Hostname: page-component-7c8c6479df-nwzlb Total loading time: 0 Render date: 2024-03-26T15:27:07.241Z Has data issue: false hasContentIssue false

Introduction to linear bicategories

Published online by Cambridge University Press:  01 April 2000

J. R. B. COCKETT
Affiliation:
Department of Computer Science, University of Calgary, 2500 University Drive, Calgary, AL, T2N 1N4, Canada. Email: robin@cpsc.ucalgary.ca
J. KOSLOWSKI
Affiliation:
Institut für Theoretische Informatik, TU Braunschweig, P.O. Box 3329, 38023 Braunschweig, Germany. Email: koslowj@iti.cs.tu-bs.de
R. A. G. SEELY
Affiliation:
Department of Mathematics, McGill University, 805 Sherbrooke St., Montréal, QC, H3A 2K6, Canada. Email: rags@math.mcgill.ca

Abstract

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 Field Theory.

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.

Type
Research Article
Copyright
© 2000 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)