What is a Categorical Model of Intuitionistic Linear Logic?

This paper re-addresses the old problem of providing a categorical model for Intuitionistic Linear Logic (ILL). In particular we compare the now standard model proposed by Seely to the lesser known one proposed by Benton, Bierman, Hyland and de Paiva. Surprisingly we nd that Seely's model is unsound in that it does not preserve equality of proofs. We shall propose how to adapt Seely's deenition so as to correct this problem and consider how this compares with the model due to Benton et al. For the rst part we shall consider only the multiplicative, exponential fragment of Intuitionistic Linear Logic (MELL). Rather than give a detailed description of the logic and associated term calculus we assume that the reader is familiar with other work 15, 5]. The sequent calculus formulation is originally due to Girard 9] and is given below.