2024-02-02: Conor McBride (MSP)

Fri, 9 Feb 2024 10:14:47 +0000

Conor McBride (MSP)

Presheaves on Purpose

Abstract

Dependently typed programming languages allow us to define indexed families of datatypes, e.g. Term (n : Nat) : Set as the type of lambda-terms with n variables free. As declared, Term is functorial only on the discrete structure of Nat, which is to say that it respects equality and nobody will faint with amazement. But by an outrageous coincidence (which I learned from Altenkirch, Hofmann and Streicher), such terms can be acted on by thinnings from n to some larger scope m, allowing us to carry terms under binders. That is, we can demonstrate by honest toil that Term is a functor from Thin to Set. I dislike honest toil, and will show you how to design it away in this and similar situations. To that end, I present a universe construction for descriptions of datatypes indexed over the objects of some category C which, by construction, extend to a functors from C to Set, a.k.a. presheaves on Cop.

2024-02-02 11:00:00 UTC
LT401 and Online