Linearly distributive functors
type: Item
instance of: scholarly article
title: Linearly distributive functors
zbMATH Open document ID: 0946.18005
DOI: 10.1016/S0022-4049(98)00110-8
author: R. A. G. Seely
author: J. R. B. Cockett
published in: Journal of Pure and Applied Algebra
publication date: 2000-10-10T00:00:00Z
review text: A linearly distributive category (called a weakly distributive category in earlier work by the same authors; the present name was suggested by \textit{M.~Barr,} to whom the paper is dedicated) is a category with two monoidal structures \(\otimes\) and \(\oplus\), together with a pair of natural transformations making each tensor strong with respect to the other. Such categories arise naturally in the construction of categorical models for the multiplicative fragment of linear logic, the two tensors being used to model the `tensor' and `par' of the latter. In the present paper, the authors argue that the appropriate notion of morphism between such categories is not a single functor but a pair of functors, each monoidal with respect to one of the tensors, and equipped with a number of natural transformations satisfying coherence conditions which are too complicated to repeat here. As evidence in support of this, they show that the exponentials and the additive connectives of linear logic can be regarded as examples of such `linearly distributive functors'; they also show that linearly distributive functors between {\(^*\)-autonomous} categories are essentially the same thing as (dual pairs of) monoidal functors, and in fact the \(2\)-category of \(^*\)-autonomous categories is coreflective in that of linearly distributive categories.
Mathematics Subject Classification ID: 18D25
Mathematics Subject Classification ID: 03B60
Mathematics Subject Classification ID: 03F07
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 03G30
Mathematics Subject Classification ID: 68Q55
Mathematics Subject Classification ID: 18A22
zbMATH DE Number: 1384147
zbMATH Keywords: distributive category
zbMATH Keywords: multiplicative fragment of linear logic
zbMATH Keywords: coherence conditions
zbMATH Keywords: \(^*\)-autonomous categories
reviewed by: Peter Johnstone
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1384147