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.
[...] 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 [...]
— 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 assigns to each ordered pair of points (p, q) in X the collection of equivalence classes of continuous paths from p to q.
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.[1] 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.[2]
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.[3]
Suppose that X is path-connected, and fix an element p of X. One can view the fundamental group π1(X, p) as a category; there is one object and the morphisms from it to itself are the elements of π1(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 π1(X, p) and the fundamental groupoid of X. More precisely, this exhibits π1(X, p) as a skeleton of the fundamental groupoid of X.[4]
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.[5] 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 Gp to each element p of X, and assigns a group homomorphism Gp → Gq 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.[6] One can define homology with coefficients in a bundle of abelian groups.[7]
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* : * → * }
The fundamental groupoid of the circle is connected and all of its vertex groups are isomorphic to (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
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.
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
J.P. 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 in nLab
fundamental infinity-groupoid in nLab
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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