On the categorical semantics of Elementary Linear Logic

We introduce the notion of elementary Seely category as a notion of categorical model of Elementary Linear Logic (ELL) inspired from Seely’s definition of models of Linear Logic (LL). In order to deal with additive connectives in ELL, we use the approach of Danos and Joinet [DJ03]. From the categorical point of view, this requires us to go outside the usual interpretation of connectives by functors. The ! connective is decomposed into a pre-connective ] which is interpreted by a whole family of functors (generated by id, ⊗ and &). As an application, we prove the stratified coherent model and the obsessional coherent model to be elementary Seely categories and thus models of ELL. Introduction The goal of implicit computational complexity is to give characterizations of complexity classes which rely neither on a particular computation model nor on explicit bounds. In linear logic (LL) [Gir87], the introduction of the exponential connectives gives a precise status to duplication and erasure of formulas (the qualitative analysis). It has been shown that putting constraints on the use of exponentials permits one to give a quantitative analysis of the cut elimination procedure of LL and to define light sub-systems of LL characterizing complexity classes (for example BLL [GSS92], LLL [Gir98] or SLL [Laf04] for polynomial time and ELL [Gir98, DJ03] for elementary time). In order to have a better understanding of the mathematical structures underlying these systems, various proposals have been made in the last years with the common goal of defining denotational models of light systems [MO00, Bai04, DLH05, LTdF06, Red07]. Our goal is to define a general categorical framework for the study of these systems. We will focus on ELL which is probably the simplest one.1 Our starting point is quite simple: starting from Seely’s notion of categorical model of LL [See89], it is natural to define models of ELL by removing the comonad structure of ! since ELL is obtained from LL by removing the dereliction and digging rules which correspond to this comonad structure. Things become more interesting when one wants to deal with the additive connectives. The usual approach to categorical logic is based roughly on the interpretation: connective 7→ functor, formula 7→ object, rule 7→ natural transformation, proof 7→ morphism, ... The non-local definition of valid proof-nets with additives for ELL given in [DJ03] is presented here by means of the preconnectives [ and ], pre-formulas and pre-proofs. Their categorical interpretation requires us to use ∗Partially supported by the French ANR “NO-CoST” project (JC05 43380). Brian Redmond has independently studied the question of categorical semantics of SLL [Red07].