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.

[...] 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)

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 (essentially) the same as the fundamental group;[1] but the notion can be useful for a space that consists of a lot of path-connected components.

Definition

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.[1]

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 groupoid of the singleton space is the trivial groupoid (a groupoid with one object * and one morphism {{{1}}}}
  • The fundamental groupoid of the circle is connected and all of its vertex groups are isomorphic to (Z, +), the additive group of integers.

Generalizations

The fundamental weak ∞-groupoid
See also: ∞-groupoid

The homotopy hypothesis
Main article: 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 (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).

See also

References
  1. May, Ch. 2, § 5.
  • Brown, Ronald (March 2006). Topology and Groupoids. North Charleston: CreateSpace. ISBN 9781419627224. OCLC 712629429.
  • May, J. A Concise Course in Algebraic Topology


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