# Morphisms and modules for poly-bicategories

## J.R.B. Cockett, J. Koslowski, and R.A.G. Seely

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.

Keywords: Bicategories, polycategories, multicategories, modules, representation theorems

2000 MSC: 18D05, 03F52, 16D90

Theory and Applications of Categories, Vol. 11, 2003, No. 2, pp 15-74.

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