R-algebroid
Definition 0.1.
If $\U0001d5a6$ is a groupoid^{} (for example, regarded as a category^{} with all morphisms^{} invertible) then we can construct an $R$-algebroid, $R\U0001d5a6$ 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 $R\U0001d5a6$ is the same as that of $\U0001d5a6$ and $R\U0001d5a6(b,c)$ is the free $R$-module on the set $\U0001d5a6(b,c)$, with composition given by the usual bilinear rule, extending the composition of $\U0001d5a6$.
Definition 0.2.
Alternatively, one can define $\overline{R}\U0001d5a6(b,c)$ to be the set of functions $\U0001d5a6(b,c)\u27f6R$ 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 $R\cong \u2102$ . 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 $\U0001d5a6$ 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.
More generally, a R-category (http://planetmath.org/RCategory) is similarly defined as an extension^{} to this R-algebroid concept.
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 |