Definition 0.1. If G is a groupoid (for example, considered as a category with all morphisms invertible) then we can construct an R-algebroid, RG as follows. The object set of RG is the same as that of G and RG(b,c) is the free R-module on the set G(b,c), with composition given by the usual bilinear rule, extending the composition of G.

Definition 0.2. Alternatively, one can define RG(b,c) to be the set of functions G(b,c)→R with finite support, and then we define the convolution product as follows:

Remark 0.1.

• As it is very well known, only the second construction is natural for the topological case, when one needs to replace ’function’ by ’continuous function with compact support’ (or locally compact support for the QFT extended symmetry sectors), and in this case Rℂ . The point made here is that to carry out the usual construction and end up with only an algebra rather than an algebroid, is a procedure analogous to replacing a groupoid G by a semigroup G′ = G ∪{0} in which the compositions not defined in G are defined to be 0 in G′. We argue that this construction removes the main advantage of groupoids, namely the spatial component given by the set of objects.
• More generally, an R-category is similarly defined as an extension to this R- algebroid concept.