KAN EXTENSIONS ALONG PROMONOIDAL FUNCTORS BRIAN DAY AND ROSS STREET

ABSTRACT. Strong promonoidal functors are defined. Left Kan extension (also called “existential quantification”) along a strong promonoidal functor is shown to be a strong monoidal functor. A construction for the free monoidal category on a promonoidal category is provided. A Fourier-like transform of presheaves is defined and shown to take convolution product to cartesian product. Let V be a complete, cocomplete, symmetric, closed, monoidal category. We intend that all categorical concepts throughout this paper should be V-enriched unless explicitly declared to be “ordinary”. A reference for enriched category theory is [10], however, the reader unfamiliar with that theory can read this paper as written with V the category of sets and ⊗ for V as cartesian product; another special case is obtained by taking all categories and functors to be additive and V to be the category of abelian groups. The reader will need to be familiar with the notion of promonoidal category (used in [2], [6], [3], and [1]): such a category A is equipped with functors P: Aop⊗Aop⊗A−→V, J: A−→V, together with appropriate associativity and unit constraints subject to some axioms. Let C be a cocomplete monoidal category whose tensor product preserves colimits in each variable. If A is a small promonoidal category then the functor category [A, C] has the convolution monoidal structure given b