# groupoid category

###### Definition 0.1.

*Groupoid categories ^{}*, or categories of groupoids

^{}, can be defined simply by considering a groupoid

^{}as a category

^{}${\mathcal{G}}_{1}$ with all invertible morphisms, and objects defined by the groupoid class or set of groupoid elements; then, the groupoid category, ${\mathcal{G}}_{2}$, is defined as the

*$\mathrm{2}$-category*whose objects are ${\mathcal{G}}_{1}$ categories (groupoids), and whose morphisms are functors

^{}of ${\mathcal{G}}_{1}$ categories consistent with the definition of groupoid homomorphisms, or in the case of topological groupoids

^{}, consistent as well with topological groupoid homeomorphisms

^{}(http://planetmath.org/Homeomorphism).

Example 0.1 :
The $2$-category of Lie groupoids is an example of a groupoid category, or *$\mathrm{2}$-category of groupoids*.

###### Definition 0.2.

The *$\mathrm{2}$-category of Lie groupoids ${\mathrm{G}}_{L}$* has Lie groupoids as objects, and for any two such objects ${\mathbf{G}}_{\mathbf{L}}$ and ${\mathbf{H}}_{\mathbf{L}}$ there is a hom-category

$$hom({\mathbf{G}}_{\mathbf{L}},{\mathbf{H}}_{\mathbf{L}})=BB({\mathbf{G}}_{\mathbf{L}},{\mathbf{H}}_{\mathbf{L}}),$$ |

where $BB({\mathbf{G}}_{\mathbf{L}},{\mathbf{H}}_{\mathbf{L}}),$ is a category whose objects are ${\mathbf{G}}_{\mathbf{L}}$–${\mathbf{H}}_{\mathbf{L}}$ bibundles of the Lie groupoids ${\mathbf{G}}_{\mathbf{L}}$ and ${\mathbf{H}}_{\mathbf{L}}$, respectively over $M$ and $N$, and whose morphisms are arrows $f:E\to {E}^{\prime}$ between such bibundles $E$ and ${E}^{\prime}$ that commute with the bundles ${\pi}_{1}:E\to M$ and ${\pi}_{2}:{E}^{\prime}\to N:$

$$\text{xymatrix}E\text{ar}{[rr]}^{f}\text{ar}{[dr]}_{{\pi}_{1}}\mathrm{\&}\mathrm{\&}{E}^{\prime}\text{ar}{[dl]}^{{\pi}_{2}}\mathrm{\&}M\mathrm{\&}$$ |

$$\text{xymatrix}E\text{ar}{[rr]}^{f}\text{ar}{[dr]}_{{\pi}_{{1}^{\prime}}}\mathrm{\&}\mathrm{\&}{E}^{\prime}\text{ar}{[dl]}^{{\pi}_{{2}^{\prime}}}\mathrm{\&}N\mathrm{\&},$$ |

consistent respectively with the ${\mathbf{G}}_{\mathbf{L}}$– and ${\mathbf{H}}_{\mathbf{L}}$– actions. Moreover, the composition^{} of two bibundles is given by the Hilsum-Skandalis product^{}.

Remark 0.1 :
The 2-category of groupoids^{} ${\mathcal{G}}_{2}$, plays a central role in the generalised, categorical Galois theory involving fundamental groupoid functors.

Title | groupoid category |

Canonical name | GroupoidCategory |

Date of creation | 2013-03-22 18:12:00 |

Last modified on | 2013-03-22 18:12:00 |

Owner | bci1 (20947) |

Last modified by | bci1 (20947) |

Numerical id | 55 |

Author | bci1 (20947) |

Entry type | Topic |

Classification | msc 55U05 |

Classification | msc 55U35 |

Classification | msc 55U40 |

Classification | msc 18G55 |

Classification | msc 18B40 |

Synonym | category of groupoids |

Synonym | 2-category of groupoids |

Synonym | category of groupoids and groupoid homomorphisms/homeomorphisms |

Related topic | 2Category |

Related topic | FundamentalGroupoid |

Related topic | Homeomorphism |

Related topic | HigherDimensionalAlgebraHDA |

Related topic | GroupoidAndGroupRepresentationsRelatedToQuantumSymmetries |

Related topic | GeneralizedVanKampenTheoremsHigherDimensional |

Related topic | Groupoids |

Related topic | GroupoidHomomorphisms |

Related topic | QuantumFundamentalGroupoids |

Related topic | CategoryTh |

Defines | groupoid category |

Defines | groupoid 2-category |

Defines | 2-category of groupoids |

Defines | 2-category of Lie groupoids |