The category of 1-cat groups, which is equivalent to the category of crossed modules, has internal object actions which are representable (by internal automorphism groups). Moreover, it is known that the crossed module, corresponding to the representing object [X] = Aut(X) $ associated with a 1-cat group X, must be isomorphic to the Norrie actor of the crossed module corresponding to X. We recall the description of Aut(X) from the author's PhD thesis, and construct that isomorphism explicitly.
Keywords: crossed module, action of an object, $ 1 $-cat group, internal automorphism, actor
2010 MSC: 18G55, 18D15, 18D05, 20L05
Theory and Applications of Categories, Vol. 34, 2019, No. 36, pp 1165-1178.