nLab
fundamental groupoid

Idea

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.

Definition

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.

Remarks

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 ${\Pi }_1(X)$:

$${\pi }_1(X,x)={\mathrm{Aut}}_{{\Pi }_1(X)}(x).$$\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 ${\Pi }_1(X)$ 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.)

Topologizing the fundamental groupoid

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.

When $X$ 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 $(s,t):\mathrm{Mor}({\Pi }_1(X))\to X\times X$ 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.

${\Pi }_1(X)$ 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 ${\Pi }_1(X,A)$, defined to be the full subgroupoid of ${\Pi }_1(X)$ on a set $A$ of base points, 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 $U,V,W$. 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.

Ronnie Brown is a big booster of ${\Pi }_1(X,A)$, which is fundamental to his development of homotopy theory in Elements of Modern Topology (1968).

Notice that ${\Pi }_1(X,X)$ recovers the full fundamental groupoid, while ${\Pi }_1(X,\{a\})$ is the delooping of the fundamental group ${\pi }_1(X,a)$.

In higher category theory

See fundamental ∞-groupoid.

Related concepts

• fundamental groupoid, fundamental ∞-groupoid

• fundamental category, fundamental (∞,1)-category

• fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos / of a locally ∞-connected (∞,1)-topos

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 $({\pi }_{1}X)/N$ where $N$ is a normal, totally disconnected subgroupoid of ${\pi }_{1}X$, and $X$ admits a universal cover).

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)