# How to define par-operations for given tensors

**Jürgen Koslowski**
### Abstract

We present two construction principles that allow a large class of
bicategories to be equipped with new associative 1-cell
compositions. Each of these so-called par-operations together with
the original 1-cell composition as tensor constitutes the structure
of a linear bicategory, except possibly for the existence of
par-units. This allows the interpretation of most of the positive
fragment of non-symmetric linear logic in these bicategories. Both
construction principles utilize local pushouts and are parametrized,
either by ``oplaxly pointed lax endo-functors'', or by modules on
the identity functor. In the first case one always obtains linear
structures satisfying the MIX-rule. In particular, a given
bicategory may carry several linear structures with the same tensor.

Already for the category of sets with cartesian product as tensor
several non-trivial par operations with unit exist that yield a linearly
distributive category (that is, a linear bicategory with one
object). By means of a matrix construction, one then also obtains
several linear structures on the bicategory of spans with ordinary
span-composition as tensor.