Groupoid
From Groupprops
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Contents
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 of points.
- A collection of paths , with two maps and , termed the source and terminus maps. Denote by the paths with source and terminus .
- For points , a map .
satisfying the following:
- Associativity: For every , and every , .
- Identity element: For every , there exists a map such that for all with , and for all with .
- Inverses: For every , there exists a such that and . Such a is denoted by .
Further term:connected
A connected groupoid is a groupoid where is nonempty for any points .
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 to , the group of paths from to itself is isomorphic to the group of paths from 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.