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Derived crossed modules. (English) Zbl 1431.55019

Crossed modules are used in many contexts, and have been defined in [J. H. C. Whitehead, Ann. Math. (2) 47, 806–810 (1946; Zbl 0060.41104) and Bull. Am. Math. Soc. 55, 453–496 (1949; Zbl 0040.38801)] in the investigation of the properties of second relative homotopy groups. Some algebraic properties of crossed modules such as actor and automorphisms were studied in [K. Norrie, Bull. Soc. Math. Fr. 118, No. 2, 129–146 (1990; Zbl 0719.20018)]. Also in [R. Brown and C. B. Spencer, Nederl. Akad. Wet., Proc., Ser. A 79, 296–302 (1976; Zbl 0333.55011)] the categorical equivalence of the category of crossed modules and group-groupoids was proved and here the notion of homotopy morphism of group-groupoids was given using homotopy of crossed modules [W. H. Cockcroft, Proc. Camb. Philos. Soc. 48, 521–532 (1952; Zbl 0048.02102)]. In [T. Porter, Proc. Edinb. Math. Soc., II. Ser. 30, 373–381 (1987; Zbl 0595.18006)], it was proved that the category of crossed modules and internal categories in some algebraic categories are equivalent, which is more general than the former categorical equivalence.
In the present paper the results of [J. H. C. Whitehead, Ann. Math. (2) 49, 610–640 (1948; Zbl 0041.10102)] and [K. Norrie, loc. cit.] are applied to a more general one, the category of crossed modules in the category of groups with operation, \(\mathbf{C}\). First of all, in Section 2, preliminaries on basic definitions and theorems are given. In Section 3, using results of Cockcroft and Brown and Spencer, homotopy of internal groupoid morphisms in the category \(\mathbf{C}\) is defined. Then the notion of homotopy morphism of crossed modules in \(\mathbf{C}\) is characterized. Derivations of crossed modules have been defined by Whitehead, which are homotopies between crossed module endomorphisms and the identity crossed module morphism. A regular derivation is a derivation with an inverse. In Section 4, homotopy morphisms of crossed modules in \(\mathbf{C}\) are developed. In Proposition 4.6, for a crossed module \((A,B,\alpha)\) in \(\mathbf{C}\) and a derivation \(d\) of it, a split extension, hence a new set of derived actions of \(B\) on \(A\) is characterized by regular derived actions. As a result, in Proposition 4.8 a new crossed module called derived crossed module is obtained. For a crossed module \((A,B,\alpha)\) and a regular derivation of it, a new regular derivation is constructed (Proposition 4.12) and therefore isomorphic crossed modules are obtained.

MSC:

55U35 Abstract and axiomatic homotopy theory in algebraic topology
18C40 Structured objects in a category (group objects, etc.)
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
18A23 Natural morphisms, dinatural morphisms
16W25 Derivations, actions of Lie algebras
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References:

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