R-algebroid

Definition 0.1.

If $\mathsf{G}$ is a groupoid    (for example, regarded as a category  with all morphisms  invertible) then we can construct an $R$-algebroid, $R\mathsf{G}$ 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 $R\mathsf{G}$ 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 $\mathsf{G}$.

Definition 0.2.

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 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 natural for the topological case, when one needs to replace the general concept of ‘function’ by the topological-analytical concept of ‘continuous function   with compact support’ (or alternatively, with ‘locally compact support  ’) for all quantum field theory (QFT) extended symmetry  sectors; in this case, one has that $R\cong\mathbb{C}$ . 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 presence of the spatial component given by the set of objects of the groupoid.

References

• 1 R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales–Bangor, Maths Preprint, 1986.
• 2 G. H. Mosa: Higher dimensional algebroids and Crossed complexes, PhD thesis, University of Wales, Bangor, (1986). (supervised by R. Brown).
 Title R-algebroid Canonical name Ralgebroid Date of creation 2013-03-22 18:14:19 Last modified on 2013-03-22 18:14:19 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 25 Author bci1 (20947) Entry type Definition Classification msc 81T10 Classification msc 81P05 Classification msc 81T05 Classification msc 81R10 Classification msc 81R50 Synonym groupoid-derived algebroids Synonym double groupoid   dual of an algebroid Related topic Module Related topic RCategory Related topic Algebroids Related topic HamiltonianAlgebroids Related topic RSupercategory Related topic SuperalgebroidsAndHigherDimensionalAlgebroids Related topic CategoricalAlgebras Defines $R$-module Defines convolution product Defines R-algebroid