1 Algebroid structures and Quantum Algebroid Extended Symmetries.
Definition 1.1. An algebroid structure A 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 R-algebroid A on a set
of “objects” A0 is a directed graph over A0 such that for each x,y ∈ A0, A(x,y) has an
R-module structure and there is an R-bilinear function
(a,b)
a∘b 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 1x ∈ A(x,x) is not assumed. For example,
if A0 has exactly one object, then an R-algebroid A over A0 is just an R-algebra. An ideal
in A is then an example of a pre-algebroid.
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) ×𝒜(c,d)→𝒜(b,d) is
R–bilinear, or is a morphism of R-modules 𝒜(b,c) ⊗R𝒜(c,d)→𝒜(b,d).
If G is a groupoid (or, more generally, a category) 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.
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:
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
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 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.
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 §, 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).
