In algebraic topology, the fundamental groupoid of a topological space is a generalization of the fundamental group. It is a topological invariant, and so can be used to distinguish non-homeomorphic spaces. The fundamental groupoid captures information about both the connectedness and homotopy type of the space.
The homotopy hypothesis, an important conjecture in homotopy theory formulated by Alexander Grothendieck, states that a suitable generalization of the fundamental groupoid captures all information about the space up to homotopy equivalence.
[...] 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 [...]
Historically, the notion was introduced to generalize or give a natural presentation of van Kampen's theorem. For a path-connected space, the fundamental groupoid is (essentially) the same as the fundamental group; but the notion can be useful for a space that consists of a lot of path-connected components.
The fundamental groupoid of a (reasonable) topological space X is the groupoid (a category with invertible morphisms) where the objects are the points of X and the morphisms the homotopy classes of paths between two points.
The fundamental weak ∞-groupoid
In homotopy type theory
In intensional intuitionistic type theory (ITT), types have the structure of weak ∞-groupoids (for details and references, see Homotopy type theory#History). This observation led to the development of homotopy type theory, in which weak ∞-groupoids are a primitive or synthetic notion (meaning they are not defined within the theory).
- May, Ch. 2, § 5.
- 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