Definition
An R-algebroid, , is constructed from a groupoid as follows. The object set of is the same as that of and is the free R-module on the set , with composition given by the usual bilinear rule, extending the composition of . [1]
In mathematics, R-algebroids are constructed starting from groupoids. These are more abstract concepts than the Lie algebroids that play a similar role in the theory of Lie groupoids to that of Lie algebras in the theory of Lie groups. (Thus, a Lie algebroid can be thought of as 'a Lie algebra with many objects ').
An R-algebroid, , is constructed from a groupoid as follows. The object set of is the same as that of and is the free R-module on the set , with composition given by the usual bilinear rule, extending the composition of . [1]
A groupoid can be regarded as a category with invertible morphisms. Then an R-category is defined as an extension of the R-algebroid concept by replacing the groupoid in this construction with a general category C that does not have all morphisms invertible.
One can also define the R-algebroid, , to be the set of functionswith finite support , and with the convolution product defined as follows: . [2]
Only this second construction is natural for the topological case, when one needs to replace 'function' by 'continuous function with compact support', and in this case .
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This article incorporates material from Algebroid Structures and Algebroid Extended Symmetries on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
