Inner automorphisms of groupoids

Richard Garner – 13 March 2019

The inner automorphisms of a group G are those given by conjugation by an element a of G. George Bergman has characterised them abstractly as the automorphisms of G which admit a functorial extension along any group homomorphism G --> H to an automorphism of H.

This notion of extended inner automorphism makes sense for objects of any category C. The purpose of this talk is to characterise the inner automorphisms of groupoids. We show that they are exactly the automorphisms given by conjugation by bisections. The twist is that this has to be done not in the category of groupoids and functors, but in the category of groupoids and cofunctors.