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. #### The category of pers Let $\mathbf{E} \to \mathbf{B}$ be an ordered [[regular fibration]]. A partial equivalence relation over the type $X \in \mathbf{B}$ is relation $R$ on $X$ that is symmetric and transitive: $$ \begin{aligned} R(x_1, x_2) & \Rightarrow R(x_2, x_1) \\ R(x_1, x_2), R(x_2, x_3) & \Rightarrow R(x_1, x_3) \end{aligned} $$ A morphism $(R, X) \to (S, Y)$ of pers is a relation $F \colon X, Y$ that satisfies $$ \begin{aligned} F(x,y) & \Rightarrow R(x,x) \wedge S(y,y) \\ F(x_1,y_1) \wedge R(x_1, x_2) \wedge S(y_1, y_2) & \Rightarrow F(x_2,y_2) \\ F(x,y_1) \wedge F(x,y_2) & \Rightarrow S(y_1,y_2) \\ R(x,x) & \Rightarrow \exists \upsilon . F(x, \upsilon) \end{aligned} $$ These form a category $Per(\mathbf{E} \to \mathbf{B})$. The effective topos is $Per$ of the effective tripos. #### The exact completion * allegory $Rel(\mathbf{Asm})$ is unitary, tabular. * effective completion is splitting of equivalences $Rel(\mathbf{Asm})[\check{\mathrm{eqv}}]$. * exact completion is category of maps in this. #### Allegories vs. bicategories of relations A unitary pre-tabular allegory is the same thing as a [[nlab:bicategory of relations]]. This extends to an equivalence of 2-categories. We define a _framed bicategory of relations_ to be a (locally ordered) [[nlab:framed bicategory]] that is a bicategory of relations in which all cartesian-product structure morphisms are 'tight' (or 'representable') maps. Then +-- {: .un_theorem} ###### Theorem The 2-categories of ordered regular fibrations and of framed bicategories of relations are equivalent. =-- ...