Introduction to linear bicategories
| Autor: | R. A. G. Seely, J. Koslowski, J. R. B. Cockett |
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| Rok vydání: | 2000 |
| Předmět: | |
| Zdroj: | Mathematical Structures in Computer Science. 10:165-203 |
| ISSN: | 1469-8072 0960-1295 |
| DOI: | 10.1017/s0960129520003047 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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