*-Autonomous Categories and Linear Logic

The subject of linear logic has recently become very important in theoretical computer science. It is apparent that the ∗-autonomous categories studied at length in [Barr, 1979] are a model for a large fragment of linear logic, although not quite for the whole thing. Since the main reference is out of print and since large parts of that volume are devoted to results highly peripheral to the matter at hand, it seemed reasonable to provide a short introduction to the subject. I include here the formal construction of my student Po-Hsiang Chu that appears as an appendix to that volume. At the time, the formal construction appeared not to have substantial mathematical interest, but it appears to be the most interesting part in the present context. It turns out, for example, that one can in many cases construct models of the full linear logic from Chu’s construction applied to a cartesian closed category. One problem in this subject is that having two very different sources—from linear logic as well as category theory—two very different and essentially inconsistent notations have arisen. Without much hope, I will attempt here to introduce a terminology and notation that draws from both subjects and is as consistent as possible with the two different traditions. To begin with, there is the name ∗-autonomous category that gives the ∗ notation a privileged position. (It also gives librarians fits, which is a good reason to keep using it.) The notation using ⊥ is certainly suggestive, but the name “perp-autonomous” categories is graceless. Thus I will continue to call them ∗-autonomous categories for the time being. The notations 0, 1,+,× for categorical initial object, final object, sum and product are too hallowed by use to even think of changing. Everyone agrees on ⊗ for the tensor. Then there is the dual construction, ∗ In the preparation of this paper, I have been assisted by a grant from the NSERC of Canada. I would also like to thank McGill University for a sabbatical leave and the University of Pennsylvania for a very congenial setting in which to spend that leave.