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\section*{fundamental groupoid}

\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{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{relationship_to_fundamental_group}{Relationship to fundamental group}\dotfill \pageref*{relationship_to_fundamental_group} \linebreak
\noindent\hyperlink{topologizing_the_fundamental_groupoid}{Topologizing the fundamental groupoid}\dotfill \pageref*{topologizing_the_fundamental_groupoid} \linebreak
\noindent\hyperlink{_with_a_chosen_set_of_basepoints}{$\Pi_1(X)$ with a chosen set of basepoints}\dotfill \pageref*{_with_a_chosen_set_of_basepoints} \linebreak
\noindent\hyperlink{in_higher_category_theory}{In higher category theory}\dotfill \pageref*{in_higher_category_theory} \linebreak
\noindent\hyperlink{simplicial_version}{Simplicial version}\dotfill \pageref*{simplicial_version} \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}

The \textbf{fundamental groupoid} of a space $X$ is a [[groupoid]] whose objects are the points of $X$ and whose morphisms are [[paths]] in $X$, identified up to endpoint-preserving [[homotopy]].

In parts of the literature the fundamental groupoid, and more generally the [[fundamental ∞-groupoid]], is called the \textbf{Poincar\'e{} groupoid}.

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

The \textbf{fundamental groupoid} $\Pi_1(X)$ of a topological space $X$ is the [[groupoid]] whose set of objects is $X$ and whose morphisms from $x$ to $y$ are the homotopy-classes $[\gamma]$ of continuous maps $\gamma : [0,1] \to X$ whose endpoints map to $x$ and $y$ (which the homotopies are required to fix). Composition is by concatenation (and reparametrization) of representative maps. Under the [[homotopy]]-[[equivalence relation]] this becomes an associative and unital composition with respect to which every morphism has an inverse; hence $\Pi_1(X)$ is a groupoid.

The use of the fundamental groupoid of a manifold for describing the monodromy principle on the extension of local morphisms is discussed in the paper by Brown/Mucuk listed below.

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\hypertarget{relationship_to_fundamental_group}{}\subsubsection*{{Relationship to fundamental group}}\label{relationship_to_fundamental_group}

For any $x$ in $X$ the first homotopy group $\pi_1(X,x)$ of $X$ based at $x$ arises as the [[automorphism group]] of $x$ in $\Pi_1(X)$:

\begin{displaymath}
\pi_1(X,x) = Aut_{\Pi_1(X)}(x)
  \,.
\end{displaymath}
So the fundamental groupoid gets rid of the choice of basepoint for the fundamental group, and this is valuable for some applications. The set of connected components of $\Pi_1(X)$ is precisely the set $\Pi_0(X)$ of path-components of $X$. (This is not to be confused with the set of connected components of $X$, sometimes denoted by the same symbol. Of course they are the same when $X$ is locally path-connected.)

\hypertarget{topologizing_the_fundamental_groupoid}{}\subsubsection*{{Topologizing the fundamental groupoid}}\label{topologizing_the_fundamental_groupoid}

The fundamental groupoid $\Pi_1(X)$ can be made into a [[topological groupoid]] (i.e. a [[internal groupoid|groupoid internal]] to [[Top]]) when $X$ is [[path-connected space|path-connected]], [[locally path-connected space|locally path-connected]] and [[semi-locally simply connected space|semi-locally simply connected]]. This construction is closely linked with the construction of a [[universal covering space]] for a path-connected pointed space. The object space of this groupoid is just the space $X$.

[[Mike Shulman]]: Could you say something about what topology you have in mind here? Is the space of objects just $X$ with its original topology?

[[David Roberts]]: The short answer is that it is propositions 4.17 and 4.18 in my [[davidroberts:HomePage|thesis]], but I will put it here soon.

Regarding topology on the fundamental groupoid for a general space; it inherits a topology from the path space $X^I$, but there is also a topology (unless I've missed some subtlety) as given in 4.17 mentioned above, but the extant literature on the topological fundamental group uses the first one.

[[Ronnie Brown]]: See the account in ``Topology and Groupoids'' referred to below. But there is also an account using path spaces in Proposition 6.2 of Mackenzie's 1987 book ``Lie groupoids and Lie algebroids in differential geometry''.

When $X$ is not semi-locally simply connected, the set of arrows of the fundamental groupoid inherits the [[quotient space|quotient topology]] from the path space such that the fibres of $(s,t):Mor(\Pi_1(X)) \to X\times X$ are not discrete, which is an obstruction to the above-mentioned source fibre's being a covering space. However the composition is no longer continuous. When $X$ is not locally path-connected, $\Pi_0(X)$ also inherits a non-discrete topology (the quotient topology of $X$ by the relation of path connections).

In circumstances like these more sophisticated methods are appropriate, such as [[shape theory]]. This is also related to the [[fundamental group of a topos]], which is in general a [[progroup]] or a [[localic group]] rather than an ordinary group.

\hypertarget{_with_a_chosen_set_of_basepoints}{}\subsubsection*{{$\Pi_1(X)$ with a chosen set of basepoints}}\label{_with_a_chosen_set_of_basepoints}

An improvement on the fundamental group and the total fundamental groupoid relevant to the [[van Kampen theorem]] for computing the fundamental group or groupoid is to use $\Pi_1(X,A)$, defined for a set $A$ to be the full subgroupoid of $\Pi_1(X)$ on the set $A\cap X$, thus giving a set of base points which can be chosen according to the geometry at hand. Thus if $X$ is the union of two open sets $U,V$ with intersection $W$ then we can take $A$ large enough to meet each path-component of $U,V,W$; note that by the above definition we can write $\Pi_1(U,A)$, etc. If $X$ has an action of a group $G$ then $G$ acts on $\Pi_1(X,A)$ if $A$ is a union of orbits of the action. Thus $\Pi_1(X,A)$ can represent some symmetry of a given situation.

The notion of $\Pi_1(X,A)$ was introduced in 1967 by [[Ronnie Brown]] to give a version of the Seifert-van Kampen Theorem which allowed the determination of the fundamental group of a connected space which is the union of connected subspaces with nonconnected intersection, such as the circle, a space which is, after all, THE basic example in topology.

Grothendieck writes in his 1984 [[Esquisse d'un Programme]] (English translation):

`` ..,people still obstinately persist, when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under the symmetries of the situation, which thus get lost on the way. In certain situations (such as descent theorems for fundamental groups `a la van Kampen) it is much more elegant, even indispensable for understanding something, to work with fundamental groupoids with respect to a suitable packet of base points,..''.

Notice that $\Pi_1(X,X)$ recovers the full fundamental groupoid, while $\Pi_1(X,\{x\})$ is simply the [[fundamental group]] $\pi_1(X,x)$.

Basically, $\Pi_1(X,A)$ allows for the \emph{computation of homotopy 1-types}; the theory was developed in \emph{Elements of Modern Topology} (1968), now available as \emph{Topology and Groupoids} (2006). These accounts show the use of the algebra of groupoids in 1-dimensional [[homotopy theory]], for example for [[covering space]]s, and, in the later edition, for [[orbit space]]s. Another text in English which covers this notion is by Philip Higgins, see below.

\hypertarget{in_higher_category_theory}{}\subsubsection*{{In higher category theory}}\label{in_higher_category_theory}

See [[fundamental ∞-groupoid]].

\hypertarget{simplicial_version}{}\subsubsection*{{Simplicial version}}\label{simplicial_version}

See [[simplicial fundamental groupoid]].

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

\begin{itemize}%
\item \textbf{fundamental groupoid}, [[fundamental ∞-groupoid]]


\item [[simplicial fundamental groupoid]]


\item [[fundamental category]], [[fundamental (∞,1)-category]]


\item [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] / [[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos|of a locally ∞-connected (∞,1)-topos]]



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

\begin{itemize}%
\item R. Brown, Groupoids and Van Kampen's theorem, \emph{Proc. London Math. Soc}. (3) 17 (1967) 385-401.


\item R. Brown and G. Danesh-Naruie, The fundamental groupoid as a topological groupoid, \emph{Proc. Edinburgh Math. Soc.} 19 (1975) 237-244.


\item R. Brown and O. Mucuk, The monodromy groupoid of a Lie groupoid, \emph{Cah. Top. G'eom. Diff. Cat}. 36 (1995) 345-369.


\item R. Brown, \emph{Topology and Groupoids}, Booksurge (2006). (See particularly 10.5.8, using lifted topologies to topologise $(\pi_1 X)/N$ where $N$ is a normal, totally disconnected subgroupoid of $\pi_1 X$, and $X$ admits a universal cover). (\href{http://pages.bangor.ac.uk/~mas010/topgpds.html}{more info})



\end{itemize}
Relations with group theory are discussed in:

\begin{itemize}%
\item P.J. Higgins, \emph{Notes on Categories and Groupoids}, Mathematical Studies, Volume 32. Van Nostrand Reinhold Co. London (1971); \emph{Reprints in Theory and Applications of Categories}, No. 7 (2005) pp 1--195.

\end{itemize}
Discussion from the point of view of [[Galois theory]] is in

\begin{itemize}%
\item [[Luis Javier Hernández-Paricio]], \emph{Fundamental pro-groupoids and covering projections}, Fund. Math. (1998), (\href{http://matwbn.icm.edu.pl/ksiazki/fm/fm156/fm15611.pdf}{pdf})


\item The use of many base points is discussed at this (\href{http://mathoverflow.net/questions/40945/compelling-evidence-that-two-basepoints-are-better-than-one}{mathoverflow page}).



\end{itemize}
Discussion of the fundamental groupoid (for good [[topological spaces]] and for [[noetherian schemes]]) as the [[costack]] (via the [[Seifert-van Kampen theorem]]) characterized as being 2-[[terminal object|terminal]] is in

\begin{itemize}%
\item [[Ilia Pirashvili]], \emph{The fundamental groupoid as a terminal costack} (\href{https://arxiv.org/abs/1406.4419}{arXiv:1406.4419})


\item [[Ilia Pirashvili]], \emph{The \'E{}tale Fundamental Groupoid as a Terminal Costack} (\href{https://arxiv.org/abs/1412.5473}{arXiv:1412.5473})



\end{itemize}
A recent paper in the area of dynamical systems which uses fundamental groupoids on many base points is:

\begin{itemize}%
\item Paul, E. and Ramis, J.-P. Dynamics on Wild Character Varieties, \emph{SIGMA} 11 (2015), 068, 21 pages. arXiv:1508.03122.

\end{itemize}
[[!redirects fundamental groupoid]] [[!redirects fundamental groupoids]]

[[!redirects Poincaré groupoid]] [[!redirects Poincaré groupoids]] [[!redirects Poincaré-groupoid]] [[!redirects Poincaré-groupoids]] [[!redirects Poincare groupoid]] [[!redirects Poincare groupoids]] [[!redirects Poincare-groupoid]] [[!redirects Poincare-groupoids]]



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