Inalgebraic topology, thefundamentalgroupoid is a certaintopological invariant of atopological space. It can be viewed as an extension of the more widely-knownfundamental group; as such, it captures information about thehomotopy type of a topological space. In terms ofcategory theory, the fundamental groupoid is a certainfunctor from the category of topological spaces to the category ofgroupoids.

[...] 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)

LetX be atopological space. Consider the equivalence relation oncontinuous paths inX in which two continuous paths are equivalent if they arehomotopic with fixed endpoints. The fundamental groupoidΠ(X), orΠ₁(X), assigns to each ordered pair of points(p,q) inX the collection of equivalence classes of continuous paths fromp toq. More generally, the fundamental groupoid ofX on a setS restricts the fundamental groupoid to the points which lie in bothX andS. This allows for a generalisation of theVan Kampen theorem using two base points to compute the fundamental group of the circle.^[1]

As suggested by its name, the fundamental groupoid ofX naturally has the structure of agroupoid. In particular, it forms a category; the objects are taken to be the points ofX and the collection of morphisms fromp toq is the collection of equivalence classes given above. The fact that this satisfies the definition of a category amounts to thestandard 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), thefundamental group ofX based atp.

Given a topological spaceX, thepath-connected components ofX are naturally encoded in its fundamental groupoid; the observation is thatp andq are in the same path-connected component ofX if and only if the collection of equivalence classes of continuous paths fromp toq is nonempty. In categorical terms, the assertion is that the objectsp andq are in the same groupoid component if and only if the set of morphisms fromp toq is nonempty.^[4]

Suppose thatX is path-connected, and fix an elementp ofX. One can view the fundamental groupπ₁(X,p) as a category; there is one object and the morphisms from it to itself are the elements ofπ₁(X,p). The selection, for eachq inM, of a continuous path fromp toq, allows one to use concatenation to view any path inX as a loop based atp. This defines anequivalence of categories betweenπ₁(X,p) and the fundamental groupoid ofX. More precisely, this exhibitsπ₁(X,p) as askeleton of the fundamental groupoid ofX.^[5]

The fundamental groupoid of a (path-connected)differentiable manifoldX is actually aLie groupoid, arising as the gauge groupoid of theuniversal cover ofX.^[6]

Given a topological spaceX, alocal system is afunctor from the fundamental groupoid ofX to a category.^[7] As an important special case, abundle of (abelian) groups onX is a local system valued in the category of (abelian) groups. This is to say that a bundle of groups onX assigns a groupG_p to each elementp ofX, and assigns agroup homomorphismG_p →G_q to each continuous path fromp toq. 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 definehomology with coefficients in a bundle of abelian groups.^[9]

WhenX satisfies certain conditions, a local system can be equivalently described as alocally constant sheaf.

• The fundamental groupoid of thesingleton space is the trivial groupoid (a groupoid with one object * and one morphismHom(*, *) = { id_* : * → *}
• The fundamental groupoid of thecircle is connected and all of itsvertex groups are isomorphic to${\displaystyle (\mathbb{Z},+)}$ , theadditive group ofintegers.

Thehomotopy hypothesis, a well-knownconjecture inhomotopy theory formulated byAlexander Grothendieck, states that a suitablegeneralization of the fundamental groupoid, known as the fundamental∞-groupoid, capturesall information about a topological spaceup toweak homotopy equivalence.

• Homotopy category of an ∞-category
• core of a category (while the fundamental groupoid is a left adjoint to the inclusion of Gprd, the core is a right adjoint)

• Ronald Brown.Topology and groupoids. Third edition ofElements 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

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