Algebroid structures and Quantum Algebroid Extended Symmetries.

Definition 1.1   An algebroid structure will be specifically defined to mean either a ring, or more generally, any of the specifically defined algebras, but with several objects instead of a single object, in the sense specified by Mitchell (1965). Thus, an algebroid has been defined (Mosa, 1986a; Brown and Mosa 1986b, 2008) as follows. An -algebroid on a set of “objects" is a directed graph over such that for each has an -module structure and there is an -bilinear function

called “composition" and satisfying the associativity condition, and the existence of identities.

Definition 1.2   A pre-algebroid has the same structure as an algebroid and the same axioms except for the fact that the existence of identities is not assumed. For example, if has exactly one object, then an -algebroid over is just an -algebra. An ideal in is then an example of a pre-algebroid.

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

If is a groupoid (or, more generally, a category) then we can construct an -algebroid as follows. The object set of is the same as that of and is the free -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:

(1.1)

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  . 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 in which the compositions not defined in are defined to be 0 in . 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, -algebroids, double algebroids , and so on. A `category' of -categories is however a super-category , or it can also be viewed as a specific example of a metacategory (or -supercategory, in the more general case of multiple operations–categorical `composition laws' being defined within the same structure, for the same class, ).