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Inner automorphisms of groupoids

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Richard Garner

Bergman has given the following abstract characterisation of the
inner automorphisms of a group G: they are exactly those
automorphisms of G which can be extended functorially along any
homomorphism G → H to an automorphism of H. This leads
naturally to a definition of "inner automorphism" applicable to
the objects of any category. Bergman and Hofstra-Parker-Scott have
computed these inner automorphisms for various structures including
k-algebras, monoids, lattices, unital rings, and
quandles-showing that, in each case, they are given by an obvious
notion of conjugation.

In this paper, we compute the inner automorphisms of groupoids,
showing that they are exactly the automorphisms induced by
conjugation by a bisection. The twist is that this result is
*false* in the category of groupoids and functors; to make
it true, we must instead work with the less familiar category of
groupoids and *cofunctors* in the sense of Higgins and
Mackenzie. Besides our main result, we also discuss generalisations
to topological and Lie groupoids, to categories and to partial
automorphisms, and examine the link with the theory of inverse
semigroups.

Keywords:
inner automorphisms, bisections, cofunctors

2020 MSC:
18B40

*Theory and Applications of Categories,*
Vol. 42, 2024,
No. 5, pp 84-101.

Published 2024-07-05.

http://www.tac.mta.ca/tac/volumes/42/5/42-05.pdf

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