
R-algebroid

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" ').
Definition.
An R-algebroid, formula_1, is constructed from a groupoid formula_2 as follows. The object set of formula_1 is the same as that of formula_2 and formula_5 is the free R-module on the set formula_6, with composition given by the usual bilinear rule, extending the composition of formula_2.
R-category.
A groupoid formula_2 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 formula_2 in this construction with a general category C that does not have all morphisms invertible.
R-algebroids "via" convolution products.
One can also define the R-algebroid, formula_10, to be the set of functions formula_11 with finite support, and with the convolution product defined as follows:
formula_12 .
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 formula_13.