Paper on linear bicategories (and non-commutative linear logic)
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.
participants (1)
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Robert A.G. Seely