groupoid category
Definition 0.1.
Groupoid categories, or categories of groupoids
, can be defined
simply by considering a groupoid
as a category
with all invertible morphisms, and objects
defined by the groupoid class or set of groupoid elements; then, the groupoid category, ,
is defined as the -category whose objects are categories (groupoids), and whose morphisms are functors
of 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 -category of Lie groupoids is an example of a groupoid category, or -category of groupoids.
Definition 0.2.
The -category of Lie groupoids has Lie groupoids as objects, and for any two such objects and there is a hom-category
where is a category whose objects are – bibundles of the Lie groupoids and , respectively over and , and whose morphisms are arrows between such bibundles and that commute with the bundles and
consistent respectively with the – and – actions. Moreover, the composition of two bibundles is given by the Hilsum-Skandalis product
.
Remark 0.1 :
The 2-category of groupoids , 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 |