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.
Keywords: module, bicategory, transformation
2000 MSC: 18D05,16D90
Theory and Applications of Categories,
Vol. 11, 2003,
No. 17, pp 375-396.