# groupoid category

###### Definition 0.1.

Groupoid categories, or , 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 (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.

 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