Linear bicategories are a generalization of bicategories 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 non-commutative, they provide a suggestive source of models
for non-commutative 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 *ast;-linear bicategory and the coherence conditions that it must satisfy. Cyclic
adjoints also give rise to linear monads: these are, essentially, the appropriate generalization
(to the linear setting) of Frobenius algebras and the ambialgebras of Topological Quantum
A number of examples of linear bicategories arising from different sources are described,
and a number of constructions that result in linear bicategories are indicated.