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, $$R\mathsf{G}$$, is constructed from a groupoid $$\mathsf{G}$$ as follows. The object set of $$R\mathsf{G}$$ is the same as that of $$\mathsf{G}$$ and $$R\mathsf{G}(b,c)$$ is the free R-module on the set $$\mathsf{G}(b,c)$$, with composition given by the usual bilinear rule, extending the composition of $$\mathsf{G}$$.

R-category
A groupoid $$\mathsf{G}$$ 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 $$\mathsf{G}$$ 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, $${\bar R}\mathsf{G}:=R\mathsf{G}(b,c)$$, to be the set of functions $$\mathsf{G}(b,c){\longrightarrow}R$$ with finite support, and with the convolution product defined as follows: $$\displaystyle (f*g)(z)= \sum \{(fx)(gy)\mid z=x\circ y \}$$.

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 $$R\cong \mathbb{C}$$.

Examples

 * Every Lie algebra is a Lie algebroid over the one point manifold.
 * The Lie algebroid associated to a Lie groupoid.