# PlanetPhysics/Algebroid Structures and Extended Symmetries

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## Algebroid structures and Quantum Algebroid Extended Symmetries.Edit

An *\htmladdnormallink{algebroid* {http://planetphysics.us/encyclopedia/Algebroids.html} structure} will be specifically defined to mean
either a ring, or more generally, any of the specifically defined algebras, but \emph{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 R R</math>-bilinear
function called ``composition" and satisfying the
associativity condition, and the existence of identities.

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.
Let be a commutative ring.

An * -category * **Failed to parse (unknown function "\A"): {\displaystyle \A}**
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 **Failed to parse (unknown function "\A"): {\displaystyle \A}**
, the set **Failed to parse (unknown function "\A"): {\displaystyle \A(b,c)}**
is given the structure of an -module, and composition **Failed to parse (unknown function "\A"): {\displaystyle \A(b,c) \times \A(c,d) \lra \A(b,d)<math> is }**
R R \A(b,c) \otimes_R \A(c,d) \lra \A(b,d)</math>.

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 **Failed to parse (unknown function "\lra"): {\displaystyle \mathsf{G}(b,c)\lra R}**
with finite support, and
then we define the *\htmladdnormallink{convolution* {http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html} 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 \emph{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 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, ).