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.
[...] 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, [,,,]
Let X be a topological space. Consider the equivalence relation on continuous paths in X in which two continuous paths are equivalent if they are homotopic with fixed endpoints. The fundamental groupoid assigns to each ordered pair of points (p, q) in X the collection of equivalence classes of continuous paths from p to q. More generally, the fundamental groupoid of X on a set S restricts the fundamental groupoid to the points which lie in both X and S. 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 X naturally has the structure of a groupoid. In particular, it forms a category; the objects are taken to be the points of X and the collection of morphisms from p to q 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 (p, p), the fundamental group of X based at p.
Given a topological space X, the path-connected components of X are naturally encoded in its fundamental groupoid; the observation is that p and q are in the same path-connected component of X if and only if the collection of equivalence classes of continuous paths from p to q is nonempty. In categorical terms, the assertion is that the objects p and q are in the same groupoid component if and only if the set of morphisms from p to q is nonempty.[4]
Suppose that X is path-connected, and fix an element p of X. One can view the fundamental group π1(X, p) as a category; there is one object and the morphisms from it to itself are the elements of π1(X, p). The selection, for each q in M, of a continuous path from p to q, allows one to use concatenation to view any path in X as a loop based at p. This defines an equivalence of categories between π1(X, p) and the fundamental groupoid of X. More precisely, this exhibits π1(X, p) as a skeleton of the fundamental groupoid of X.[5]
The fundamental groupoid of a (path-connected) differentiable manifold X is actually a Lie groupoid, arising as the gauge groupoid of the universal cover of X.[6]
Given a topological space X, a local system is a functor from the fundamental groupoid of X to a category.[7] As an important special case, a bundle of (abelian) groups on X is a local system valued in the category of (abelian) groups. This is to say that a bundle of groups on X assigns a group Gp to each element p of X, and assigns a group homomorphism Gp → Gq to each continuous path from p to q. 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 X satisfies certain conditions, a local system can be equivalently described as a locally constant sheaf.
- The fundamental groupoid of the singleton space is the trivial groupoid (a groupoid with one object * and one morphism Hom(*, *) = { id* : * → * }
- The fundamental groupoid of the circle is connected and all of its vertex groups are isomorphic to , the additive group of integers.
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.