Groupoid

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Definition

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

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 $f^{- 1}$ .

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

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.