the basic fact cited above), so is their intersection, which is $H\times H$. Since $\bar{H}\times \bar{H}\to \bar{H}$ is an [[epimorphism]], so is $H\times H\to\bar{H}$. But this map factors through $H\rightarrowtail \bar{H}$ (since $H$ is itself a subgroup of $G$), so that inclusion is also epic. But it is also a [[regular monomorphism]], and hence an [[isomorphism]]; thus $H$ is closed.

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

\begin{itemize}%
\item The [[Elephant]], chapter C5.


\item Isbell, Ki, Pultr and Rosick\'y{}, \emph{Remarks on Localic Groups}



\end{itemize}
[[!redirects localic groups]]



\end{document}