R-algebroid

Definition 0.1.

If $𝖦$ is a groupoid (for example, regarded as a category with all morphisms invertible)then we can construct an $R$-algebroid, $R𝖦$ as follows. Let us consider first a module over a ring $R$, also called a $R$-module, that is, a module (http://planetmath.org/Module) $M_R$ that takes its coefficients in a ring $R$. Then, 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 $𝖦$.

Definition 0.2.

Alternatively, one can define $\overline{R}𝖦(b,c)$ to be the set of functions $𝖦(b,c)⟶R$ with finite support, and then one defines the convolution product as follows:

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

Remark 0.1.

As it is very well known, only the second construction is naturalfor the topological case, when one needs to replace the general concept of ‘function’ bythe topological-analytical concept of ‘continuous function with compact support’ (or alternatively, with ‘locallycompact support’) for all quantum field theory (QFT) extended symmetry sectors; in this case, one has that $R\cong ℂ$ .The point made here is that to carry out the usual construction and end up with only an algebrarather than an algebroid, is a procedure analogous to replacing agroupoid $𝖦$ 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, namely the presence of the spatial component given by the set of objects of the groupoid.

More generally, a R-category (http://planetmath.org/RCategory) is similarly defined as an extension to this R-algebroidconcept.