groupoid

A groupoid is a category in which every morphism is invertible. It consists of a collection of objects and a collection of arrows (morphisms) between objects, together with composition, identity arrows for each object, source and target maps s and t, and an inversion operation that assigns to each arrow its inverse. The composition is defined when the target of one arrow matches the source of the next, and every arrow has an inverse arrow such that their composition yields the appropriate identity at the source or target object.

Groups are special cases of groupoids: a groupoid with a single object is precisely a group, while a groupoid with multiple objects allows morphisms between different objects, each invertible. If there are no arrows between distinct objects, the groupoid decomposes as a disjoint union of groups, one for each object (the automorphism group of that object).

Examples include: the fundamental groupoid π1(X) of a topological space X has points of X as objects and homotopy classes of paths between points as arrows; composition is path concatenation, and inverses correspond to path reversal. The pair groupoid on a set X has objects X and a unique arrow from y to x for every pair (x, y); this is the coarse or indiscrete groupoid on X. Lie groupoids are groupoids in the category of smooth manifolds with smooth structure maps, generalizing Lie groups and playing a central role in differential geometry and foliation theory.

Terminology: for an object a, the automorphism group Aut(a) consists of arrows from a to a; the orbits of the groupoid partition the objects according to connectivity. Groupoids also appear in algebraic topology, algebraic geometry, and representation theory, and they can be described equivalently as small categories where all morphisms are isomorphisms or as groupoid objects in a given category.