category theory

# Contents

## 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 ${\Pi }_{1}\left(X\right)$ 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 $\left[\gamma \right]$ of continuous maps $\gamma :\left[0,1\right]\to X$ 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 ${\Pi }_{1}\left(X\right)$ is a groupoid.

## Remarks

### Relationship to fundamental group

For any $x$ in $X$ the first homotopy group ${\pi }_{1}\left(X,x\right)$ of $X$ based at $X$ arises as the automorphism group of $x$ in ${\Pi }_{1}\left(X\right)$:

${\pi }_{1}\left(X,x\right)={\mathrm{Aut}}_{{\Pi }_{1}\left(X\right)}\left(x\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 ${\Pi }_{1}\left(X\right)$ is precisely the set ${\Pi }_{0}\left(X\right)$ 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 ${\Pi }_{1}\left(X\right)$ 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 $\left(s,t\right):\mathrm{Mor}\left({\Pi }_{1}\left(X\right)\right)\to X×X$ 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, ${\Pi }_{0}\left(X\right)$ 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.

### ${\Pi }_{1}\left(X\right)$ 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 ${\Pi }_{1}\left(X,A\right)$, defined to be the full subgroupoid of ${\Pi }_{1}\left(X\right)$ 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 ${\Pi }_{1}\left(X,A\right)$ if $A$ is a union of orbits of the action.

Ronnie Brown is a big booster of ${\Pi }_{1}\left(X,A\right)$, which is fundamental to his development of homotopy theory in Elements of Modern Topology (1968).

Notice that ${\Pi }_{1}\left(X,X\right)$ recovers the full fundamental groupoid, while ${\Pi }_{1}\left(X,\left\{a\right\}\right)$ is the delooping of the fundamental group ${\pi }_{1}\left(X,a\right)$.

## 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 $\left({\pi }_{1}X\right)/N$ 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

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