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# groupoid category

###### Definition 0.1.

*Groupoid categories*, or categories of groupoids, can be defined
simply by considering a groupoid as a category $\mathsf{\mathcal{G}}_{1}$ with all invertible morphisms, and objects
defined by the groupoid class or set of groupoid elements; then, the groupoid category, $\mathsf{\mathcal{G}}_{2}$,
is defined as the *$2$-category* whose objects are $\mathsf{\mathcal{G}}_{1}$ categories (groupoids), and whose morphisms are functors of $\mathsf{\mathcal{G}}_{1}$ categories consistent with the definition of groupoid homomorphisms, or in the case of topological groupoids, consistent as well with topological groupoid
homeomorphisms.

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

###### Definition 0.2.

The *$2$-category of Lie groupoids $\mathsf{\mathcal{G}}_{L}$* has Lie groupoids as objects, and for any two such objects ${\bf G_{L}}$ and ${\bf H_{L}}$ there is a hom-category

$hom({\bf G_{L}},{\bf H_{L}})=BB({\bf G_{L}},{\bf H_{L}}),$ |

where $BB({\bf G_{L}},{\bf H_{L}}),$ is a category whose objects are ${\bf G_{L}}$–${\bf H_{L}}$ bibundles of the Lie groupoids ${\bf G_{L}}$ and ${\bf H_{L}}$, 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 $\pi_{2}:E^{{\prime}}\to N:$

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

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

consistent respectively with the ${\bf G_{L}}$– and ${\bf H_{L}}$– actions. Moreover, the composition of two bibundles is given by the Hilsum-Skandalis product.

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

## Mathematics Subject Classification

55U05*no label found*55U35

*no label found*55U40

*no label found*18G55

*no label found*18B40

*no label found*

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