1/24 Point-free Descriptive Set Theory and Algorithmic Randomness Alex Simpson University of Ljubljana, Slovenia incorporating j.w.w. Antonin Delpeuch (Univ. Oxford) CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 2/24 σ -frames . . . A σ -frame O ( X ) is a partially-ordered set with: • countable joins � (including the empty join ∅ ), • finite meets ∧ (including the empty meet X ), • satisfying the countable distributive law: � � U ∧ ( V i ) = U ∧ V i . i i A morphism p : O ( Y ) → O ( X ), between σ -frames is a function that preserves countable joins and finite meets. We write σ Frm for the category of σ -frames. CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 3/24 . . . and σ -locales A σ -locale X is given by a σ -frame O ( X ). A map f : X → Y , between σ -locales X, Y , is given by a morphism f − 1 : O ( Y ) → O ( X ) of σ -frames. We write σ Loc for the category of σ -locales. (N.B., σ Loc ≃ σ Frm op .) CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 4/24 Example σ -frames • O ( X ) = the lattice of open subsets of a topological space. • O ( X ) = the lattice of Borel subsets of a topological space. • O ( X ) = the lattice of Σ α -subsets of a topological space, for any ordinal α . CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 5/24 Full subcategories of σ Loc σ Loc is the category of maps between σ -locales. • The category of continuous functions between hereditarily Lindel¨ of sober topological spaces. • The category of Borel-measurable functions between standard Borel spaces. CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 6/24 Complements and Boolean algebras The complement (if it exists) of an element u in a distributive lattice P is the (necessarily unique) element u ∈ P satisfying u ∧ u = ⊥ u ∨ u = ⊤ If p : P → Q is a homomorphism of distributive lattices and u, u are complements in P then p ( u ) , p ( u ) are complements in Q . A distributive lattice P is a Boolean algebra if and only if every element of P has a complement. Every distributive-lattice homomorphism p : P → Q between Boolean algebras is a Boolean-algebra homomorphism. CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 7/24 σ -Boolean algebras For any σ -frame P , define a σ -Boolean algebra B ( P ) and ✲ B ( P ) via the following universal property. homomorphism i : P • for every homomorphism p : P → Q , where Q is a σ -Boolean algebra, there exists a unique homomorphism q such that q ✲ Q B ( P ) ✲ ✻ i p P Theorem (CLASS). For any quasi-Polish space X , it holds that B ( O ( X )) ∼ = B or ( X ). CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 8/24 The ‘jump’ functor For any σ -frame P , define a σ -frame S ( P ) and homomorphism ✲ S ( P ) via the following universal property. j : P 1. Every element in the image of j has a complement in S ( P ); and 2. for every homomorphism p : P → Q , where every element in the image of p has a complement in Q , there exists a unique homomorphism q such that q ✲ Q S ( P ) ✲ ✻ j p P CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 9/24 The Borel hierarchy Theorem (CLASS). For any quasi-Polish space X , it holds that S n ( O ( X )) ∼ = Σ n +1 ( X ). • The classical result should generalise to ordinal-indexed iterations. • It should hold constructively that B is the free monad over the functor S . • By interpreting the definition in suitable realizability toposes, it should be possible to obtain connections with Turing degrees, the arithmetic hierarchy and the lightface hierarchy. • . . . CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 10/24 Probability valuations Write − − → [0 , 1] for the set of ‘reals’ defined as limits of ascending sequences of rationals in [0 , 1]. A probability ( σ -)valuation on a σ -frame O ( X ) is a function µ : O ( X ) → − − → [0 , 1] satisfying • µ ( ∅ ) = 0 and µ ( X ) = 1. • µ ( u ∨ v ) + µ ( u ∧ v ) = µ ( u ) + µ ( v ). • u ≤ v implies µ ( u ) ≤ µ ( v ). • ( u i ) i ascending implies µ ( � i u i ) = sup i µ ( u i ). CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 11/24 The Cantor locale Define O ( 2 N ) to be the free σ -frame on countably many complemented generators ( c i ) i . Intuitively, the generator c i represents the clopen set { α ∈ { 0 , 1 } ω | α i = 1 } Proposition . There is a unique probability valuation λ : O ( 2 N ) → − − → [0 , 1] such that λ ( c i ) = 1 2 for every i . We are endowing the Cantor ( σ -)locale 2 N with the uniform probability valuation. CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 12/24 σ -sublocales A map f : X → Y between σ -locales is said to be an embedding if f − 1 : O ( Y ) → O ( X ) is surjective. The embeddings determine the notion of σ -sublocale. The σ -sublocales of a σ -locale X are in 1–1 correspondence with congruences on O ( X ). The σ -sublocales of X form a complete lattice S ub ( X ) under the embedding order. CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 13/24 Open σ -sublocales For v ∈ O ( X ) define a congruence relation ≈ o ( v ) on O ( X ) by u ≈ o ( v ) u ′ ⇔ u ∧ v = u ′ ∧ v It holds that ∼ O ( X ) / ≈ o ( v ) = ↓ v We call o ( v ) the open σ -sublocale determined by v . CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 14/24 The σ -locale of random sequences For u, v ∈ O ( 2 N ), define: u ≈ v ⇔ λ ( u ) = λ ( u ∧ v ) = λ ( v ) Define O (Ran) = O ( 2 N ) / ≈ . Theorem 1. Ran is the intersection of all outer-measure-1 σ -sublocales of 2 N . 2. Ran is the intersection of all measure-1 open σ -sublocales of 2 N . 3. Ran itself has outer measure 1. (We say a σ -sublocale X ⊆ 2 N has outer measure 1 if, for every open σ -sublocale X ⊆ o ( u ) ⊆ 2 N , it holds that λ ( u ) = 1.) CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 15/24 Points The terminal σ -locale 1 is given by defining O ( 1 ) to be the free σ -frame on no generators. A point of a σ -locale X is a map from the terminal σ -locale 1 to X . That is, points are given by σ -frame homomorphisms from O ( X ) to O ( 1 ). Proposition Ran has no points. CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 16/24 Classical points In our intuitionistic development, there is a potentially weaker notion of classical point of a σ -locale X : a map from 1 c to X where O ( 1 c ) = Ω ¬¬ = { p ∈ Ω | ¬¬ p ⇒ p } Under classical logic, 1 c ∼ = 1 , so classical points coincide with points. If LEM fails, they may differ. We can view this difference by interpreting the development in Hyland’s effective topos E ff . [Hyland 1981] CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 17/24 Interpretation in E ff The objects in our development all produce assemblies. |O ( 2 N ) | = { U ⊆ { 0 , 1 } ω | U c.e. open } � n r U ⇔ n codes a sequence ( C i ) i of cylinders s.t. U = C i i |− − → [0 , 1] | = { x ∈ [0 , 1] | x left c.e. } n r x ⇔ n codes a sequence ( q i ) i of rationals s.t. x = sup q i i |O ( 1 ) | = { 0 , 1 } n r a ⇔ ( a = 1 ∧ n ∈ K ) ∨ ( a = 0 ∧ n ∈ K ) |O ( 1 c ) | = { 0 , 1 } n r a ⇔ true CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 18/24 Theorem (CLASS) 1. The points of 2 N in E ff are in 1–1 correspondence with computable sequences α ∈ { 0 , 1 } ω . 2. The classical points of 2 N in E ff are in 1–1 correspondence with arbitrary sequences α ∈ { 0 , 1 } ω . 3. Ran in E ff has no points. 4. The classical points of Ran in E ff are in 1–1 correspondence with Kurtz random sequences α ∈ { 0 , 1 } ω . A sequence α ∈ { 0 , 1 } ω is Kurtz random if it is contained in every measure-1 c.e. open subset U ⊆ { 0 , 1 } ω . [Kurtz 1981] CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 19/24 Revisiting constructive point-free descriptive set theory There are two possible approaches to generating a σ -frame Σ α +1 ( X ) from Σ α ( X ). 1. Obtain Σ α +1 ( X ) as the free σ -frame that adds complements to every element of Σ α ( X ). This is the ‘jump’ operation from earlier. 2. Obtain Σ α +1 ( X ) by extending the σ -coframe Π α ( X ) = (Σ α ( X )) op with countable joins. Approach 1 seems the ‘correct’ approach to obtaining a rich constructive point-free descriptive set theory. But we now follow approach 2. CCC, Nancy, June 2017
Point-free Descriptive Set Theory and Algorithmic Randomness 20/24 The σ -frame Σ 2 ( 2 N ) op considered as a Define Σ 2 ( 2 N ) to be the free σ -frame over O ( 2 N ) distributive lattice. • There is a distributive-lattice homomorphism op → Σ 2 ( 2 N ) c : O ( 2 N ) • It further holds that c preserves countable meets. We call elements of Σ 2 ( 2 N ) in the image of c closed. CCC, Nancy, June 2017
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