If is a groupoid (for example, regarded as a category with all morphisms invertible) then we can construct an -algebroid, as follows. Let us consider first a module over a ring , also called a -module, that is, a module (http://planetmath.org/Module) that takes its coefficients in a ring . Then, the object set of is the same as that of and is the free -module on the set , with composition given by the usual bilinear rule, extending the composition of .
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 . 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 by a semigroup in which the compositions not defined in are defined to be in . 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.
- 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).
|Date of creation||2013-03-22 18:14:19|
|Last modified on||2013-03-22 18:14:19|
|Last modified by||bci1 (20947)|
|Synonym||double groupoid dual of an algebroid|