fundamental groupoid

Let {{mvar|X}} be a topological space. Consider the equivalence relation on continuous paths in {{mvar|X}} in which two continuous paths are equivalent if they are homotopic with fixed endpoints. The fundamental groupoid {{math|Π(X)}}, or {{math|Π₁(X)}}, assigns to each ordered pair of points {{math|(p, q)}} in {{mvar|X}} the collection of equivalence classes of continuous paths from {{mvar|p}} to {{mvar|q}}. More generally, the fundamental groupoid of {{mvar|X}} on a set {{mvar|S}} restricts the fundamental groupoid to the points which lie in both {{mvar|X}} and {{mvar|S}}. This allows for a generalisation of the Van Kampen theorem using two base points to compute the fundamental group of the circle.{{Cite book |last=Brown |first=Ronald |url=https://www.worldcat.org/oclc/712629429 |title=Topology and Groupoids. |date=2006 |publisher=CreateSpace |others=Academic Search Complete |isbn=978-1-4196-2722-4 |location=North Charleston |oclc=712629429 |author-link=Ronald Brown (mathematician)}}

As suggested by its name, the fundamental groupoid of {{mvar|X}} naturally has the structure of a groupoid. In particular, it forms a category; the objects are taken to be the points of {{mvar|X}} and the collection of morphisms from {{mvar|p}} to {{mvar|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.Spanier, section 1.7; Lemma 6 and Theorem 7. 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.Spanier, section 1.7; Theorem 8.

Note that the fundamental groupoid assigns, to the ordered pair {{math|(p, p)}}, the fundamental group of {{mvar|X}} based at {{mvar|p}}.

Given a topological space {{mvar|X}}, the path-connected components of {{mvar|X}} are naturally encoded in its fundamental groupoid; the observation is that {{mvar|p}} and {{mvar|q}} are in the same path-connected component of {{mvar|X}} if and only if the collection of equivalence classes of continuous paths from {{mvar|p}} to {{mvar|q}} is nonempty. In categorical terms, the assertion is that the objects {{mvar|p}} and {{mvar|q}} are in the same groupoid component if and only if the set of morphisms from {{mvar|p}} to {{mvar|q}} is nonempty.Spanier, section 1.7; Theorem 9.

Suppose that {{mvar|X}} is path-connected, and fix an element {{mvar|p}} of {{mvar|X}}. One can view the fundamental group {{math|π₁(X, p)}} as a category; there is one object and the morphisms from it to itself are the elements of {{math|π₁(X, p)}}. The selection, for each {{mvar|q}} in {{mvar|M}}, of a continuous path from {{mvar|p}} to {{mvar|q}}, allows one to use concatenation to view any path in {{mvar|X}} as a loop based at {{mvar|p}}. This defines an equivalence of categories between {{math|π₁(X, p)}} and the fundamental groupoid of {{mvar|X}}. More precisely, this exhibits {{math|π₁(X, p)}} as a skeleton of the fundamental groupoid of {{mvar|X}}.May, section 2.5.

The fundamental groupoid of a (path-connected) differentiable manifold {{mvar|X}} is actually a Lie groupoid, arising as the gauge groupoid of the universal cover of {{mvar|X}}.{{Cite book |last=Mackenzie |first=Kirill C. H. |url=https://www.cambridge.org/core/books/general-theory-of-lie-groupoids-and-lie-algebroids/DA70C6FAF52F88FB471F62DD68049608 |title=General Theory of Lie Groupoids and Lie Algebroids |date=2005 |publisher=Cambridge University Press |isbn=978-0-521-49928-6 |series=London Mathematical Society Lecture Note Series |location=Cambridge |doi=10.1017/cbo9781107325883}}

Given a topological space {{mvar|X}}, a local system is a functor from the fundamental groupoid of {{mvar|X}} to a category.Spanier, chapter 1; Exercises F. As an important special case, a bundle of (abelian) groups on {{mvar|X}} is a local system valued in the category of (abelian) groups. This is to say that a bundle of groups on {{mvar|X}} assigns a group {{math|G_p}} to each element {{mvar|p}} of {{mvar|X}}, and assigns a group homomorphism {{math|G_p → G_q}} to each continuous path from {{mvar|p}} to {{mvar|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.Whitehead, section 6.1; page 257. One can define homology with coefficients in a bundle of abelian groups.Whitehead, section 6.2.

When {{mvar|X}} satisfies certain conditions, a local system can be equivalently described as a locally constant sheaf.

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

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.

• Homotopy category of an ∞-category

{{Reflist}}

• Ronald Brown. [http://groupoids.org.uk/topgpds.html 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. [http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf 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}}

• The website of Ronald Brown, a prominent author on the subject of groupoids in topology: http://groupoids.org.uk/
• {{nlab|id=fundamental+groupoid|title=fundamental groupoid}}
• {{nlab|id=fundamental+infinity-groupoid|title=fundamental infinity-groupoid}}

{{Category theory}}

Category:Higher category theory

Category:Algebraic topology