Two bicategory structures $(\otimes,\top)$ and $(\parr,\bot)$ on these hom-categories. Thus we have two compositions $\otimes,\parr : B(y,z) \times B(x,y) \rightrightarrows B(x,z)$ and two units $\top_x,\bot_x \in B(x,x)$, each coherently associative and unital.

Linear distributivities $(X \parr Y) \otimes Z \to X \parr (Y\otimes Z)$ and $X \otimes (Y \parr Z) \to (X\otimes Y) \parr Z$, satisfying the usual coherence laws for a linearly distributive category.

Any allegory whose hom-sets are Boolean algebras is a linear bicategory, with $\otimes$ the usual composition and with $X \parr Z = \neg (\neg X^\circ \otimes \neg Y^\circ)^\circ$. In particular, this includes the bicategory of relations in any Boolean category, such as Set (assuming classical logic).

Any ordinary bicategory can be regarded as a linear bicategory with $\otimes = \parr$.

If $B$ is a linear bicategory (such as the delooping of a linearly distributive category) such that $(B,\otimes,\top)$ has local coproducts and $(B,\parr,\bot)$ has local products, then the bicategory $Mat(B)$ (whose objects are families of objects of $B$ and whose morphisms are matrices of 1-cells in $B$) is again a linear bicategory. For instance, if $B$ is the Boolean algebra$\mathbf{2}$ of truth values, then $Mat(\mathbf{2}) \cong Rel(Set)$.

Linear adjoints

Definition

A linear adjunction in a linear bicategory consists of 1-cells $f:X\to Y$ and $g:Y\to X$ along with a unit $\eta : \top_X \to g \parr f$ and $\epsilon : f \otimes g \to \bot_Y$ satisfying versions of the usual triangle identities that include the linear distributivities, as for dual objects in a linearly distributive category

A linear bicategory in which every 1-cell has both a linear right adjoint and a linear left adjoint is a horizontal categorification of a non-symmetric star-autonomous category?. But in Cockett-Koslowski-Seely this is called a “closed linear bicategory”, with the term “$\ast$-linear bicategory” reserved for something stronger analogous to a cyclic star-autonomous category?.

References

Cockett and Koslowski and Seely, Introduction to linear bicategories, Mathematical Structures in Computer Science, 10 (2), 2000 (165 - 203)