A talk given at The Graduate Center on May 22, 2024.

Abstract: In categorical database theory, profunctors are ubiquitous. For example, they are used to define schemas in the algebraic data model. However, they can also be used to query and migrate data. In this talk, we will discuss an interesting phenomenon that arises when trying to model profunctors in a computer. We will introduce two notions of profunctor presentations: the UnCurried and Curried presentations. They are modeled on thinking of profunctors as functors P: C^op x D -- Set and as functors P: C^op -- Set^D, respectively. Semantically of course, these are equivalent, but their syntactic properties are quite different. The UnCurried presentations are more intuitive and easier to work with, but they carry a fatal flaw: there does not exist a semantics-preserving composition operation of UnCurried presentations that also preserves finiteness. Therefore we introduce the Curried presentations and show that they remedy this flaw. In the process, we characterize which UnCurried Presentations can be made Curried, and discuss some applications. This talk will be based off of this recent preprint which is joint work with Gabriel Goren Roig and Joshua Meyers.