# Groupoid

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This is a variation of group|Find other variations of group | Read a survey article on varying group

## 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.