Subject: categories: Paper on linear bicategories (and non-commutative linear logic)

From: "Robert A.G. Seely" <[email-masked]>
Date: Thu, 25 Feb 1999 00:01:23 -0500

I would like to announce the following paper, which has been posted on
my www site <http://www.math.mcgill.ca/~rags>.  (The McGill ftp site is
currently not functionning, but as soon as it is restored, this paper
ought to appear on the Hypatia mirror site as well.)

The abstract follows.

===================================

Introduction to linear bicategories

by J.R.B. Cockett, J. Koslowski, R.A.G. 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.

This paper is dedicated to Jim Lambek, as part of the celebration of his
75th birthday.