# natural equivalence of ${C}_{G}$ and ${C}_{M}$ categories

###### Theorem 0.1.

(with proof by Verdier [1])
The category^{} ${\mathrm{C}}_{G}$ of categorical groups and functorial homomorphisms^{} between categorical groups, and the category ${\mathrm{C}}_{M}$ of crossed modules of groups and homomorphisms between them, are naturally equivalent.

## References

- 1 Jean-Louis Verdier, Des cat$\mathrm{\xe9}$gories d$\mathrm{\xe9}$riv$\mathrm{\xe9}$es des cat$\mathrm{\xe9}$gories ab$\mathrm{\xe9}$liennes, Ast$\mathrm{\xe9}$risque, vol. 239, Soci$\mathrm{\$\u0301}et\text{\xe9}Math\mathrm{\$\u0301}ematiquedeFrance,1996(inFrench).$

Title | natural equivalence of ${C}_{G}$ and ${C}_{M}$ categories |
---|---|

Canonical name | NaturalEquivalenceOfCGAndCMCategories |

Date of creation | 2013-03-22 18:25:47 |

Last modified on | 2013-03-22 18:25:47 |

Owner | bci1 (20947) |

Last modified by | bci1 (20947) |

Numerical id | 13 |

Author | bci1 (20947) |

Entry type | Theorem |

Classification | msc 55M05 |

Classification | msc 18E05 |

Classification | msc 18-00 |

Related topic | HomotopyGroupoidsAndCrossComplexesAsNonCommutativeStructuresInHigherDimensionalAlgebraHDA |

Related topic | EquivalenceOfCategories2 |

Related topic | FunctorCategory2 |

Related topic | GroupCohomology |

Related topic | IndexOfCategories |