# Equivalence of categories of directed complete posets

I asked this question there: https://math.stackexchange.com/questions/700975/equivalence-of-categories-of-directed-complete-posets. Since I had no answer, I try here.

In the book Domains and Lambda-Calculi'' by Amadio and Curien, there is the following exercise: Define an equivalence between the category of partial morphisms generated by $(\mathcal{M}_S, \textbf{Dcpo})$ and the category $\textbf{sCpo}$.

The category $\textbf{Dcpo}$ has as objects directed complete posets i.e. partially ordered sets such that any directed subset has a least upper bound and as morphisms continuous functions for the Scott topology (Scott opens of a dcpo $D$ are subsets $O$ of $D$ such that (1) $x \in O \textit{ and } x \leq y \Rightarrow y \in O$ and (2) $\Delta$ directed and $\bigvee \Delta \in O \Rightarrow O \cap \Delta \not= \emptyset$).

The category $\textbf{sCpo}$ is a subcategory of $\textbf{Dcpo}$: objects are dcpo's with a least element $\bot$ and morphisms are continuous functions $f$ such that $f(\bot) = \bot$.

For any dcpo $C$, we denote by $C_\bot$ the object of $\textbf{sCpo}$ obtained from $C$ by adding a new element $\bot$, which is the least element of $C_\bot$.

The "admissible family of monos" $\mathcal{M}_S$ associates with every object $A$ of $\textbf{Dcpo}$ the class of monomorphisms $\mathcal{M}_S(A)$ such that if $m \in \mathcal{M}_S(A)$, then (1) $m$ is a monomorphism $D \rightarrow A$ for some $D$ and (2) $\textit{im}(m)$ is a Scott open of $A$.

The category of partial morphisms generated by $(\mathcal{M}_S, \textbf{Dcpo})$, denoted below by $\textbf{pC}$, has as objects dcpo's and as morphisms from $D$ to $D'$ equivalence classes $[m, f]$ of representatives of partial morphisms $(m, f)$, where a representative $(m, f)$ for a partial morphism from $A$ to $B$ is a pair of morphisms in $\textbf{Dcpo}$ with $m : D \rightarrow A \in \mathcal{M}_S(A)$ and $f \in \textbf{Dcpo}(D, B)$ and $(m : D \rightarrow A, f: D \rightarrow B)$ is equivalent to $(m': D' \rightarrow A, f': D' \rightarrow B)$ iff there is an isomorphism $i : D \rightarrow D'$ such that $m' \circ i = m$ and $f' \circ i = f$.

I had the idea to define the following functor $F: \textbf{pC} \rightarrow \textbf{sCpo}$: for any object $D$, $F(D) = D_\bot$; for any $[m, f] \in \textbf{pC}(D, D')$, $F([m, f])$ is the morphism $g: D_\bot \rightarrow D'_\bot$ defined by: $g(y) = f(x)$ with $m(x) =y$ if $y \in \textit{im}(m)$; $g(y) = \bot$ if $y \notin \textit{im}(m)$. But it seems that it does not work. Indeed, consider the following dcpo's: $D = (\{ a, b \}, \leq)$ with $a$ and $b$ not comparable; $D' = (\{ a', b' \}, \leq')$ with $a' <' b'$. I denote by $m$ the monomorphism $D \rightarrow D'$ defined by $m(a) = a'$ and $m(b) = b'$. Notice that $[m, m] \not= [id_{D'}, id_{D'}]$, since all the morphisms from $D'$ to $D$ are constant. But $F([m, m]) = id_{D'_\bot} = F([id_{D'}, id_{D'}])$, hence $F$ is not faithful.

So I am not able to solve the exercise.

(Assume that $\mathcal{M}_T$ associates with every object $A$ of $\textbf{Dcpo}$ the class of monomorphisms $\mathcal{M}_T(A)$ such that if $m \in \mathcal{M}_T(A)$, then (1) $m$ is a monomorphism $D \rightarrow A$ for some $D$, (2) $\textit{im}(m)$ is a Scott open of $A$ and (3) $m(x) \leq m(y) \Rightarrow x \leq y$. Consider the category $\textbf{pC'}$ of partial morphisms generated by $(\mathcal{M}_T, \textbf{Dcpo})$. Then it seems that the functor $F' : \textbf{pC'} \rightarrow \textbf{sCpo}$ defined as $F$ (for any object $D$, $F'(D) = D_\bot$; for any $[m, f] \in \textbf{pC'}(D, D')$, $F'([m, f])$ is the morphism $g: D_\bot \rightarrow D'_\bot$ defined by: $g(y) = f(x)$ with $m(x) =y$ if $y \in \textit{im}(m)$; $g(y) = \bot$ if $y \notin \textit{im}(m)$) is an equivalence of categories.)