Definition 0.1   If is a groupoid (for example, regarded as a category with all morphisms invertible) then we can construct an $R$ -algebroid, as follows. Let us consider first a module over a ring $R$ , also called a $R$ -module, that is, a module $M_R$ that takes its coefficients in a ring $R$ . Then, 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 .

Definition 0.2   Alternatively, one can define to be the set of functions with finite support, and then one defines 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 the general concept of `function' by the topological-analytical concept of `continuous function with compact support' (or alternatively, with `locally compact support') for all quantum field theory (QFT) extended symmetry sectors; in this case, one has that $R \cong \mathbb{C}$ . 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 by a semigroup $G'=G\cup \{0\}$ 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 presence of the spatial component given by the set of objects of the groupoid.

More generally, a R-category is similarly defined as an extension to this R-algebroid concept.