Let $P$ be a poset with directed joins.
For $a, b \in P$, we say that $a$ is way below $b$ and write $a \ll b$ if whenever $S \subseteq P$ is a directed subset and $b \leq \bigvee S$ (where $\bigvee S$ denotes the join of $S$), then there exists $s \in S$ with $a \leq s$.
We say that $P$ is continuous if for every $a\in P$, the subset
is directed and has join $a$.
If $P$ also has finite joins, hence is a suplattice, then $\Downarrow(a)$ is automatically directed, so the condition reduces to $\bigvee (\Downarrow(a)) = a$; in this case $P$ is called a continuous lattice.
Let $Idl(P)$ denote the poset of ideals in $P$, i.e. subsets that are downward-closed and upwards-directed.
A poset $P$ has directed joins if and only if the principal-ideal map $\downarrow : P \to Idl(P)$, defined by
has a left adjoint, which must be $\bigvee$.
A poset $P$ with directed joins is continuous if and only if $\bigvee: Idl(P) \to P$ has its own left adjoint, which must be $\Downarrow$.
This characterization generalizes directly to the notion of continuous category.
Since a continuous poset must have directed joins, the obvious morphisms in a category whose objects are continuous posets would be the Scott-continuous functions, that is those that preserve directed joins. (These always preserve the order; that is, they are monotone functions.)
Between continuous lattices, we may use the same morphisms; or we may more generally use those Scott-continuous functions that preserve the semilattice structure of finitary joins, in other words, the suplattice morphisms that preserve all joins. However, because a suplattice is a complete lattice, another common choice is to use the Scott-continuous functions that are also inflattice morphisms, that is those that also preserve all meets. Another choice might be the complete-lattice morphisms, those that preserve all meets and all joins.
Continuous lattices are those complete lattices for which taking suprema of directed subsets commutes with taking infima of arbitrary subsets.
The forgetful functor $U$ from the category of continuous lattices to the category of sets is monadic if we use Scott-continuous inflattice morphisms. Here the left adjoint of $U$ takes a set $X$ to the lattice of filters on $X$ (that is filters in the power set Boolean algebra $P X$). For more, see filter monad.
The category of continuous lattices is cartesian closed if we use all Scott-continuous functions. This category was used by Dana Scott to construct models of the untyped lambda calculus.
Every continuous lattice is a Baire lattice.
In Freyd08 in constructive mathematics an interval coalgebra is a continuous poset and the unit interval is the terminal interval coalgebra.
A locale is called locally compact just when the corresponding frame is a continuous lattice. This is equivalently to being an exponentiable in the category of locales. A continuous map between such locales is proper iff its direct image function (which is always an inflattice morphism) is Scott-continuous.
As a consequence the lattice of open subsets of a topological space is a continuous lattice if and only if the sobrification of the topological space is locally compact (i.e. the topology has a basis of compact neighborhoods).
Continuous posets can be generalized to continuous algebras for any lax-idempotent 2-monad.
Rudolf-E. Hoffmann, Continuous posets and adjoint sequences, Semigroup Forum 18 (1979) pp. 173-188. (gdz)
Karl H. Hofmann, A note on Baire spaces and continuous lattices 1980. Bulletin of the Australian Mathematical Society, 21(2), pp. 265-279.
See section 30 of
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