Pointed complete partial order ($\perp \le x$)

A directed-complete partial order with a least element. Here directed suprema concern nonempty directed sets. Morphisms are Scott continuous maps and need not preserve the least element. Requiring strictness would change the category and its cartesian-closure argument.

For Scott continuous maps $e:D\to E$, $p:E\to D$, these two identities say that $D$ is embedded in $E$ and that projection followed by embedding approximates from below. Such pairs lift to endomorphism spaces by $f\mapsto efp$ and $g\mapsto pge$. Their composite on the smaller space is the identity, and the other composite is $g\mapsto (ep)g(ep)\le g$.

For $D_{n+1}=[D_n\to D_n]$ with recursively lifted embedding-projection pairs, form coherent sequences $x_n=p_n{x}_{n+1}$, ordered componentwise. Stage projections $P_n$ and embeddings $E_n$ satisfy $P_n{E}_n=\mathrm{id}$, $E_n{P}_n\uparrow \mathrm{id}$. A coherent family $f_n\in [D_n\to D_n]$ induces $f={\sup }_n{E}_n{f}_n{P}_n$; conversely $f_n=P_{n}f{E}_n$. These constructions are inverse. Removing the initial coordinate of the coherent sequence identifies this function-space limit with the original domain limit.

The Scott continuous maps from $D$ to $E$, ordered pointwise. A nonempty directed family has pointwise supremum, and the constant-bottom function is least. Joint evaluation is Scott continuous: for a directed family of pairs $(f_i,x_i)$, directedness makes diagonal values cofinal among all $f_i(x_j)$. Evaluation and currying make the category of pointed complete partial orders and non-strict Scott continuous maps a Cartesian closed category.

For a directed set of Scott continuous maps $f_i:D\to E$, its least upper bound is $f(x)={\sup }_i{f}_i(x)$. The two directed suprema over $i$ and over inputs commute, proving that $f$ remains Scott continuous. Thus the function space of complete partial orders is a complete partial order.