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Algebraic models of 3-types and automorphism structures for crossed modules. (English) Zbl 0645.18007

It is standard that sets correspond to homotopy 0-types, and groups, which arise as automorphisms of sets, correspond to (connected) 1-types. Associated to the automorphism group Aut G of a group G is the homomorphism \(\chi\) : \(G\to Aut G\) that sends \(g\in G\) to the inner automorphism \(x\mapsto g^{-1}xg\). This homomorphism is one of the standard examples of a crossed module. Now crossed modules are algebraic models of 2-types: there is a classifying space functor B: (crossed modules)\(\to (CW\) complexes) such that, if \(\partial: M\to P\) is a crossed module, then B(M\(\to P)\) has fundamental group coker \(\partial\), second homotopy group ker \(\partial\) and all other higher homotopy groups trivial [cf. J.-L. Loday, J. Pure Appl. Algebra 24, 179-202 (1982; Zbl 0491.55004)]. Since groups are algebraic models of 1-types, we see that the crossed module \(\chi\) : \(G\to Aut G\) furnishes an automorphism structure for a model of a 1-type that is naturally considered as a model of a 2-type. The original motivation for the present work was to investigate whether there could be found, for crossed odules, an automorphism structure which could be considered as an algebraic model of a 3-type.
We are able to derive such a structure by embedding the category of crossed modules into the category of crossed complexes, which was shown by the first author and P. J. Higgins [J. Pure Appl. Algebra 47, 1- 33 (1987; Zbl 0621.55009)] to be monoidal closed (with tensor product \(\otimes\) and internal hom-functor \(CRS(-,-)).\) Then for any crossed complex C, \(END(C)=CRS(C,C)\) is a monoid object with respect to \(\otimes\) in the category of crossed complexes, and END(C) has a submonoid which may reasonably be labelled AUT(C). If C is in fact a crossed module, regarded as a 2-truncated crossed complex, then END(C) is also a crossed module, and its monoid structure endows END(C) with additional properties of being “braided” and “semiregular”. The automorphism object AUT(C) inherits from END(C) a braiding and a stronger internal symmetry, making it braided and “regular”. (The term “braiding” derives from the preprint “Braided monoidal categories” by A. Joyal and R. Street.)
We establish the role of AUT(C) as an algebraic model of a 3-type, and link the construction to classical algebraic homotopy theory, in our main result. Theorem. The category of braided, regular crossed modules is equivalent to the category of simplicial groups with Moore complex of length two. D. Conduché [J. Pure Appl. Algebra 34, 155-178 (1984; Zbl 0554.20014)] has shown that the category of simplicial groups with Moore complex of length two is also equivalent to the category of 2- crossed modules. This equivalence allows a succinct description of AUT(C) in terms of derivations, as considered by 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)]. It is also convenient for the computation of the homotopy groups of the corresponding 3-type in certain special cases.
The equivalence, due to the first author and P. J. Higgins [J. Pure Appl. Algebra 22, 11-41 (1981; Zbl 0475.55009)], between the categories of crossed complexes and of \(\omega\)-groupoids, shows that our derivation of an automorphism structure using crossed complexes is an instance of the use of multiple groupoids to study symmetry properties. The crossed module \(\chi\) : \(G\to Aut G\) encodes symmetries of the group G and a notion of “homotopy” between symmetries given by conjugation. The automorphism structure AUT(C) of a crossed module encodes the symmetries of the crossed module C in dimension zero, homotopies in dimension one, and homotopies between homotopies in dimension two.
Reviewer: R.Brown

MSC:

18G55 Nonabelian homotopical algebra (MSC2010)
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
20F28 Automorphism groups of groups
55U99 Applied homological algebra and category theory in algebraic topology
55P15 Classification of homotopy type
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