# 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, [,,,]

## Definition[edit]

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 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[edit]

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[edit]

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} → *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[edit]

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

## The homotopy hypothesis[edit]

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[edit]

**^**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[edit]

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