Séminaire Logique et Interactions

We exploit the analogy between the notion of symmetric monoidal closed category and that of commutative ring. We say that a presentable monoidal closed category is a 2-rig (a rig is like a ring, but the addition may not have an inverse). A category enriched over a 2-rig is like a module over a ring. The 2-category of symmetric 2-rigs is similar to the category of commutative rings. Polynomial functors are like polynomial functions. The category of polynomial functors is cartesian closed (by a result of Hyland Fiore, Gambino, Winskel). We extend this result to the category of bimodules between operads.