Groupoid

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Definition

Definition in terms of categories

A groupoid is a (locally small) category where every morphism is invertible.

Definition in basic terms

A groupoid is the following:

  • A collection X of points.
  • A collection P of paths f, with two maps s:P \to X and t:P \to X, termed the source and terminus maps. Denote by P(x,y) the paths with source x and terminus y.
  • For points x,y,z \in X, a map \circ : P(y,z) \times P(x,y) \to P(x,z).

satisfying the following:

  • Associativity: For every x,y,z,w \in X, and every f \in P(x,y), g \in P(y,z), h \in P(z,w), h \circ (g \circ f) = (h \circ g) \circ f.
  • Identity element: For every x \in X, there exists a map e_x \in P(x,x) such that e_x \circ f = f for all f with t(f) = x, and f \circ e_x = f for all f with s(f) = x.
  • Inverses: For every f \in P(x,y), there exists a g \in P(y,x) such that f \circ g = e_y and g \circ f = e_x. Such a g is denoted by f^{-1}.

Further term:connected

A connected groupoid is a groupoid where P(x,y) is nonempty for any points x,y \in X.

Particular cases

  • For any groupoid, the paths from any point to itself form a group under composition.
  • The notion of groupoid with one point is equivalent to the notion of group, where the group is simply all the paths from that point to itself under composition.
  • If there is a path from x to y, the group of paths from x to itself is isomorphic to the group of paths from y to itself. Further, the isomorphism is uniquely specified up to (possibly) conjugation in the source group (or equivalently, up to conjugation in the target group).
  • Building on the above, we get a homomorphism from automorphism group of connected groupoid to outer automorphism group at a point.