nLab continuous poset

Continuous posets

Continuous posets


Let PP be a poset with directed joins.

Elementary definition


For a,bPa, b \in P, we say that aa is way below bb and write aba \ll b if whenever SPS \subseteq P is a directed subset and bSb \leq \bigvee S (where S\bigvee S denotes the join of SS), then there exists sSs \in S with asa \leq s.


We say that PP is continuous if for every aPa\in P, the subset

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

is directed and has join aa.

If PP also has finite joins, hence is a suplattice, then (a)\Downarrow(a) is automatically directed, so the condition reduces to ((a))=a\bigvee (\Downarrow(a)) = a; in this case PP is called a continuous lattice.

Adjoint characterization

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


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

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

has a left adjoint, which must be \bigvee.


A poset PP with directed joins is continuous if and only if :Idl(P)P\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.




  • 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)

Last revised on October 17, 2022 at 16:49:48. See the history of this page for a list of all contributions to it.