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# algebroid structures and extended symmetries

# 0.1 Algebroid Structures and Algebroid Extended Symmetries

###### Definition 0.1.

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” $A_{0}$
is a directed graph over $A_{0}$ such that for each $x,y\in A_{0},\;A(x,y)$ has an $R$-module structure and there is an $R$-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.

###### Definition 0.2.

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 $\mathcal{A}$ 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 $\mathcal{A}$, the set $\mathcal{A}(b,c)$ is given the structure of an $R$-module, and composition $\mathcal{A}(b,c)\times\mathcal{A}(c,d){\longrightarrow}\mathcal{A}(b,d)$ is $R$–bilinear, or is a morphism of $R$-modules $\mathcal{A}(b,c)\otimes_{R}\mathcal{A}(c,d){\longrightarrow}\mathcal{A}(b,d)$.

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 $\mathsf{G}(b,c){\longrightarrow}R$ with finite support, and
then we define the *convolution product* as follows:

$(f*g)(z)=\sum\{(fx)(gy)\mid z=x\circ y\}~{}.$ | (0.1) |

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 $R\cong\mathbb{C}$ . 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 $G^{{\prime}}=G\cup\{0\}$ in which the
compositions not defined in $G$ are defined to be $0$ in $G^{{\prime}}$. 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 $\mathbb{S}$, 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, $C$).

# References

- 1
I. C. Baianu , James F. Glazebrook, and Ronald Brown. 2009. Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review.
*SIGMA*5 (2009), 051, 70 pages. $arXiv:0904.3644$, $doi:10.3842/SIGMA.2009.051$, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

## Mathematics Subject Classification

81T25*no label found*81T18

*no label found*81T13

*no label found*81T10

*no label found*81T05

*no label found*81R50

*no label found*55U35

*no label found*

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