An R-algebroid, ${\displaystyle R{\mathsf {G}}}$ , is constructed from a groupoid ${\displaystyle {\mathsf {G}}}$  as follows. The object set of ${\displaystyle R{\mathsf {G}}}$  is the same as that of ${\displaystyle {\mathsf {G}}}$  and ${\displaystyle R{\mathsf {G}}(b,c)}$  is the free R-module on the set ${\displaystyle {\mathsf {G}}(b,c)}$ , with composition given by the usual bilinear rule, extending the composition of ${\displaystyle {\mathsf {G}}}$ .^[1]

A groupoid ${\displaystyle {\mathsf {G}}}$  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 ${\displaystyle {\mathsf {G}}}$  in this construction with a general category C that does not have all morphisms invertible.

One can also define the R-algebroid, ${\displaystyle {\bar {R}}{\mathsf {G}}:=R{\mathsf {G}}(b,c)}$ , to be the set of functions ${\displaystyle {\mathsf {G}}(b,c){\longrightarrow }R}$  with finite support, and with the convolution product defined as follows: ${\displaystyle \displaystyle (f*g)(z)=\sum \{(fx)(gy)\mid z=x\circ y\}}$  .^[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 ${\displaystyle R\cong \mathbb {C} }$ .

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

This article incorporates material from Algebroid Structures and Algebroid Extended Symmetries on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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