Title: “Sheaf representation of monoidal categories”
Abstract: Wouldn’t it be great if monoidal categories were nice and easy? They are! We will discuss how a monoidal category embeds into a “nice” one, and how a “nice” monoidal category consists of global sections of a sheaf of “easy” monoidal categories. Here “nice” means that the idempotent subobjects of the tensor unit have joins that are respected by tensor products, and “easy” means that the topological space of which these subobjects are the opens is local. This theorem subsumes sheaf representation results for toposes, its proof is entirely concrete, and it cleanly separates “spatial” and “temporal” directions of monoidal categories. We will focus mostly on explaining these statements and placing them in context. Joint work with Rui Soares Barbosa and others.