This paper studies lax higher dimensional structure over bicategories.
The general notion of a module between two morphisms of bicategories is
described. These modules together with their (multi-)2-cells, which we
call modulations, organize themselves into a multi-bicategory. The
usual notion of a module can be recovered from this general notion by
simply choosing the domain bicategory to be the terminal or final
bicategory.
The composite of two such modules need not exist. However, when the
domain bicategory is small and the codomain bicategory is locally
cocomplete then the composite of any two modules does exist and has a
simple construction using the local colimits. These modules and their
modulations then give rise to a bicategory.
Recall that neither transformations nor optransformations (respectively
lax natural transformations and oplax natural transformations) between
morphisms of bicategories give rise to a smooth 3-dimensional structure.
However, there is a smooth 3-dimensional structure for modules, and
both transformations and optransformations give
rise to associated modules. Furthermore, the modulations between two
modules associated with transformations can then be described directly
as a new sort of modification between the transformations. This
provides a locally full and faithful homomorphism from transformations
and modifications into the bicategory of modules.
Finally, if each 1-cell component of a transformation is a left-adjoint
then the right-adjoints provide an optransformation. In the module
bicategory the module associated with this optransformation is
right-adjoint to the module associated with the transformation.
Therefore the inclusion of transformations whose 1-cells have left
adjoints into the (multi-)bicategory of modules provides a source of
proarrow equipment.
