Generalized Stone

Tristan Bice joint work with Charles Starling

Institute of of the Polish Academy of Sciences

Facets of Irreversibility Oslo University, December 4th 2017 Non-Commutative Stone Dualities

I In recent years various non-commutative Stone dualities considered by Kudryavtseva, Lawson, Lenz, Resende, etc. I These extend one of two commutative dualities – 1. Classic Stone duality (Stone 1936):

Stone spaces ↔ Boolean algebras.

( = 0-dimensional compact Hausdorff space) 2. Point-free topology (late 50’s-):

sober spaces ↔ spatial frames.

I Our goal: find a happy medium, a more faithful extension of classic Stone duality to non-0-dimensional spaces/groupoids.

I Somewhat similar to Shirota (1952)/De Vries (1962) duality: compact Hausdorff spaces ↔ R-lattices ↔ compingent algebras. Classic Stone Duality

I Given a 0-dimensional compact Hausdorff space X , we consider its clopen sets S ordered by inclusion, i.e. O ≤ N ⇔ O ⊆ N.

I This poset S has a minimum 0 = ∅ and a maximum 1 = X as well as meets, joins and complements given by O ∧ N = O ∩ N. O ∨ N = O ∪ N. Oc = X \ O.

I In fact, S is a Boolean algebra as it also satisfies M ∧ (N ∨ O) = (M ∧ N) ∨ (M ∧ O). (Distributivity)

Theorem (Stone 1937) Every Boolean algebra arises in this way. Hofmann-Lawson Duality

I Given locally compact X , consider all open sets S again with O ≤ N ⇔ O ⊆ N.

I S is a complete and in fact a frame, satisfying _ _ O ∧ N = O ∧ N. (Infinite Distributivity) N∈N I Can recover compact containment b as way-below : O b N ⇔ ∃ compact C (O ⊆ C ⊆ N). \ _ O  N ⇔ O ∈ {I ⊆ S : I is an with N ≤ I }

I Moreover, S is continuous, i.e. for all O, _ O = N. NO

Theorem (Hoffman-Lawson 1978) Every continuous frame arises in this way. Pluses and Minuses

(+) Continuous frames describe general locally compact spaces that are not 0-dimensional or even Hausdorff. (−) Spaces often have uncountably many open sets, even when they have countable bases, e.g. the Cantor space {0, 1}N. (−) Frames are 2nd order but Boolean algebras are 1st order, i.e.

I frames rely on joins of arbitrary infinite subsets while st I Boolean algebras are defined by a finite list of 1 order sentences involving finitary operations and relations (i.e. ≤). I So classical model theory only applies Boolean algebras, e.g. 1. Lo´s’stheorem: ultraproducts of Boolean algebras are Boolean. 2. L¨owenheim-Skolem theorem: arbitrarily large Boolean algebras have countable elementary submodels. 3. Fra¨ıssetheory: homogeneous Fra¨ısselimits exist for suitable subclasses of Boolean algebras. st I Goal: find a 1 order extension of classic Stone duality to spaces that are not 0-dimensional. Bases

I Given compact Hausdorff X , take a basis S of open sets

I Problem: (S, ⊆) does not always determine X .

I E.g. Consider the Boolean algebra B of regular open subsets of the unit inverval [0, 1] with its usual topology.

I This has a countable basis, which generates a countable subalgebra S of B.

I X has no isolated points so S has no atoms.

I All countable atomless Boolean algebras are isomorphic.

I Thus S ≈ clopen subsets of the Cantor space {0, 1}N.

I i.e. arbitrary bases fail to distinguish [0, 1] and {0, 1}N. I Solution: either 1. restrict to certain bases with enough (1st order) conditions so compact Hausdorff spaces are distinguished, or 2. add more structure, e.g. add compact containment as an extra primitive relation (i.e.  is not defined by ≤)

I Shirota/De Vries does a bit of both. We go for option 1. ∪-Bases

I Given compact Hausdorff X , consider a ∪-basis S of open sets, i.e. require S to be closed under finite unions [ O, N ∈ S ⇒ O ∪ N ∈ S (and ∅ = ∅ ∈ S).

I If every O ∈ S is clopen then S must be the entire clopen lattice as in the classical Stone duality. (Proof: any clopen C ⊆ X has a cover in S, as S is a basis, which can be made finite, as C is compact, so the union C is in S, as S is ∪-closed.) I Problem: compact containment b is not way-below . I E.g. let S = clopen subsets of the Cantor space X = {0, 1}N. I So b is just ⊆. In particular, X b X . I Take any x ∈ X and consider the ideal I = {O ∈ S : x ∈/ O}. S W I Then I = X \{x}. As x is not isolated I = X in S. I But X ∈/ I so X 6 X . Rather Below

I Assumption: S is a ∪-basis of compact Hausdorff X . I In particular, S is bounded: min(S) = ∅ and max(S) = X . I Can then recover compact containment b as rather below:

O b N ⇔ ∃M ∈ S (O ∩ M = ∅ and N ∪ M = X ).

I Proof (⇒): As X is Hausdorff,

O b N ⇔ O ⊆ N.

∴ for each x ∈ X \ N we have x ∈ Ox ∈ S with Ox ∩ O = ∅. As X \ N is compact, we have a finite subcover, whose union M is in S, as S is a ∪-basis. Note O ∩ M = ∅ and X = N ∪ M. I Proof (⇐): If O ∩ M = ∅ then O ∩ M = ∅. Thus X = N ∪ M implies O ⊆ N. I Can show that points of X correspond to maximal b-filters. I Thus X can be recovered from (S, ⊆). I In particular, ∪-bases distinguish compact Hausdorff spaces. Rather Distributive

I Goal: Characterize ∪-bases S of compact Hausdorff X (considering S as a poset where ≤ is ⊆ and ≺ is b) Proposition Any ∪-basis S satisfies ≺-distributivity:

a ≤ b ∨ c ⇔ ∀a0 ≺ a ∃b0 ≺ b ∃c0 ≺ c (a0 ≺ b0 ∨ c0 ≺ a).

0 I Note ≤-distributivity reduces to a = a case, i.e.

a ≤ b ∨ c ⇔ ∃b0 ≤ b ∃c0 ≤ c (a = b0 ∨ c0),

which is the usual notion of distributivity for ∨-semilattices. Theorem (B.-Starling 2017) Every (bounded) ≺-distributive ∨-semilattice arises in this way.

I Classic Stone duality recovered when ≺ is reflexive/≺ = ≤. Generalizations

I Given locally compact Hausdorff X , consider a ∪-basis S of relatively compact open sets. Then S has no maximum but ∀O ∈ S ∃N ∈ S (O b N). (b-round) O b N ⇔ ∀P ⊇ N ∃M ∈ S (O ∩ M = ∅ and N ∪ M ⊇ P). Theorem (B.-Starling 2017) Every ≺-round ≺-distributive ∨-semilattice (with 0) arises this way. I For locally compact locally Hausdorff X , take a basis S 3 ∅: ∃ open Hausdorff M c O, N ∈ S ⇔ O ∪ N ∈ S. (∪) I Note ⇐ means every O ∈ S is compactly contained in some open Hausdorff set (in particular O itself is Hausdorff). I Such a ∪-basis S is only a conditional ∨-semilattice, i.e. a, b ≤ c ⇒ a ∨ b exists.

I Problem: being a conditional ∨-semilattice is not sufficient to characterize such ∪-bases. I Solution: find a way to express (∪) order theoretically. The Hausdorff Relation I Given a poset S, define the Hausdorff relation ` by 0 0 0 0 0 0 a ` b ⇔ ∀a ≺ a ∀b ≺ b ∃c ≺ a, b ∀c ≤ a , b (c ≺ c).

I If S is a ≺-distributive ∧-semilattice then 0 0 0 0 a ` b ⇔ ∀a ≺ a ∀b ≺ b (a ∧ b ≺ a ∧ b). Proposition If S is a ∪-basis of some locally compact locally Hausdorff X then

O ∪ N is Hausdorff ⇔ O ` N holds in S.

Theorem (B.-Starling 2017) Every ≺-round ≺-distributive poset (with minimum 0) satisfying

0 0 a ≺ a ` b b ⇒ a ∨ b exists comes from a ∪-basis of some locally compact locally Hausdorff X . Non-Commutative Extension

I Given locally compact locally Hausdorff etal´egroupoid G, consider a basis S of open Hausdorff bisections such that O ∈ S ⇒ O−1 ∈ S. O, N ∈ S ⇒ O · N ∈ S. ∃ open Hausdorff bisection M c O, N ∈ S ⇔ O ∪ N ∈ S. I S is a ≺-round ≺-distributive inverse semigroup with 0 and a ≺ a0 ∼ b0 b ⇒ a ∨ b exists ` a ≺ b ⇒ a−1a ≺ b−1b. Theorem (B.-Starling 2017) Every such inverse semigroup arises in this way.

I Final note: these dualities are also functorial etal´egroupoids ↔ inverse semigroups. continuous ↔ relational morphisms.