Homomorphism from automorphism group of connected groupoid to outer automorphism group at a point

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Statement

Consider a connected groupoid with X its collection of points and P(x,y) denoting the paths between points x,y \in X. Let x \in X be a point. Denote by A the group of automorphisms of the groupoid X, and by G_x the group P(x,x) under composition. Then, there is a canonical homomorphism:

A \to \operatorname{Out}(G_x)

defined as follows:

  • Given g \in A, pick a path from x to y = g.x. Call this path p.
  • Consider now the map sending h \in G_x to p^{-1} \circ (g.h) \circ p. Note that this map is an automorphism of the group G_x.
  • Now, the automorphism we got depended on the choice of path from x to y. 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 \operatorname{Out}(G_x).