R-algebroid

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  

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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