# PlanetPhysics/Algebroid Structures and Extended Symmetries

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## Algebroid structures and Quantum Algebroid Extended Symmetries.

An \htmladdnormallink{algebroid {http://planetphysics.us/encyclopedia/Algebroids.html} structure} $A$  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 $R$ -algebroid $A$  on a set of objects" $A_{0}$  is a directed graph over $A_{0}$  such that for each $x,y\in A_{0},\;A(x,y)$hasan$ R$-modulestructureandthereisan$ R$-bilinear function $\circ :A(x,y)\times A(y,z)\to A(x,z)$  $(a,b)\mapsto a\circ b$  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 $1_{x}\in A(x,x)$  is not assumed. For example, if $A_{0}$  has exactly one object, then an $R$ -algebroid $A$  over $A_{0}$  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 $\displaystyle \A$ 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 $\displaystyle \A$ , the set $\displaystyle \A(b,c)$ is given the structure of an $R$ -module, and composition $\displaystyle \A(b,c) \times \A(c,d) \lra \A(b,d)$is$ R$--bilinear,orisa[[../TrivialGroupoid/|morphism]]of$ R$-modules$ \A(b,c) \otimes_R \A(c,d) \lra \A(b,d)$.

If ${\mathsf {G}}$  is a groupoid (or, more generally, a category) then we can construct an $R$ -algebroid $R{\mathsf {G}}$  as follows. The object set of $R{\mathsf {G}}$  is the same as that of ${\mathsf {G}}$  and $R{\mathsf {G}}(b,c)$  is the free $R$ -module on the set ${\mathsf {G}}(b,c)$ , with composition given by the usual bilinear rule, extending the composition of ${\mathsf {G}}$ .

Alternatively, one can define ${\bar {R}}{\mathsf {G}}(b,c)$  to be the set of functions $\displaystyle \mathsf{G}(b,c)\lra R$ with finite support, and then we define the \htmladdnormallink{convolution {http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html} product} as follows:

$(f*g)(z)=\sum \{(fx)(gy)\mid z=x\circ y\}~.$

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 $R\cong \mathbb {C}$ ~. 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 ${\mathsf {G}}$  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, $C$ ).