In parts of the literature the fundamental groupoid, and more generally the fundamental ∞-groupoid, is called the Poincaré groupoid.
The fundamental groupoid of a topological space is the groupoid whose set of objects is and whose morphisms from to are the homotopy-classes of continuous maps whose endpoints map to and (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 is a groupoid.
For any in the first homotopy group of based at arises as the automorphism group of in :
So the fundamental groupoid is an improvement on the idea of the fundamental group, which gets rid of the choice of basepoint. The set of connected components of is precisely the set of path-components of . (This is not to be confused with the set of connected components of , sometimes denoted by the same symbol. Of course they are the same when is locally path-connected.)
The fundamental groupoid can be made into a topological groupoid (i.e. a groupoid internal to Top) when 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 .
Mike Shulman: Could you say something about what topology you have in mind here? Is the space of objects just with its original topology?
Regarding topology on the fundamental groupoid for a general space; it inherits a topology from the path space , 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.
When is not semi-locally simply connected, the arrows of the fundamental groupoid inherits the quotient topology from the path space such that the fibres of 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 is not locally path-connected, also inherits a non-discrete topology (the quotient topology of 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 this relevant to the van Kampen theorem for computing the fundamental group or groupoid is to take , defined to be the full subgroupoid of on a set of base points, chosen according to the geometry at hand. Thus if is the union of two open sets with intersection then we can take large enough to meet each path-component of . If has an action of a group then acts on if is a union of orbits of the action.
fundamental groupoid, fundamental ∞-groupoid
R. Brown and G. Danesh-Naruie, The fundamental groupoid as a topological groupoid, Proc. Edinburgh Math. Soc. 19 (1975) 237-244.
R. Brown, Topology and groupoids, Booksurge (2006). (See particularly 10.5.8, using lifted topologies to topologise where is a normal, totally disconnected subgroupoid of , and admits a universal cover).
Discussion from the point of view of Galois theory is in