Fundamental groupoid


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.

Motivation

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 the same as the fundamental group; but the notion can be useful for a space that consists of a lot of path-connected components.

Definition

The fundamental groupoid of a topological space X is the groupoid where the objects are the points of X and the morphisms the homotopy classes of paths between two points.
For each point x of X, the automorphism group of x is precisely the group of classes of loops at x, commonly known as the fundamental group of X at x.

Examples

The fundamental weak ∞-groupoid

The homotopy hypothesis

The homotopy hypothesis is an important conjecture in homotopy theory formulated by Alexander Grothendieck. It states that a suitable generalization of the fundamental groupoid captures all information about the space up to homotopy equivalence.

In homotopy type theory

In intensional intuitionistic type theory, types have the structure of weak ∞-groupoids. This observation led to the development of homotopy type theory, in which weak ∞-groupoids are a primitive or synthetic notion.