Seminario de Categorías

Expositor: David Spivak (Topos Institute).

Resumen: One can imagine a database schema as a category and an instance or state of that database as a functor I: C-->Set; the category of these is denoted C-Set. One can think of a data-migration functor, a way of moving data between schemas C and D, as a parametric right adjoint C-Set --> D-Set. In database speak, these are D-indexed "unions of conjunctive queries".

Scene change. The usual semiring of polynomials in one variable with cardinal coefficients, polynomials such as p = y^3 + 3y + 2, can be categorified to Poly, the category of polynomial functors, where + and x are the categorical coproduct and product. Composition of polynomials (p o q) gives a monoidal operation on this category for which the identity polynomial, y, is the unit. Ahman and Uustalu showed in 2016 that, up to isomorphism, the comonoids in (Poly, o, y) are precisely categories (!), and Garner sketched a proof in a recent video that bimodules between polynomial comonoids are parametric right adjoints between copresheaf categories. Recall that these are precisely the data-migration functors described above. In the talk, I will describe this circle of ideas.

I propose that in 2021 a great transition is upon us; distances that were measured in days are now measured in zoom-hiccups. The speed of data migration—if that's indeed a valid way to model it—is much faster than ever, driving dominance into the hands of those who move data: roughly speaking, computerized processes. Researchers use biomimicry to formalize as many aspects of human intelligence as they can, much of which is then installed as automated software systems that run constantly. I call the automated speed-up of bio-inspired intellectual processes AI, and I'm not judging it as good or bad, but I do consider it immensely important. I propose that we as mathematicians have the ability to shape the course of AI. Mathematics becomes technology, and I believe we'll fare better if that technology is based on elegant principles rather than made ad-hoc. Polynomial functors are my entry point, and this talk can serve as an invitation to others to join in whatever capacity appeals to them. To respect the standards of academic talks, I will mainly restrict my discussion to mathematics and its applications, rather than to speculation.

https://www.youtube.com/watch?v=sulssc31XRw&feature=youtu.be