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Actions and automorphisms of crossed modules. (English) Zbl 0719.20018
The reviewer suggested in his paper [in Low Dimensional Topology, Lond. Math. Soc. Lect. Note Ser. 48, 215-238 (1982; Zbl 0484.55007)] that crossed modules could be regarded as ‘2-dimensional groups’. In this direction, the paper under review extends some aspects of the theory of automorphisms of groups to automorphisms of crossed modules. Some of the basic ideas come from J. H. C. Whitehead [Ann. Math., II. Ser. 49, 610-640 (1948; Zbl 0041.101)] and A. S.-T. Lue [Bull. Lond. Math. Soc. 11, 8-16 (1979; Zbl 0416.20030)].
Section 1 defines the actor \({\mathcal A}(T,G,\partial)\) of a crossed module \(\partial: T\to G\) to be the crossed module \(\Delta\) : D(G,T)\(\to Aut(T,G,\partial)\) where D(G,T) is Whitehead’s (loc. cit.) group of derivations \(G\to T\), and Aut(T,G,\(\partial)\) is the group of automorphisms of the crossed module. There is a morphism of crossed modules \(<\eta,\gamma >: (T,G,\partial)\to {\mathcal A}(T,G,\partial)\), whose image is called the inner actor of the crossed module, and whose kernel is the centre of the crossed module. Section 2 defines actions and semidirect products of crossed modules. Section 3 shows that the above morphism \(<\eta,\gamma >\) forms part of a crossed square in the sense of D. Guin-Waléry and J.-L. Loday [Algebraic K-theory, Proc. Conf., Lect. Notes Math. 854, 179-216 (1981; Zbl 0461.18007)]. This helps to explain the relation between crossed squares and 2-cat-groups, as in J.-L. Loday [J. Pure Appl. Algebra 24, 179-202 (1982; Zbl 0491.55004)], since a crossed square is also a crossed module internal to the category of crossed modules. This result is also related to work of R. Brown and N. D. Gilbert on automorphism structures for crossed modules, obtained from a different viewpoint [in Proc. Lond. Math. Soc., III. Ser. 59, 51-73 (1989; Zbl 0645.18007)]. Section 4 considers the analogues for crossed modules of various results on completeness of groups, for example results of Burnside and of J. Rose.
{Results of this type have been applied by L. Breen [Théorie de Schreier supérieure (Prepublication Université Paris-Nord 91-3)]. They fit into the notion of ‘Higher order symmetry’, as suggested by Brown and Gilbert (loc. cit.).}
Reviewer: R.Brown (Bangor)

MSC:
20F28 Automorphism groups of groups
18D35 Structured objects in a category (MSC2010)
55Q05 Homotopy groups, general; sets of homotopy classes
55U99 Applied homological algebra and category theory in algebraic topology
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
18G55 Nonabelian homotopical algebra (MSC2010)
19D55 \(K\)-theory and homology; cyclic homology and cohomology
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