Finn Lawler thesis outline (Rev #2)

This page is for material relevant to my Ph.D. thesis — I will try to sketch the important bits here as I write up the document itself.

Chapter 2

There are two quite different ways to construct the effective topos: we start with the Kleene algebra $(\mathbb{N}, \cdot)$ of partial recursive functions, and then

1. construct the effective tripos,

   • this is the hyperdoctrine over $Set$ whose predicates over $X$ are $\mathbb{N}$-valued sets $X \to P\mathbb{N}$, ordered by $\phi \leq \psi$ if there exists a partial recursive function $\Phi$ such that for any $x \in X$ and $n \in \phi x$, $\Phi n$ is defined and contained in $\psi x$.

   and then take its category of partial equivalence relations; or

2. construct the regular category of assemblies over $\mathbb{N}$

   • …

   and take its exact completion.

We show how the two are related.