Automorphisms and homotopies of groupoids and crossed modules.
Summary:
This paper is concerned with the algebraic structure
of groupoids and crossed modules of groupoids.
We describe the group structure of the automorphism group of a
finite groupoid as a quotient of a semidirect product.
We then show that the automorphism group of a crossed module of groupoids
C, in the case when the range groupoid is connected,
may be determined from that of the crossed module of groups C_u
formed by restricting to a single object u.
We pay particular attention to the conjugation automorphisms
\wedge c_{p,q} of C.
Finally, we investigate the homotopies of a crossed module of groupoids
over a single object.
