## Abstract for: Introduction to linear bicategories

#### by Cockett, Koslowski, and Seely

Linear bicategories are a generalization of the notion of a
bicategory, in which the one horizontal composition is replaced by
two (linked) horizontal compositions. These compositions provide a
semantic model for the tensor and par of linear logic: in
particular, as composition is fundamentally noncommutative, they
provides a suggestive source of models for noncommutative linear
logic.
In a linear bicategory, the logical notion of complementation
becomes a natural linear notion of adjunction. Just as ordinary
adjoints are related to (Kan) extensions, these linear adjoints are
related to the appropriate notion of linear extension.

There is also a stronger notion of complementation, which arises,
for example, in cyclic linear logic. This sort of complementation
is modelled by cyclic adjoints. This leads to the notion of a
$*$-linear bicategory and the coherence conditions which it must
satisfy. Cyclic adjoints also give rise to linear monads: these
are, essentially, the appropriate generalization (to the linear
setting) of Frobenius algebras.

A number of examples of linear bicategories arising from different
sources are described, and a number of constructions which result in
linear bicategories are indicated.