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R-algebroid (Definition)

R-algebroid

Definition 0.1   If $\mathsf{G}$ is a groupoid (for example, considered as a category with all morphisms invertible) 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 $\mathsf{G}$.
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 we define the convolution product as follows:
$\displaystyle (f*g)(z)= \sum \{(fx)(gy)\mid z=x\circ y \} ~.$ (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 '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 $G'=G\cup \{0\}$ 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.
  • More generally, an R-category is similarly defined as an extension to this R- algebroid concept.

Bibliography

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).



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See Also: groupoid, 2-groupoid, superalgebroids in higher dimensions, R-category, n-groupoids, R-supercategory

Other names:  dual of R-groupoid or ringoid
Also defines:  ringoid
Keywords:  R-algebroid

Cross-references: concept, R-category, semigroup, algebroid, QFT, topological, convolution, functions, composition, object, morphisms, category, groupoid
There is 1 reference to this object.

This is version 2 of R-algebroid, born on 2009-01-31, modified 2009-01-31.
Object id is 455, canonical name is RAlgebroid.
Accessed 923 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
Pending Errata and Addenda
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