Homomorphism from automorphism group of connected groupoid to outer automorphism group at a point
Consider a connected groupoid with its collection of points and denoting the paths between points . Let be a point. Denote by the group of automorphisms of the groupoid , and by the group under composition. Then, there is a canonical homomorphism:
defined as follows:
- Given , pick a path from to . Call this path .
- Consider now the map sending to . Note that this map is an automorphism of the group .
- Now, the automorphism we got depended on the choice of path from to . Choosing a different path gives an automorphism that differs from the original one by an inner automorphism. Hence, we get a well-defined element of .