nLab continuous poset

Definitions

Let $P$ be a poset with directed joins.

Elementary definition

Definition

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 $a \leq s$.

Definition

We say that $P$ is continuous if for every $a\in P$, the subset

$$\Downarrow (a) \coloneqq \{ b \in P | b \ll a \}$$

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.

Adjoint characterization

Let $Idl(P)$ denote the poset of ideals in $P$, i.e. subsets that are downward-closed and upwards-directed.

Proposition

A poset $P$ has directed joins if and only if the principal-ideal map $\downarrow : P \to Idl(P)$, defined by

$$\downarrow(a) \coloneqq \{ b \in P | b \leq a \} ,$$

has a left adjoint, which must be $\bigvee$.

Theorem

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.

Morphisms

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.

Properties

• 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.

Examples

• 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. every point of the sobrification has a neighborhood base made of compact subsets).

Related pages

• continuous category

• locally compact locale

• Continuous posets can be generalized to continuous algebras for any lax-idempotent 2-monad.

References

• 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

• Peter Freyd, Algebraic real analysis, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (tac:20-10)