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 (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