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 equivalence classes of homotopy of homotopy relative to $\partial I$ $[\gamma]$ 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.

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 $\Pi_1(X)$:

$$\pi_1(X,x) = Aut_{\Pi_1(X)}(x) \,.$$

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.)

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 is a special case of (Brown 06, 10.5.8). 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 $X$.

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.

$\Pi_1(X)$ with a chosen set of basepoints

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. spring

In higher category theory

See fundamental ∞-groupoid.

Simplicial version

See simplicial fundamental groupoid.

Related concepts

• fundamental group,

• fundamental 2-groupoid, fundamental ∞-groupoid

• fundamental theorem of covering spaces

• homotopy of smooth paths relative to their endpoints

• simplicial fundamental groupoid

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

• equivariant fundamental groupoid

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

References

A detailed treatment is available in

• Nicolas Bourbaki, Topologie Algébrique, Chapitres 1 à 4, Springer (1998, 2016) doi:10.1007/978-3-662-49361-8, ISBN 978-3-662-49361-8.

Monograph:

• Philip J. Higgins, §6 of: Categories and Groupoids, Mathematical Studies 32, van Nostrand New York (1971), Reprints in Theory and Applications of Categories 7 (2005) 1-195 [tac:tr7, pdf]

Review and Exposition:

• Jesper Møller, The fundamental group and covering spaces (2011) [arXiv:1106.5650, pdf, pdf]

• Alberto Santini, Topological groupoids (2011) [pdf, pdf]

  (about groupoids in topology, notably fundamental groupoids – not about topological groupoids)

• Introduction to Topology – 2

See also:

• 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.

• Ronnie 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)

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)

Discussion in the context of dynamical systems:

• Emmanuel Paul, Jean-Pierre Ramis, Dynamics on Wild Character Varieties, SIGMA 11 (2015) 068 [arXiv:1508.03122, doi:10.3842/SIGMA.2015.068]