##
**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.

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.

Reviewer: Serap Demir (Kayseri)

### 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 |

### Citations:

Zbl 0060.41104; Zbl 0040.38801; Zbl 0719.20018; Zbl 0333.55011; Zbl 0048.02102; Zbl 0595.18006; Zbl 0041.10102### References:

[1] | H. F. Akız, O. Mucuk, N. Alemdar and T. S ̧ahan, Coverings of internal groupoids and crossed modules in the category of groups with operations, Georgian Math. J. 20 (2) (2013), 223-238. |

[2] | J. C. Baez and D. Stevenson, The Classifying Space of a Topological 2-Group, pages 1-31. Algebraic Topology. Abel Symposia. Springer, 2009. · Zbl 1182.55015 |

[3] | R. Brown, Groupoids and crossed objects in algebraic topology Homol. Homotopy Appl. 1 (1999), 1-78. · Zbl 0920.55002 |

[4] | R. Brown and J. Huebschmann, Identities among relations, pages 153-202. Lon- don Mathematical Society Lecture Note Series. Cambridge University Press, 1982. |

[5] | R. Brown and C. B. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group, Indagat. Math. 79 (4) (1976), 296-302. · Zbl 0333.55011 |

[6] | W. H. Cockcroft, On the homomorphisms of sequences, Math. Proc. Cambridge 48 (4) (1952), 521-532. · Zbl 0048.02102 |

[7] | T. Datuashvili, Cohomologically trivial internal categories in categories of groups with operations, Appl. Categor. Struct. 3 (3) (1995), 221-237. · Zbl 0844.18004 |

[8] | T. Datuashvili. Categorical, homological and homotopical properties of algebraic objects. Dissertation, Georgian Academy of Science, 2006. |

[9] | J. Huebschmann, Crossed n-folds extensions of groups and cohomology, Comment. Math. Helv. 55 (1980), 302-313. · Zbl 0443.18019 |

[10] | J.-L. Loday, Cohomologie et groupe de steinberg relatifs, J. Algebra 54 (1) (1978), 178-202. · Zbl 0391.20040 |

[11] | A. S.-T. Lue, Cohomology of groups relative to a variety, J. Algebra 69 (1) (1981), 155-174. · Zbl 0468.20044 |

[12] | O. Mucuk and H.F. Akız, Monodromy groupoid of an internal groupoid in topological groups with operations, Filomat 29 (10), (2015), 2355-2366. · Zbl 1458.22001 |

[13] | O. Mucuk and H. C ̧ akallı, G-connectedness in topological groups with operations, 1079-1089 Filomat 32 (3), (2018), 1079-1089. |

[14] | O. Mucuk and T. S ̧ahan, Coverings and crossed modules of topological groups with operations, Turk. J. Math. 38 (5) (2014), 833-845. · Zbl 1364.54024 |

[15] | K. Norrie, Actions and automorphisms of crossed modules, Bull. Soc. Math. Fr. 118 (2) (1990), 129-146. · Zbl 0719.20018 |

[16] | G. Orzech, Obstruction theory in algebraic categories, I, J. Pure. Appl. Algebra 2 (4) (1972), 287-314. · Zbl 0251.18016 |

[17] | G. Orzech, Obstruction theory in algebraic categories, II, J. Pure. Appl. Algebra 2 (4) (1972), 315-340. · Zbl 0251.18017 |

[18] | A. Patchkoria, Crossed semimodules and schreier internal categories in the cat- egory of monoids, Georgian Math. J. 5 (6) (1998), 575-581. · Zbl 0915.18002 |

[19] | T. Porter, Extensions, crossed modules and internal categories in categories of groups with operations P. Edinburgh. Math. Soc. 30 (3) (1987), 373-381. · Zbl 0595.18006 |

[20] | T. S ̧ahan, Further remarks on liftings of crossed modules, Hacet. J. Math. Stat., Retrieved 4 Mar. 2018, from Doi: 10.15672/HJMS.2018.554. |

[21] | S. Temel, Topological crossed semimodules and schreier internal categories in the category of topological monoids, Gazi Univ. J. Sci. 29 (4) (2016), 915-921. |

[22] | S. Temel, Crossed semimodules of categories and Schreier 2-categories, Tbilisi Math. J. 11 (2) (2018), 47-57. |

[23] | J. H. C. Whitehead, Note on a previous paper entitled “on adding relations to homotopy groups”, Ann. Math. 47 (4) (1946), 806-810. · Zbl 0060.41104 |

[24] | J. H. C. Whitehead, On operators in relative homotopy groups, Ann. Math. 49 (3) (1948), 610-640. · Zbl 0041.10102 |

[25] | J. H. C. Whitehead, Combinatorial homotopy. II, Bull. Amer. Math. Soc. 55 (5) (1949), 453-496. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.