Double algebroids
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The definition of a double algebroid specified above was introduced by Brown and Mosa [1]. Two functors can be then constructed, one from the category of double algebroids to the category of crossed modules of algebroids, whereas the reverse functor is the unique adjoint (up to natural equivalence). The construction of such functors requires the following definition.
A morphism
of double algebroids is then
defined as a morphism of truncated cubical sets which commutes
with all the algebroid structures. Thus, one can construct a
category
of double algebroids and their morphisms.
The main construction in this subsection is that of two functors
from this category
to the category
of crossed modules of algebroids.
Let
be a double algebroid. One can associate to
a
crossed module
. Here
will consist
of elements
of
with boundary of the form:
0 1
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(0.5) |
One can extend the above notion of double algebroid to cubic and higher dimensional algebroids.
The concepts
of 2-algebroid, 3-algebroid,...,
-algebroid and superalgebroid are however quite distinct from those of double, cubic,..., n-tuple algebroid, and have technically less complicated
definitions.
As of this snapshot date, this entry was owned by bci1.