Categorical models of Linear Logic with fixed points of formulas

TL;DR

This paper introduces a categorical semantics for a linear logic with fixed points, extending existing models and demonstrating normalization properties through two concrete instances.

Contribution

It develops a novel denotational semantics for muLL using Seely categories and strong functors, with two models illustrating different fixed point interpretations.

Findings

Fixed points interpreted uniformly in sets and relations

Distinct fixed point interpretations in totality spaces

Proof normalization demonstrated in the new model

Abstract

We develop a denotational semantics of muLL, a version of propositional Linear Logic with least and greatest fixed points extending David Baelde's propositional muMALL with exponentials. Our general categorical setting is based on the notion of Seely category and on strong functors acting on them. We exhibit two simple instances of this setting. In the first one, which is based on the category of sets and relations, least and greatest fixed points are interpreted in the same way. In the second one, based on a category of sets equipped with a notion of totality (non-uniform totality spaces) and relations preserving them, least and greatest fixed points have distinct interpretations. This latter model shows that muLL enjoys a denotational form of normalization of proofs.