Fundamental groupoid

In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a topological space. In terms of category theory, the fundamental groupoid is a certain functor from the category of topological spaces to the category of groupoids.

[...] 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 à 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, [,,,]
— Alexander Grothendieck, Esquisse d'un Programme (Section 2, English translation)

Definition

Let $X$ be a topological space. Consider the equivalence relation on continuous paths in $X$ in which two continuous paths are equivalent if they are homotopic with fixed endpoints. The fundamental groupoid $Π(X)$ , or $Π 1 (X)$ , assigns to each ordered pair of points $(p, q)$ in $X$ the collection of equivalence classes of continuous paths from $p$ to $q$ . More generally, the fundamental groupoid of $X$ on a set $S$ restricts the fundamental groupoid to the points which lie in both $X$ and $S$ . This allows for a generalisation of the Van Kampen theorem using two base points to compute the fundamental group of the circle.^[1]

As suggested by its name, the fundamental groupoid of $X$ naturally has the structure of a groupoid. In particular, it forms a category; the objects are taken to be the points of $X$ and the collection of morphisms from $p$ to $q$ is the collection of equivalence classes given above. The fact that this satisfies the definition of a category amounts to the standard fact that the equivalence class of the concatenation of two paths only depends on the equivalence classes of the individual paths.^[2] Likewise, the fact that this category is a groupoid, which asserts that every morphism is invertible, amounts to the standard fact that one can reverse the orientation of a path, and the equivalence class of the resulting concatenation contains the constant path.^[3]

Note that the fundamental groupoid assigns, to the ordered pair $(p, p)$ , the fundamental group of $X$ based at $p$ .

Basic properties

Given a topological space $X$ , the path-connected components of $X$ are naturally encoded in its fundamental groupoid; the observation is that $p$ and $q$ are in the same path-connected component of $X$ if and only if the collection of equivalence classes of continuous paths from $p$ to $q$ is nonempty. In categorical terms, the assertion is that the objects $p$ and $q$ are in the same groupoid component if and only if the set of morphisms from $p$ to $q$ is nonempty.^[4]

Suppose that $X$ is path-connected, and fix an element $p$ of $X$ . One can view the fundamental group $π 1 (X, p)$ as a category; there is one object and the morphisms from it to itself are the elements of $π 1 (X, p)$ . The selection, for each $q$ in $M$ , of a continuous path from $p$ to $q$ , allows one to use concatenation to view any path in $X$ as a loop based at $p$ . This defines an equivalence of categories between $π 1 (X, p)$ and the fundamental groupoid of $X$ . More precisely, this exhibits $π 1 (X, p)$ as a skeleton of the fundamental groupoid of $X$ .^[5]

The fundamental groupoid of a (path-connected) differentiable manifold $X$ is actually a Lie groupoid, arising as the gauge groupoid of the universal cover of $X$ .^[6]

Bundles of groups and local systems

Given a topological space $X$ , a local system is a functor from the fundamental groupoid of $X$ to a category.^[7] As an important special case, a bundle of (abelian) groups on $X$ is a local system valued in the category of (abelian) groups. This is to say that a bundle of groups on $X$ assigns a group $G p$ to each element $p$ of $X$ , and assigns a group homomorphism $G p \to G q$ to each continuous path from $p$ to $q$ . In order to be a functor, these group homomorphisms are required to be compatible with the topological structure, so that homotopic paths with fixed endpoints define the same homomorphism; furthermore the group homomorphisms must compose in accordance with the concatenation and inversion of paths.^[8] One can define homology with coefficients in a bundle of abelian groups.^[9]

When $X$ satisfies certain conditions, a local system can be equivalently described as a locally constant sheaf.

Examples

The fundamental groupoid of the singleton space is the trivial groupoid (a groupoid with one object * and one morphism $Hom(*, *) = { id * : * \to *$ }
The fundamental groupoid of the circle is connected and all of its vertex groups are isomorphic to $(\mathbb {Z} ,+)$ , the additive group of integers.

The homotopy hypothesis

The homotopy hypothesis, a well-known conjecture in homotopy theory formulated by Alexander Grothendieck, states that a suitable generalization of the fundamental groupoid, known as the fundamental ∞-groupoid, captures all information about a topological space up to weak homotopy equivalence.

References

^ Brown, Ronald (2006). Topology and Groupoids. Academic Search Complete. North Charleston: CreateSpace. ISBN 978-1-4196-2722-4. OCLC 712629429.
^ Spanier, section 1.7; Lemma 6 and Theorem 7.
^ Spanier, section 1.7; Theorem 8.
^ Spanier, section 1.7; Theorem 9.
^ May, section 2.5.
^ Mackenzie, Kirill C. H. (2005). General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press. doi:10.1017/cbo9781107325883. ISBN 978-0-521-49928-6.
^ Spanier, chapter 1; Exercises F.
^ Whitehead, section 6.1; page 257.
^ Whitehead, section 6.2.

Ronald Brown. Topology and groupoids. Third edition of Elements of modern topology [McGraw-Hill, New York, 1968]. With 1 CD-ROM (Windows, Macintosh and UNIX). BookSurge, LLC, Charleston, SC, 2006. xxvi+512 pp. ISBN 1-4196-2722-8
Brown, R., Higgins, P. J. and Sivera, R., Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids. Tracts in Mathematics Vol 15. European Mathematical Society (2011). (663+xxv pages) ISBN 978-3-03719-083-8
J. Peter May. A concise course in algebraic topology. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1999. x+243 pp. ISBN 0-226-51182-0, 0-226-51183-9
Edwin H. Spanier. Algebraic topology. Corrected reprint of the 1966 original. Springer-Verlag, New York-Berlin, 1981. xvi+528 pp. ISBN 0-387-90646-0
George W. Whitehead. Elements of homotopy theory. Graduate Texts in Mathematics, 61. Springer-Verlag, New York-Berlin, 1978. xxi+744 pp. ISBN 0-387-90336-4

External links

The website of Ronald Brown, a prominent author on the subject of groupoids in topology: http://groupoids.org.uk/
fundamental groupoid at the nLab
fundamental infinity-groupoid at the nLab

[1] Brown, Ronald (2006). Topology and Groupoids. Academic Search Complete. North Charleston: CreateSpace. ISBN 978-1-4196-2722-4. OCLC 712629429.

[2] Spanier, section 1.7; Lemma 6 and Theorem 7.

[3] Spanier, section 1.7; Theorem 8.

[4] Spanier, section 1.7; Theorem 9.

[5] May, section 2.5.

[6] Mackenzie, Kirill C. H. (2005). General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press. doi:10.1017/cbo9781107325883. ISBN 978-0-521-49928-6.

[7] Spanier, chapter 1; Exercises F.

[8] Whitehead, section 6.1; page 257.

[9] Whitehead, section 6.2.

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