groupoid category

Definition 0.1.

Groupoid categories, or categories of groupoids, can be definedsimply by considering a groupoid as a category $\\mathsf{\\mathcal{G}}_{1}$ with all invertible morphisms, and objectsdefined 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 groupoidhomeomorphisms (http://planetmath.org/Homeomorphism).

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.