Fundamental groupoid

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In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a topological space. In terms of category theory, the fundamental groupoid is a certain functor from the category of topological spaces to the category of groupoids.

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Definition

Let Template:Mvar be a topological space. Consider the equivalence relation on continuous paths in Template:Mvar in which two continuous paths are equivalent if they are homotopic with fixed endpoints. The fundamental groupoid Template:Math, or Template:Math, assigns to each ordered pair of points Template:Math in Template:Mvar the collection of equivalence classes of continuous paths from Template:Mvar to Template:Mvar. More generally, the fundamental groupoid of Template:Mvar on a set Template:Mvar restricts the fundamental groupoid to the points which lie in both Template:Mvar and Template:Mvar. This allows for a generalisation of the Van Kampen theorem using two base points to compute the fundamental group of the circle.[1]

As suggested by its name, the fundamental groupoid of Template:Mvar naturally has the structure of a groupoid. In particular, it forms a category; the objects are taken to be the points of Template:Mvar and the collection of morphisms from Template:Mvar to Template:Mvar 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.[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 Template:Math, the fundamental group of Template:Mvar based at Template:Mvar.

Basic properties

Given a topological space Template:Mvar, the path-connected components of Template:Mvar are naturally encoded in its fundamental groupoid; the observation is that Template:Mvar and Template:Mvar are in the same path-connected component of Template:Mvar if and only if the collection of equivalence classes of continuous paths from Template:Mvar to Template:Mvar is nonempty. In categorical terms, the assertion is that the objects Template:Mvar and Template:Mvar are in the same groupoid component if and only if the set of morphisms from Template:Mvar to Template:Mvar is nonempty.[4]

Suppose that Template:Mvar is path-connected, and fix an element Template:Mvar of Template:Mvar. One can view the fundamental group Template:Math as a category; there is one object and the morphisms from it to itself are the elements of Template:Math. The selection, for each Template:Mvar in Template:Mvar, of a continuous path from Template:Mvar to Template:Mvar, allows one to use concatenation to view any path in Template:Mvar as a loop based at Template:Mvar. This defines an equivalence of categories between Template:Math and the fundamental groupoid of Template:Mvar. More precisely, this exhibits Template:Math as a skeleton of the fundamental groupoid of Template:Mvar.[5]

The fundamental groupoid of a (path-connected) differentiable manifold Template:Mvar is actually a Lie groupoid, arising as the gauge groupoid of the universal cover of Template:Mvar.[6]

Bundles of groups and local systems

Given a topological space Template:Mvar, a local system is a functor from the fundamental groupoid of Template:Mvar to a category.[7] As an important special case, a bundle of (abelian) groups on Template:Mvar is a local system valued in the category of (abelian) groups. This is to say that a bundle of groups on Template:Mvar assigns a group Template:Math to each element Template:Mvar of Template:Mvar, and assigns a group homomorphism Template:Math to each continuous path from Template:Mvar to Template:Mvar. 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 define homology with coefficients in a bundle of abelian groups.[9]

When Template:Mvar satisfies certain conditions, a local system can be equivalently described as a locally constant sheaf.

Examples

The homotopy hypothesis

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.

References

Template:Reflist

Template:Category theory

  1. ↑ Template:Cite book
  2. ↑ Spanier, section 1.7; Lemma 6 and Theorem 7.
  3. ↑ Spanier, section 1.7; Theorem 8.
  4. ↑ Spanier, section 1.7; Theorem 9.
  5. ↑ May, section 2.5.
  6. ↑ Template:Cite book
  7. ↑ Spanier, chapter 1; Exercises F.
  8. ↑ Whitehead, section 6.1; page 257.
  9. ↑ Whitehead, section 6.2.
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