  Linearly distributive categories were introduced to model the tensor/par
fragment of linear logic, without resorting to the use of negation. Linear
bicategories are the bicategorical version of linearly distributive categories.
Essentially, a linear bicategory has two forms of composition, each determining
the structure of a bicategory, and the two compositions are related by a linear
distribution.
  While it is standard in the field of monoidal topology that the category of
quantale-valued relations is a bicategory, we begin by showing that if the
quantale is a Girard quantale, we obtain a linear bicategory. We further show
that the category $QRel$ for $Q$ a unital quantale is a Girard quantaloid if
and only if $Q$ is a Girard quantale. The tropical and arctic semiring
structures fit together into a Girard quantale, so this construction is likely
to have multiple applications. More generally, we define LD-quantales, which
are sup-lattices with two quantale structures related by a linear distribution,
and show that $QRel$ is a linear bicategory if $Q$ is an LD-quantale.
  We then consider several standard constructions from bicategory theory, and
show that these lift to the linear bicategory setting and produce new examples
of linear bicategories. In particular, we consider quantaloids. We first define
the notion of a linear quantaloid ${\cal Q}$ and then consider linear ${\cal
Q}$-categories and linear monads in ${\cal Q}$, where ${\cal Q}$ is a linear
quantaloid. Every linear quantaloid is a linear bicategory.
  We also consider the bicategory whose objects are locales, 1-cells are
bimodules and two-cells are bimodule homomorphisms. This bicategory turns out
to be what we call a Girard bicategory, which are in essence a closed version
of linear bicategories.
