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\begin{document}

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\section*{profinite completion of a group}

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

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{compact_objects}{}\paragraph*{{Compact objects}}\label{compact_objects}

[[!include compact object - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{profinite_completion_of_a_group}{Profinite completion of a group}\dotfill \pageref*{profinite_completion_of_a_group} \linebreak
\noindent\hyperlink{pro_completion_of_a_group}{Pro-$\mathcal{C}$ completion of a group}\dotfill \pageref*{pro_completion_of_a_group} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{profinite_completion_of_spaces}{Profinite completion of spaces}\dotfill \pageref*{profinite_completion_of_spaces} \linebreak
\noindent\hyperlink{equational_or_monadic_completions}{Equational or monadic completions}\dotfill \pageref*{equational_or_monadic_completions} \linebreak
\noindent\hyperlink{profinite_rings}{Profinite rings}\dotfill \pageref*{profinite_rings} \linebreak
\noindent\hyperlink{profinite_completions_of_homotopy_types}{Pro-finite completions of homotopy types.}\dotfill \pageref*{profinite_completions_of_homotopy_types} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\hypertarget{profinite_completion_of_a_group}{}\subsubsection*{{Profinite completion of a group}}\label{profinite_completion_of_a_group}

The \textbf{profinite completion} $\hat{G}$ of a ([[discrete group|discrete]]) [[group]] $G$ is the [[limit]] (in the [[category]] of [[topological groups]]) over the [[diagram]] with [[objects]] all the [[finite group|finite]] [[quotient groups]] $G/N_{fin}$ where $N_{fin}$ is a [[normal subgroup]] of $G$ with [[finite number|finite]] [[index of a subgroup|index]], and [[morphisms]] induced from the [[lattice]] of [[subgroups]] of $G$.

Note that the profinite completion actually is a [[profinite group]], and there is a canonical homomorphism $G \to \hat{G}$.

More formally, we note that for any group $G$, the family of its normal finite index subgroups forms a [[filtered category|cofiltered category]] under inclusion. (Denote it by $\Omega_G$.) The assignment of $G/N$ to $N$ gives a functor from $\Omega_G^{op}$ to the category of finite groups. It is thus a [[profinite group]] in the sense given in that entry, i.e. a [[pro-object]] in the category of finite groups. This pro-object is the profinite completion of $G$.

The above topological version of this, with which we started, is obtained by means of the equivalence between the category of pro-(finite groups) and that of the groups internal to profinite spaces that is by taking the limit in the category of topological groups of the diagram of (discrete) finite groups that the above construction gives one.

\begin{remark}
\label{}\hypertarget{}{}
The inclusion [[functor]] $inc$ from the category, $FinGrps$, of finite groups, into that of groups does not have a [[left adjoint]]. It does have a [[pro-left adjoint]], that is to say, it induces a functor on [[pro-objects|procategories]]

\begin{displaymath}
pro\!-\!inc : pro\!-\! FinGrps\to pro\!-\Grps
\end{displaymath}
and that functor \emph{does} have a left adjoint. If we restrict that `pro-adjoint' to the subcategory of $pro\!-\! FinGrps$ given by the `constant' pro-objects, then the result is the pro-finite completion construction that is given above. Because of this, if we think of the natural functor $C\to pro\!-\! C$ to be an inclusion, i.e. think of an object as a pro-object indexed by the one arrow category, we can give a [[universal property]] for the pro-finite completion of a group $G$. This universal property gives a universal cone from $G$ to finite groups, and just encodes the obvious fact that any homomorphism from $G$ to a finite group factors through one of its finite quotient groups. If we write $\hat{G}$ for the pro-finite completion, the universal cone is a map $G\to \hat{G}$ in $pro\!-\! Grps$.

\end{remark}
\hypertarget{pro_completion_of_a_group}{}\subsubsection*{{Pro-$\mathcal{C}$ completion of a group}}\label{pro_completion_of_a_group}

Let $\mathcal{C}$ be any class of [[finite groups]] that is closed under the formation of [[subgroups]], homomorphic [[images]] and [[group extensions]].

\begin{defn}
\label{}\hypertarget{}{}
A \emph{pro-$\mathcal{C}$ group} is an [[inverse limit]] of an inverse system of groups in the class $\mathcal{C}$ or alternatively a [[pro-object]] in the full subcategory $\mathcal{C}$ determined by the class $\mathcal{C}$.

\end{defn}
The subcategory of $pro\!-\!FinGrps$ consisting of the pro-$\mathcal{C}$ groups and the continuous homomorphisms between them will be denoted $pro\!-\!\mathcal{C}$.

This notation now has two definitions, but, as the corresponding categories are equivalent, this causes no problem.

The categories of the form $pro\!-\!\mathcal{C}$ form [[varieties]] in $prof-FinGrps$. Recall that a [[variety]] in any algebraic context means a subcategory of `algebras' closed under products, subobjects and quotients. We note the condition on $\mathcal{C}$ implies the closure of $\mathcal{C}$ under finite products, so $\mathcal{C}$ is what is called a \emph{pseudovariety}. The category $pro\!-\!FinGrps$ is [[monadic]] over the category of spaces. This means that free objects exist in all the $pro\!-\!\mathcal{C}$. A good reference for this is Gildenhuys and Kennison, (1971), see below.

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

\begin{example}
\label{}\hypertarget{}{}
Consider the profinite completion of the [[fundamental group]] of an complex [[projective variety]] $X$. Since $X$ has an underlying [[topological space]], its [[fundamental group]] of loops $\pi_1^{top}(X)$ can be defined in the usual way. But one can also define the [[algebraic fundamental group]] $\pi_1^{alg}(X)$. This is a profinite group, which is isomorphic to the profinite completion of $\pi_1^{top}(X)$.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
The [[profinite completion of the integers]] is

\begin{displaymath}
\widehat {\mathbb{Z}} \coloneqq \underset{\leftarrow}{\lim}_n \mathbb{Z}/n\mathbb{Z}
  \,.
\end{displaymath}
This is [[isomorphism|isomorphic]] to the [[product]] of the [[p-adic integers]] for all $p$

\begin{displaymath}
\widehat{\mathbb{Z}} \simeq \underset{p}{\prod} \mathbb{Z}_p
  \,.
\end{displaymath}
For more on this see at \emph{[[p-adic integers]]}, at \emph{[[adele]]} and \emph{[[idele]]}.

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

\hypertarget{profinite_completion_of_spaces}{}\subsubsection*{{Profinite completion of spaces}}\label{profinite_completion_of_spaces}

\begin{itemize}%
\item [[profinite completion of a space|Profinite completion of a space]].

\end{itemize}
(Beware there are two possible interpretations of this term. One is handled in the section above, being profinite completion of the homotopy type of a space. The entry linked to here treats another more purely topological concept.)

\hypertarget{equational_or_monadic_completions}{}\subsubsection*{{Equational or monadic completions}}\label{equational_or_monadic_completions}

Profinite completion of groups is a special case of a general process that `completes' a category together with a `forgetful functor' to some `base' category, replacing it by a category which is equational/monadic over the base.

\begin{itemize}%
\item D. Gildenhuys and J. Kennison, \emph{Equational completions, model induced triples and pro-objects}, J. Pure Applied Alg., 1(4), (1971), 317--346.

\end{itemize}
\hypertarget{profinite_rings}{}\subsubsection*{{Profinite rings}}\label{profinite_rings}

[[compact Hausdorff rings are profinite]]

\hypertarget{profinite_completions_of_homotopy_types}{}\subsubsection*{{Pro-finite completions of homotopy types.}}\label{profinite_completions_of_homotopy_types}

Artin and Mazur in their lecture note on \'e{}tale homotopy introduced a process of profinite completion, generalising that for groups in as much as the profinite completion of an Eilenberg-Mac Lane space having $G$ as fundamental group has the profinite completion of $G$ as \emph{its} fundamental group. (WARNING: This needs a bit more detail to make it true! so this part of the entry needs more work.)

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

\begin{itemize}%
\item L. Ribes and P. Zalesskii, 2000, \emph{Profinite groups} , volume 40 of \emph{Ergebnisse der Mathematik und ihrer Grenzgebiete}. 3. Folge , Springer-Verlag, Berlin.


\item J. Dixon, M. du Sautoy, A. Mann and D. Segal, 1999, \emph{Analytic pro-p groups}, volume 61 of Cambridge Studies in Advanced Mathematics , Cambridge Univ. Press.



\end{itemize}
[[!redirects profinite completion]]



\end{document}