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 meaneither a ring, or more generally, any of the specifically defined algebras, but with severalobjects 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 $R$-bilinearfunction

$$\circ :A(x,y)\times A(y,z)\to A(x,z)$$

$(a,b)\mapsto a\circ b$ called “composition” and satisfying theassociativity condition, and the existence of identities.

Definition 0.2.

A pre-algebroid has the same structure as an algebroid and the sameaxioms 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, thenan $R$-algebroid $A$ over $A_0$ is just an $R$-algebra. An idealin $A$ is then an example of a pre-algebroid.

Let $R$ be a commutative ring. An $R$-category $\mathrm{A}$ 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 themonoidal 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 $𝒜(b,c)\times 𝒜(c,d)⟶𝒜(b,d)$ is $R$–bilinear, or is a morphism of $R$-modules $𝒜(b,c){\otimes }_R𝒜(c,d)⟶𝒜(b,d)$.

If $𝖦$ is a groupoid (http://planetmath.org/Groupoids) (or, more generally, a category)then we can construct an $R$-algebroid $R𝖦$ asfollows. The object set of $R𝖦$ is the same as that of$𝖦$ and $R𝖦(b,c)$ is the free $R$-module on theset $𝖦(b,c)$, with composition given by the usualbilinear rule, extending the composition of $𝖦$.

Alternatively, one can define $\overline{R}𝖦(b,c)$ to be theset of functions $𝖦(b,c)⟶R$ with finite support, andthen 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 naturalfor the topological case, when one needs to replace ‘function’ by‘continuous function with compact support’ (http://planetmath.org/SmoothFunctionsWithCompactSupport) (or locallycompact support for the QFT (http://planetmath.org/QFTOrQuantumFieldTheories) extendedhttp://planetmath.org/?op=getobj&from=books&id=153symmetry sectors), and inthis case $R\cong ℂ$ . The point made here isthat to carry out the usual construction and end up with only an algebrarather than an algebroid, is a procedure analogous to replacing agroupoid (http://planetmath.org/Groupoids) $𝖦$ by a semigroup $G^{\prime }=G\cup \{0\}$ in which thecompositions not defined in $G$ are defined to be $0$ in $G^{\prime }$. Weargue that this construction removes the main advantage ofgroupoids (http://planetmath.org/Groupoids), namely the spatial component given by the set ofobjects.

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 (http://planetmath.org/Supercategory) $𝕊$, or it can also be viewed as a specific example of a metacategory (http://planetmath.org/AxiomsOfMetacategoriesAndSupercategories) (or$R$-supercategory, in the more general case of multiple operations–categorical ‘composition laws’– being defined within the same structure, for the same class, $C$).