The 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 Poincaré groupoid.
The 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.
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 $\Pi_1(X)$:
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.)
The fundamental groupoid $\Pi_1(X)$ can be made into a topological groupoid (i.e. a groupoid internal to Top) when $X$ is path-connected, locally path-connected and 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 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 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.
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 computation of homotopy 1-types; the theory was developed in Elements of Modern Topology (1968), now available as Topology and Groupoids (2006). These accounts show the use of the algebra of groupoids in 1-dimensional homotopy theory, for example for covering spaces, and, in the later edition, for orbit spaces. Another text in English which covers this notion is by Philip Higgins, see below.
fundamental groupoid, fundamental ∞-groupoid
fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos / of a locally ∞-connected (∞,1)-topos
R. Brown, Groupoids and Van Kampen’s theorem, Proc. London Math. Soc. (3) 17 (1967) 385-401.
R. Brown and G. Danesh-Naruie, The fundamental groupoid as a topological groupoid, Proc. Edinburgh Math. Soc. 19 (1975) 237-244.
R. Brown and O. Mucuk, The monodromy groupoid of a Lie groupoid, Cah. Top. G'eom. Diff. Cat. 36 (1995) 345-369.
R. Brown, 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). (more info)
Relations with group theory are discussed in:
Discussion from the point of view of Galois theory is in
Luis Javier Hernández-Paricio, Fundamental pro-groupoids and covering projections, Fund. Math. (1998), (pdf)
The use of many base points is discussed at this (mathoverflow page).
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 is in]
Ilia Pirashvili, The fundamental groupoid as a terminal costack (arXiv:1406.4419)
Ilia Pirashvili, The Étale Fundamental Groupoid as a Terminal Costack (arXiv:1412.5473)
A recent paper in the area of dynamical systems which uses fundamental groupoids on many base points is: