Groupoid

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.