Algebroid Structures and Algebroid Extended Symmetries

Let $R$ be a commutative ring. An $R$ -category $\A$ is a category equipped with an $R$ -module structure on each hom set such that the composition is $R$ -bilinear. More precisely, let us assume for instance that we are given a commutative ring $R$ with identity. Then a small $R$ -category-or equivalently an $R$ -algebroid- will be defined as a category enriched in the monoidal category of $R$ -modules, with respect to the monoidal structure of tensor product. This means simply that for all objects $b,c$ of $\A$ , the set $\A(b,c)$ is given the structure of an $R$ -module, and composition $\A(b,c) \times \A(c,d) \lra \A(b,d)$ is $R$ -bilinear, or is a morphism of $R$ -modules $\A(b,c) \otimes_R \A(c,d) \lra \A(b,d)$ .

If is a groupoid (or, more generally, a category) then we can construct an $R$ -algebroid as follows. 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 .

Alternatively, one can define to be the set of functions with finite support, and then we define the convolution product as follows:

\begin{equation} (f*g)(z)= \sum \{(fx)(gy)\mid z=x\circ y \} ~. \end{equation} 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 $R \cong \mathbb{C}$ . 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 spatial component given by the set of objects.

Remarks: One can also define categories of algebroids, $R$ -algebroids, double algebroids , and so on. A `category' of $R$ -categories is however a super-category $\S$ , or it can also be viewed as a specific example of a metacategory (or $R$ -supercategory, in the more general case of multiple operations-categorical `composition laws'- being defined within the same structure, for the same class, $C$ ).