# weak homotopy double groupoid

###### Definition 0.1.

*a weak homotopy double groupoid ^{} (WHDG)* of a

*compactly–generated space*${X}_{cg}$, (weak Hausdorff space) is defined through a construction method similar to that developed by R. Brown (ref. [1]) for the

*homotopy double groupoid of a Hausdorff space*. The key changes here involve replacing the regular homotopy equivalence relation

^{}from the cited ref. with the

*weak homotopy equivalence relation*in the definition of the fundamental groupoid

^{}^{}, as well as replacing the Hausdorff space by the compactly-generated space ${X}_{cg}$. Therefore, the weak homotopy

^{}data for the

*weak homotopy double groupoid*of ${X}_{cg}$, ${\bm{\rho}}^{\mathrm{\square}}({X}_{cg})$, will now be:

$$\begin{array}{c}\hfill ({\bm{\rho}}_{2}^{\mathrm{\square}}(X),{\bm{\rho}}_{1}^{\mathrm{\square}}(X),{\partial}_{1}^{-},{\partial}_{1}^{+},{+}_{1},{\epsilon}_{1}),{\bm{\rho}}_{2}^{\mathrm{\square}}(X),{\bm{\rho}}_{1}^{\mathrm{\square}}(X),{\partial}_{2}^{-},{\partial}_{2}^{+},{+}_{2},{\epsilon}_{2})\hfill \\ \hfill ({\bm{\rho}}_{1}^{\mathrm{\square}}(X),X,{\partial}^{-},{\partial}^{+},+,\epsilon ).\hfill \end{array}$$ |

## References

- 1 R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A homotopy double groupoid of a Hausdorff space, Theory and Applications of Categories 10,(2002): 71-93.

Title | weak homotopy double groupoid |

Canonical name | WeakHomotopyDoubleGroupoid |

Date of creation | 2013-03-22 18:15:12 |

Last modified on | 2013-03-22 18:15:12 |

Owner | bci1 (20947) |

Last modified by | bci1 (20947) |

Numerical id | 16 |

Author | bci1 (20947) |

Entry type | Definition |

Classification | msc 55N33 |

Classification | msc 55N20 |

Classification | msc 55P10 |

Classification | msc 55U40 |

Classification | msc 18B40 |

Classification | msc 18D05 |

Synonym | homotopy double groupoid |

Related topic | WeakHomotopyAdditionLemma |

Related topic | OmegaSpectrum |

Related topic | FEquivalenceInCategory |

Defines | higher dimensional weak homotopy |