groupoid category

Definition 0.1.

Groupoid categoriesPlanetmathPlanetmath, or categories of groupoidsPlanetmathPlanetmath, can be defined simply by considering a groupoidPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath as a categoryMathworldPlanetmath 𝒢1 with all invertible morphisms, and objects defined by the groupoid class or set of groupoid elements; then, the groupoid category, 𝒢2, is defined as the 2-category whose objects are 𝒢1 categories (groupoids), and whose morphisms are functorsMathworldPlanetmath of 𝒢1 categories consistent with the definition of groupoid homomorphisms, or in the case of topological groupoidsPlanetmathPlanetmathPlanetmathPlanetmath, consistent as well with topological groupoid homeomorphismsPlanetmathPlanetmath (

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 GL has Lie groupoids as objects, and for any two such objects 𝐆𝐋 and 𝐇𝐋 there is a hom-category


where BB(𝐆𝐋,𝐇𝐋), is a category whose objects are 𝐆𝐋𝐇𝐋 bibundles of the Lie groupoids 𝐆𝐋 and 𝐇𝐋, respectively over M and N, and whose morphisms are arrows f:EE between such bibundles E and E that commute with the bundles π1:EM and π2:EN:


consistent respectively with the 𝐆𝐋– and 𝐇𝐋– actions. Moreover, the compositionMathworldPlanetmath of two bibundles is given by the Hilsum-Skandalis productPlanetmathPlanetmath.

Remark 0.1 : The 2-category of groupoidsPlanetmathPlanetmath 𝒢2, 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