nLab
fundamental groupoid

Contents

Idea

The fundamental groupoid of a space XX is a groupoid whose objects are the points of XX and whose morphisms are paths in XX, 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.

Definition

The fundamental groupoid Π 1(X)\Pi_1(X) of a topological space XX is the groupoid whose set of objects is XX and whose morphisms from xx to yy are the homotopy-classes [γ][\gamma] of continuous maps γ:[0,1]X\gamma : [0,1] \to X whose endpoints map to xx and yy (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 Π 1(X)\Pi_1(X) is a groupoid.

Remarks

Relationship to fundamental group

For any xx in XX the first homotopy group π 1(X,x)\pi_1(X,x) of XX based at XX arises as the automorphism group of xx in Π 1(X)\Pi_1(X):

π 1(X,x)=Aut Π 1(X)(x). \pi_1(X,x) = Aut_{\Pi_1(X)}(x) \,.

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 Π 1(X)\Pi_1(X) is precisely the set Π 0(X)\Pi_0(X) of path-components of XX. (This is not to be confused with the set of connected components of XX, sometimes denoted by the same symbol. Of course they are the same when XX is locally path-connected.)

Topologizing the fundamental groupoid

The fundamental groupoid Π 1(X)\Pi_1(X) can be made into a topological groupoid (i.e. a groupoid internal to Top) when XX 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 XX.

Mike Shulman: Could you say something about what topology you have in mind here? Is the space of objects just XX 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 IX^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.

When XX 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 (s,t):Mor(Π 1(X))X×X(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 XX is not locally path-connected, Π 0(X)\Pi_0(X) also inherits a non-discrete topology (the quotient topology of XX 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.

Π 1(X)\Pi_1(X) with a chosen set of basepoints

An improvement on this relevant to the van Kampen theorem for computing the fundamental group or groupoid is to take Π 1(X,A)\Pi_1(X,A), defined to be the full subgroupoid of Π 1(X)\Pi_1(X) on a set AA of base points, chosen according to the geometry at hand. Thus if XX is the union of two open sets U,VU,V with intersection WW then we can take AA large enough to meet each path-component of U,V,WU,V,W. If XX has an action of a group GG then GG acts on Π 1(X,A)\Pi_1(X,A) if AA is a union of orbits of the action.

Ronnie Brown is a big booster of Π 1(X,A)\Pi_1(X,A), which is fundamental to his development of homotopy theory in Elements of Modern Topology (1968).

Notice that Π 1(X,X)\Pi_1(X,X) recovers the full fundamental groupoid, while Π 1(X,{a})\Pi_1(X,\{a\}) is the delooping of the fundamental group π 1(X,a)\pi_1(X,a).

In higher category theory

See fundamental ∞-groupoid.

References

  • 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 (π 1X)/N(\pi_1 X)/N where NN is a normal, totally disconnected subgroupoid of π 1X\pi_1 X, and XX admits a universal cover).

Discussion from the point of view of Galois theory is in

Revised on March 22, 2012 07:59:41 by Tim_Porter (95.147.237.122)