Define objects as points of a topological space X, and morphisms from x to y as homotopy classes of paths from x to y, with composition as path concatenation. What is the correct algebraic classification of this structure?
AA group, because path composition is associative and every path has a reverse
BA groupoid, because every path has an inverse (its reversal) but composition is only defined when the endpoint of one path matches the start of the next
CA category but not a groupoid, because paths between distinct points cannot be inverted in the homotopy sense
DNeither a group nor a groupoid, because path concatenation is not strictly associative
This is the fundamental groupoid Π₁(X). It is a groupoid — not a group — because it has multiple objects (the points of X) and morphisms only compose when endpoints match. Every path from x to y has an inverse (the path traversed backwards), making every morphism an isomorphism. It cannot be a group because a group has a single object (all elements compose with all others), whereas here composition x→y composed with z→w is undefined unless y = z.
Question 2 Multiple Choice
How does a groupoid fundamentally differ from a group?
AA groupoid relaxes associativity — composition is only associative when all morphisms have the same source
BA groupoid allows morphisms without inverses, whereas a group requires every element to have an inverse
CA groupoid has multiple objects, so composition is only defined when the target of one morphism equals the source of the next
DA groupoid requires all objects to be isomorphic, whereas a group has a single identity element
The defining difference is that a groupoid has multiple objects. In a group, there is exactly one object, so every pair of elements (morphisms) can be composed. In a groupoid, a morphism f: A → B and g: C → D can only be composed as g ∘ f when B = C. Both structures require all morphisms to be invertible — that is not the distinction. Option B is wrong because a groupoid requires all morphisms to have inverses (that is what makes it a groupoid rather than just a category).
Question 3 True / False
A group can be viewed as a special case of a groupoid — specifically, a groupoid with exactly one object.
TTrue
FFalse
Answer: True
This is a clean categorical fact. In a one-object category where every morphism is an isomorphism, the single object provides the common source and target for all morphisms, so any two morphisms compose. The set of all morphisms with composition as the binary operation satisfies the group axioms: associativity (inherited from category axioms), identity (the single object's identity morphism), and inverses (since every morphism is an isomorphism). The groupoid concept genuinely extends the group concept by allowing more than one object.
Question 4 True / False
In a groupoid, any two morphisms can be composed, just as any two elements of a group can be multiplied.
TTrue
FFalse
Answer: False
This is the core distinction between groups and groupoids. In a group (viewed as a one-object category), every morphism shares the same source and target, so all pairs compose. In a groupoid with multiple objects, morphism f: A → B and morphism g: C → D compose as g ∘ f only when B = C — the target of f must equal the source of g. This partial composition is what makes groupoids useful for structures where 'elements' only interact with compatible partners, such as paths between specific points in a space.
Question 5 Short Answer
Explain why an equivalence relation on a set can be viewed as a groupoid, and what property of the equivalence relation corresponds to invertibility of morphisms.
Think about your answer, then reveal below.
Model answer: Given an equivalence relation ~ on a set S, define a category where objects are elements of S, and there is exactly one morphism from x to y whenever x ~ y (and no morphism otherwise). Composition is forced (there is at most one morphism between any two objects, so the composition of x→y and y→z is the unique morphism x→z, which exists because ~ is transitive). This is a groupoid because symmetry of ~ guarantees that if x ~ y then y ~ x — so every morphism x→y has an inverse y→x. The identity morphisms exist because ~ is reflexive.
The three axioms of an equivalence relation correspond exactly to the categorical structure: reflexivity (x ~ x) gives identity morphisms; symmetry (x ~ y implies y ~ x) gives inverses, making every morphism an isomorphism; transitivity (x ~ y and y ~ z imply x ~ z) gives composition. This example shows how groupoids unify two seemingly different mathematical objects — equivalence relations and groups — under a single framework. A group is the special case where every element is related to every other; an equivalence relation is the special case where at most one morphism exists between any two objects.