Linear bicategories are a generalization of ordinary bicategories in
which there are two horizontal (1-cell) compositions corresponding to
the ``tensor'' and ``par'' of linear logic. Benabou's notion of a morphism
(lax 2-functor) of bicategories may be generalized to linear
bicategories, where they are called linear functors. Unfortunately, as
for the bicategorical case, it is not obvious how to organize linear
functors smoothly into a higher dimensional structure. Not only do linear
functors seem to lack the two compositions expected for a linear
bicategory but, even worse, they inherit from the bicategorical level
the failure to combine well with the obvious notion of transformation.
As we shall see, there are also problems with lifting the notion of lax
transformation to the linear setting.
One possible resolution is to step up one dimension, taking morphisms as
the 0-cell level. In the linear setting, this suggests making linear
functors 0-cells, but what structure should sit above them? Lax
transformations in a suitable sense just do not seem to work very well
for this purpose (Section \ref{S:linnattran}).
Modules provide a more promising direction, but
raise a number of technical issues concerning the composability of both
the modules and their transformations. In general the required
composites will not exist in either the linear bicategorical or ordinary
bicategorical setting. However, when these composites do exist modules
between linear functors do combine to form a linear bicategory. In
order to better understand the conditions for the existence of
composites, we have found it convenient, particularly in the linear
setting, to develop the theory of ``poly-bicategories''. In this
setting we can develop the theory so as to extract the answers to these
problems not only for linear bicategories but also for ordinary
bicategories.Poly-bicategories are 2-dimensional generalizations of Szabo's
poly-categories, consisting of objects, 1-cells, and poly-2-cells.
The latter may have several 1-cells as input and as output and can
be composed by means of cutting along a single 1-cell. While a
poly-bicategory does not require that there be any compositions
for the 1-cells, such composites are determined (up to 1-cell isomorphism)
by their universal properties. We say a poly-bicategory is representable
when there is a representing 1-cell for each of the two possible
1-cell compositions geared towards the domains and codomains of the
poly 2-cells. In this case we recover the notion of a linear bicategory.
The poly notions of functors, modules and their transformations are
introduced as well. The poly-functors between two given
poly-bicategories P and P' together with poly-modules
between
poly-functors and their transformations form a new poly-bicategory
provided P is representable and closed in the sense that every
1-cell
has both a left and a right adjoint (in the appropriate linear sense).
Finally we revisit the notion of linear (or lax) natural
transformations, which can only be defined for representable
poly-bicategories. These in fact correspond to modules having special
properties.
