groupoids

A groupoid is a category in which every morphism is invertible. It consists of a class of objects and a class of arrows (morphisms) between these objects, together with composition of arrows, identity arrows for each object, source and target maps, and for every arrow there exists an inverse. A groupoid is called small if both its object set and arrow set are sets; in general, it may have a proper class of objects and arrows.

Groups arise as a special case: a group can be viewed as a groupoid with a single object, in which all arrows correspond to group elements and composition is ordinary multiplication. The pair groupoid on a set X has as arrows all ordered pairs (x,y) in X×X, with source x and target y; composition is defined when the target of one arrow matches the source of the next. The fundamental groupoid π1(X) of a topological space has the points of X as objects and homotopy classes of paths as arrows, with composition given by path concatenation.

Key features include isotropy groups Aut(x), the automorphism group of an object x, and the ability to model equivalence relations and group actions. An equivalence relation on a set can be encoded as a groupoid with objects the elements and arrows given by related pairs. An action of a group G on a set M yields an action groupoid M⋉G.

Nerves and classifying spaces link groupoids to topology: the nerve of a groupoid is a simplicial set, and its geometric realization is the classifying space BG. The concepts extend to Lie groupoids (smooth manifolds of objects and arrows) and differentiable stacks, connecting groupoid theory to geometry and algebraic topology.