1 Continuous Monads: Continuous Lattices Revisited Ernie Manes, University of Massachusetts Amherst, Massachusetts, USA FMCS 2019 May 28, 2019
2 1. Quick Overview 2. Lattices for Computation 3. Motivating Monads 4. Conditionals 5. Power Set Monads 6. Continuous Monads 7. The Two-Element Algebra 8. Scott Monads
1 QUICK OVERVIEW 3 1 Quick Overview Usually, datatypes are posets in which every directed set has a supre- mum. We generalize directed sets to ρ -sets and define ρ -continuous posets. Main theorem part 1 : For every ρ there is a “continuous” monad T ρ whose algebras are the ρ -continuous posets. Main theorem part 2 : Certain abstract axioms characterize whether a monad is of form T ρ for some ρ .
1 QUICK OVERVIEW 4 Every ρ -continous poset has two topologies, the Sierpi´ nski topology and the canonical topology . When ρ = directed we get continuous lattices with the Scott and Law- son topologies. For a Scott monad where each ρ -set is directed, the Sierpi´ nski topology is the Scott topology. The condition that a directed set be a ρ -set is necessary and sufficient for the canonical topology to be compact Hausdorff. For general ρ , an intersection of two ρ -Scott-open sets need not be ρ -Scott-open.
1 QUICK OVERVIEW 5 The vitality of the 1980 text Compendium of Continuous Lattices stems from the fact that • Continuous lattices are data types (Scott topology). • Continuous lattices are part of the theory of compact abelian semi- groups (Lawson topology). Our aim is to keep this perspective in a wider range of structures. continuous multivariable calculus = ρ -continuous differential categories
2 LATTICES FOR COMPUTATION 6 2 Lattices for Computation In the early 70s, Dana Scott introduced the idea that computation states are partially ordered. x ≤ y means y has better information than x . The mathematical problem was to represent data types as lattices.
2 LATTICES FOR COMPUTATION 7 D ⊂ ( X, ≤ ) is consistent if D � = ∅ and if for a 1 , . . . , a m ∈ D there exists a ∈ X with all a i ≤ a . Even stronger, D ⊂ ( X, ≤ ) is directed if D � = ∅ and if for a, b ∈ D there exists c ∈ D with a, b ≤ c . As a computation proceeds, we think of its intermediate states as begin consistent, hopefully directed. The final computation is the supremum.
2 LATTICES FOR COMPUTATION 8 Theorem In analysis, all topological spaces are Hausdorff. Proof Ask my analyst friends at UMass. Early topology texts were Alexandroff and Hopf [3] in 1935 and ˘ Cech [5] written in 1955 and later rewritten in English by Frolik and Kat˘ etov. Alexandroff and Hopf credited Kolmogoroff with the invention of T0 spaces. These were studied in some depth by ˘ Cech.
2 LATTICES FOR COMPUTATION 9 Scott [19] pointed out that ˘ Cech [5, page 483] called T0 spaces “feebly semi-separated”. The Grothendieck school [11] noticed that if X is a T0 space, its “spe- cialization order” x ≤ y ⇔ x ∈ { y } yields a poset. There are many topologies whose specialization is a given poset. For such a topology • { y } = ↓ y (= { x : x ≤ y } ) ∈ ( ↓ y ) ′ so the topology is T0 • if x �≤ y , x ∈ ( ↓ y ) ′ , y / • Each closed set A is a lower set, A = ↓ A . • U is upper ⇔ U ′ is lower, so all open sets are upper sets. • The topology is not T1 unless x ≤ y ⇒ x = y .
2 LATTICES FOR COMPUTATION 10 In his seminal paper [19], Dana Scott wanted a specialization topology on ( X, ≤ ) such that when a directed set D is considered as a net in X , � D as one of its limits. it would have With this in mind, U ⊂ X is Scott open if • U = ↑ U � D ∈ U ⇒ U ∩ D � = ∅ • For directed D , We say x is way below y , written x ≪ y , if whenever D is directed � D then there exists d ∈ D with x ≤ d . and y ≤ To explain this x ≤ y ⇔ x ∈ ↓ y x ≪ y ⇔ x ∈ ( ↓ y ) o where ( · ) o is Scott interior.
2 LATTICES FOR COMPUTATION 11 First definition: a poset with directed suprema is continuous if the axiom of approximation holds, that is, � { y : y ≪ x } x = There are many texts, e.g. Abramsky and Jung [1], and Gunter [12]. Note: The Lawson topology plays no role in these texts. Complete lattices are mathematically convenient, but it is problematic to interpret the greatest element, or arbitrary binary suprema. A set X of alternatives can be represented as the flat poset X ∪ {⊥} with ⊥≤ x . This is not a complete lattice. Additionally, the fundamental example of partial functions between an input set and an output set has non-empty infima (and hence bounded suprema), but does not have binary suprema.
2 LATTICES FOR COMPUTATION 12 Scott showed that the category of continuous lattices and morphisms which preserve directed suprema is cartesian-closed. This category isomorphically embeds (via the Scott topology) as the full subcategory of injective objects in T0 spaces and continuous maps. Morphisms cannot be strict because constants are continuous. The same paper had the D ∞ -construction: an arbitrary poset can be fully embedded in a continuous lattice which is isomorphic to its own function space. This provides the first model of the type-free lambda calculus.
2 LATTICES FOR COMPUTATION 13 While the Scott topology does not have enough open sets to be Haus- dorff, we can extend this topology to the Lawson topology by now allowing ↑ x to be closed for each x . This topology is compact Hausdorff. A function between continous lattices is Lawson continuous if and only if it preserves all infima and directed suprema. This is a different and equally useful notion of morphism. So the Scott topology completely recovers the continuous lattice struc- ture ( x ≤ y ⇔ x ∈ { y } ). The Lawson topology identifies continuous lattices as a class of compact abelian semigroups. Hence the Compendium [8] has six authors.
3 MOTIVATING MONADS 14 3 Motivating Monads “Cayley theorem” for posets. ( X, ≤ ) → (2 X , ⊃ ) , x �→ ↑ x = { y : y ≥ x } x ≤ y ⇔ ↑ x ⊃ ↑ y We have already expressed interest in subsets of ( X, ≤ ), such as con- sistent sets or directed sets. These will become families of subsets of X under the Cayley map. Starting point: BX = {A : A ⊂ 2 X } is a monad in Set , the double- dualization monad induced by 2.
3 MOTIVATING MONADS 15 Say that two subsets A, D ⊂ ( X, ≤ ) are mutually cofinal if ∀ a ∈ A ∃ d ∈ D a ≤ d ∀ d ∈ D ∃ a ∈ A d ≤ a For example, if ( a n ) is a chain then ( a 2 n ) and ( a 3 n ) are mutually cofinal and have ther same supremum. It is natural to seek a version of the Cayley representation which identifies these.
3 MOTIVATING MONADS 16 With this in mind, consider these definitions: • For A ∈ BX , A c = { D ⊂ X : ∃ A ∈ A D ⊃ A } . Note: For A ⊂ (2 X , ⊃ ), A c = ↓A . • B c X = {A ∈ BX : A = A c } B c is a submonad of B . The familification map Υ for subsets of a poset is Υ A = {↑ a : a ∈ A } c ∈ B c X A ⊂ ( X, ≤ ) �→ So A, D are mutually cofinal ⇔ Υ A = Υ D . We always have Υ A = Υ( ↓ A ).
3 MOTIVATING MONADS 17 We are heading to a generalization ρ of “consistent” and “directed”. We call such a ρ a conditional (for suprema), and it assigns to each poset a collection of subsets called ρ -sets. Define T ρ X = {A ∈ B c X : A is a ρ -set in (2 X , ⊃ ) } As we shall see, the axioms on ρ will force T ρ to be a submonad of B c whose algebras are the ρ -continuous posets.
3 MOTIVATING MONADS 18 In my 1982 paper A class of fuzzy theories about Kleisli categories I noticed that NX = {F : F is a filter with non-empty intersection } was a submonad of the filter monad. I wondered what the algebras were. Trying to find out recently , I got started on all of this. It turns out that N is a Scott monad with NX = T ρ X if a ρ -set is a directed and bounded set. The N -algebras are a reasonable category of data types, called continuous posets by some (e.g. [7, 17]). They include • Continuous lattices • Flat posets • Pfn ( A, B ) I believe it is new that NX is the free continuous poset.
3 MOTIVATING MONADS 19 The monad point of view leads to the following fundamental diagram which has at least three useful interpretations: ✲ 2 2 X TX ξ ❄ X 1. The two-element poset generates the whole variety. 2. The Sierpi´ nski topology on 2 induces the “Scott topology”. 3. The discrete topology on 2 induces the “Lawson topology”. More details later.
3 MOTIVATING MONADS 20 Define L to be the category of posets with non-empty infima and mor- phisms which preserve these. L -splitting lemma : Surjective morphisms in L split. Proof . For surjective f define g g f Y − → X − → Y � { x : fx = y } . by gy = Has anybody seen this in the literature?
4 CONDITIONALS 21 4 Conditionals Before giving axioms for ρ -sets, let us set down the intended semantics. There are two versions, as follows. A ρ -poset has all non-empty infima and has suprema for all ρ -sets. Morphisms preserve non-empty infima and ρ -suprema. A � ρ -poset is a ρ -poset with a greatest element. Morphisms must also preserve the greatest element.
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