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.

Definition

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

As suggested by its name, the fundamental groupoid of naturally has the structure of a groupoid. In particular, it forms a category; the objects are taken to be the points of and the collection of morphisms from to 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. 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.

Note that the fundamental groupoid assigns, to the ordered pair , the fundamental group of based at .

Basic properties

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

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

The fundamental groupoid of a (path-connected) differentiable manifold is actually a Lie groupoid, arising as the gauge groupoid of the universal cover of .

Bundles of groups and local systems

Given a topological space , a '' local system'' is a functor from the fundamental groupoid of to a category. As an important special case, a ''bundle of (abelian) groups'' on is a local system valued in the category of (abelian) groups. This is to say that a bundle of groups on assigns a group to each element of , and assigns a group homomorphism to each continuous path from to . 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. One can define homology with coefficients in a bundle of abelian groups.Whitehead, section 6.2.

When 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
* The fundamental groupoid of the circle is connected and all of its vertex groups are isomorphic to $(\mathbb,+)$, 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

* Ronald Brown
Topology and groupoids.
Third edition of ''Elements of modern topology'' cGraw-Hill, New York, 1968 With 1 CD-ROM (Windows, Macintosh and UNIX). BookSurge, LLC, Charleston, SC, 2006. xxvi+512 pp.
* 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) ''

*J. Peter May
A concise course in algebraic topology.
Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1999. x+243 pp.
* Edwin H. Spanier. Algebraic topology. Corrected reprint of the 1966 original. Springer-Verlag, New York-Berlin, 1981. xvi+528 pp.
* George W. Whitehead. Elements of homotopy theory. Graduate Texts in Mathematics, 61. Springer-Verlag, New York-Berlin, 1978. xxi+744 pp.

External links

* The website of Ronald Brown, a prominent author on the subject of groupoids in topology: http://groupoids.org.uk/
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Higher category theory
Algebraic topology

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