Polymorphic Linear Logic and Topos Models

We give a deenition of a \linear bration", which is a hyperdoctrine model of polymorphic linear logic, and show how to internalise the bration, generating topos models. This gives a constructive set theoretical context for the logic of Petri nets, as recently developed by N. Mart-Oliet and J. Meseguer. Also, we sketch how this can be further extended to include the exponential operator !. In this context, the topos model we construct can be embedded in the model constructed by A.M. Pitts. 0 Introduction In 4], it is shown how to enrich the logic of Petri nets with gedanken states and processes, by embedding it into linear logic. Recently, Mart-Oliet and Meseguer have asked how to extend this even further to include polymorphism. It turns out that the process is fairly straightforward, and for maximal impact (so as to include constructive set theory), can be internalised to give topos models of polymorphic linear logic. Here we give the necessary deenitions, and sketch the outline of the construction. These notes should be thought of as a sequel to 7], extending the categorical semantics of linear logic to include polymorphism. As there, we (sketchily) describe the interpretation of polymorphic-calculus as the (indexed) Kleisli category induced by the !-cotriple. However, once in this context, we have another well-known model, viz. the internal full subcategory of a presheaf topos as constructed by A.M. Pitts 5]. Our construction does not give fullness, perhaps fortunately, but it will turn out that our model does embed faithfully into Pitts' model. (In essence his model consists of certain families, and ours corresponds to the \constant" families.) 1 Deenitions 1.1 Linear brations We begin with a deenition: