Groupoid - Knowlpedia

Reflexive: a ∼ a a\sim a a ∼ a .
Symmetric: if a ∼ b a\sim b a ∼ b , then b ∼ a b\sim a b ∼ a .
Transitive: if a ∼ b a\sim b a ∼ b and b ∼ c b\sim c b ∼ c , then a ∼ c a\sim c a ∼ c .

Definition

A groupoid is a category $\mathcal G$ in which every morphism is an isomorphism. Thus for every $f:x\to y$ there is a morphism $f^{-1}:y\to x$ satisfying $f^{-1}f=\operatorname{id}_x$ and $ff^{-1}=\operatorname{id}_y$ .

Groups and equivalence relations

A group is the same thing as a groupoid with one object: its morphisms are the group elements and composition is multiplication. An equivalence relation on a set determines a groupoid having one arrow $x\to y$ exactly when $x$ and $y$ are equivalent. General groupoids allow several arrows between two objects and nontrivial automorphism groups.

Geometric role

Groupoids record objects together with reversible identifications. They arise from group actions, atlases, moduli problems, and the isomorphisms inside any category. Keeping the arrows retains symmetry information that the set of isomorphism classes discards.

References

Ronald Brown, Topology and Groupoids, BookSurge, 2006. Author-hosted record. Relevant: Chapters 1–2.