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 1. construct the regular category of _assemblies_ over $\mathbb{N}$ * ... and take its _exact completion_. We show how the two are related.