Questions: The Fundamental Groupoid of a Space

Question 1

What are the morphisms from point x to point y in the fundamental groupoid Π₁(X) of a topological space X?

• A. All continuous functions from x to y in the space X
• B. All continuous paths from x to y, counted without any identification
• C. Homotopy classes of continuous paths from x to y, where two paths are identified if one can be continuously deformed into the other while keeping endpoints fixed
• D. The group of loops at x tensored with the group of loops at y

Answer: (C)

Morphisms in Π₁(X) are homotopy classes of paths — not individual paths, because two paths related by a homotopy (a continuous deformation fixing endpoints) are identified as the same morphism. This quotienting is essential: without it, composition (path concatenation) would not be strictly associative, only associative up to homotopy. The identification ensures the groupoid axioms hold on the nose.

Question 2

A space X consists of two disjoint circles (disconnected). How does the fundamental groupoid Π₁(X) capture the disconnection better than the fundamental group π₁(X, x₀) based at a point in one circle?

• A. The groupoid contains strictly more morphisms than the fundamental group, encoding more loop information
• B. The fundamental group at x₀ captures only loops in one component, missing the other component entirely; the groupoid has no morphisms between components, directly encoding the disconnection in its structure
• C. The fundamental group can be based simultaneously at points in both circles, capturing the full space
• D. Groupoids always contain more information than groups regardless of the space

Answer: (B)

To use the fundamental group, you must choose a basepoint in one component. That component's loops are captured, but the other component is invisible — there are no paths connecting them. The fundamental groupoid avoids this problem: it has objects in both components, morphisms within each component (the loop information), and crucially no morphisms between components — directly encoding the disconnection as an absence of morphisms. No basepoint choice is required.

Question 3

The fundamental group π₁(X, x₀) at a basepoint x₀ is exactly the automorphism group of the object x₀ in the fundamental groupoid Π₁(X).

• T. True
• F. False

Answer: True

In Π₁(X), the morphisms from x₀ to x₀ are homotopy classes of loops based at x₀ — exactly the elements of π₁(X, x₀). Composition of morphisms in the groupoid corresponds to loop concatenation, and the group structure of π₁ matches the automorphism group structure at x₀. The fundamental groupoid thus packages the fundamental groups at all basepoints simultaneously, with the automorphism groups being the 'diagonal' of the structure.

Question 4

For a path-connected space, the fundamental groupoid contains strictly more topological information than the fundamental group at any single basepoint.

• T. True
• F. False

Answer: False

For a path-connected space, all automorphism groups Aut_{Π₁(X)}(x) are isomorphic to each other (conjugate via any path between basepoints), so the groupoid and the fundamental group at any single point contain the same information up to group isomorphism. The advantage of the groupoid is not more information but greater naturality: no arbitrary basepoint choice, cleaner functoriality, and more natural statements of theorems like van Kampen. For disconnected spaces, the groupoid does encode strictly more information.

Question 5

Why is the fundamental groupoid considered more natural than the fundamental group for studying topology? What concrete advantage does it provide for disconnected spaces?

Model answer: The fundamental groupoid Π₁(X) requires no basepoint choice — it treats all points of X symmetrically as objects. Any continuous map f: X → Y induces a functor Π₁(f): Π₁(X) → Π₁(Y) without fixing any basepoint, making the construction fully functorial. The fundamental group, by contrast, requires choosing a basepoint, and a map induces a group homomorphism only between groups at corresponding basepoints. For disconnected spaces, the groupoid encodes the full connectivity structure: components with no path between them have no morphisms between them in the groupoid, directly capturing disconnection. The fundamental group at a single basepoint sees only its own component.

The groupoid's naturality also shows up in the van Kampen theorem: the groupoid version (Π₁(X∪Y) is the pushout of Π₁(X) and Π₁(Y) over Π₁(X∩Y)) is cleaner and more general than the group version, which requires basepoint conditions and separate handling of path-connected intersections.