Equideductive Logic and CCCs with Subspaces Paul Taylor Domains Workshop IX U of Sussex, Tuesday, 23 September 2008 www.PaulTaylor.EU / ASD
Abstract Stone Duality ◮ Lattice part: ⊤ , ⊥ , ∧ , ∨ for open sets, = for discrete spaces, � for Hausdor ff , ∀ for compact and ∃ for overt ones. ◮ Categorical part: λ -calculus for Σ ( − ) , and the adjunction Σ ( − ) ⊣ Σ ( − ) is monadic: gives definition by description, Dedekind completeness and Heine–Borel. The categorical part only handles locally compact spaces. It needs to be generalised. We will get a CCC, but that’s not important, because the exponential Y X is tested by incoming maps, but its topology by outgoing ones. We certainly need products, Σ ( − ) and equalisers.
CCCs with all finite limits E .................................... > > b i ∧ ˜ α a > > Σ Y Γ > X > ∧ ∧ ∧ ˜ β E × Y ............................. > i × id Y α × Y > ˜ a × id Y > Σ Y × Y Γ × Y > X × Y > ˜ β × Y α ( a , y ) = β ( a , y ) α β ∨ ∨ > ev > Σ < Want to write E = { x | ∀ y . α xy = β xy } .
Equideductive logic ⊢ ⊤ x : 0 ⊢ p p , q ⊢ p & q p & q ⊢ p p & q ⊢ q Γ , x : A , p ( x ) ⊢ α x = β x ∀ I Γ ⊢ ∀ x : A . p ( x ) = ⊲ α x = β x Γ ⊢ a : A , p ( a ) Γ ⊢ ∀ x : A . p ( x ) = ⊲ α x = β x ∀ E Γ ⊢ α a = β a All the variables on the left of = ⊲ must be bound by ∀ . Maybe add some dependent types later. Must have subsitution (cut) for free variable x .
Interpretation of equideductive logic ◮ The obvious set-theoretic one — the construction to follow will give Dana Scott’s equilogical spaces. ◮ In locales — but I’m not sure whether this works (Does ( − ) × X preserve epis? I have both a proof and a counterexample!) ◮ In Formal Topology, if this works. ◮ Proof-theoretic, taking the rules just as they are (as we shall do for most of this lecture). ◮ In another type theory such as Coquand’s Calculus of Constructions or Coq. ◮ With additional axioms of our choosing.
Interaction with the lattice structure The implication = ⊲ in equideductive logic depends on the categorical structure (equalisers and Σ ( − ) ). If Σ also has lattice structure, with induced order ⇒ , then these interact very nicely. That is, if we assume the Phoa principle. In the Gentzen style, this is x : X , α x = ⊤ ⊢ β x = ⊤ x : X , β x = ⊥ ⊢ α x = ⊥ = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = and x : X ⊢ α x ⇒ β x x : X ⊢ α x ⇒ β x which we rewrite as ( ∀ x .α x = ⊤ = ⊲ β x = ⊤ ) ⊳ = = ⊲ ( ∀ x .α x ⇒ β x ) ( ∀ x .β x = ⊥ = ⊲ α x = ⊥ ) ⊳ = = ⊲ This is also the definition of α � β .
Interaction with topological structure Similarly, equality = N in a discrete space N is a special case of general equality of terms: n = m ⊳ = = ⊲ ( n = N m ) = ⊤ , whilst h = k ⊳ = = ⊲ ( h � H k ) = ⊥ in a Hausdor ff space H . The universal quantifier U in a compact space is related to ∀ : ( ∀ x . φ x = ⊤ ) ⊳ = = ⊲ ( U x . φ x ) = ⊤ Similarly ( ∀ x . φ x = ⊥ ) ⊳ = = ⊲ ( ∃ x . φ x ) = ⊥ in an overt space. See Foundations for Computable Topology , § 12, for more discussion: www.Paul Taylor.EU/ASD/foufct
Equideductive spaces Urtypes: generated from 0 , 1 and N by + , × and (( − ) → Σ ). Combinators, including I : ( A → Σ ) → A → Σ , K : ( A → Σ ) → B → A → Σ , � � � � C : ( B → Σ ) → ( C → Σ ) → ( A → Σ ) → ( B → Σ ) → ( A → Σ ) → C → Σ T : 1 , ν 0 : A → ( A + B ) , ν 1 : B → ( A + B ) , � � π 0 : ( A + B ) → Σ ) → A → Σ , π 1 : ( A + B ) → Σ ) → B → Σ , � � � � �� : ( C → Σ ) → A → Σ → ( C → Σ ) → B → Σ → ( C → Σ ) → ( A + B ) → Σ . � � A : (( A → Σ ) + A ) → Σ → 1 → Σ , � � L : (( A + B ) → Σ ) → 1 → Σ → ( A → Σ ) → ( B → Σ ) → Σ . with appropriate equational axioms, such as ∀ MN φ c . C NM φ c = N ( M φ ) c , without = ⊲ .
Equideductive spaces An equideductive space X is ( A , p , q ) where A is an urtype, p is an urstatement on Σ A and q one on A , for which φ, ψ : Σ A , p ( φ ) , ∀ a : A . q ( a ) = ⊲ φ a = ψ a ⊢ p ( ψ ) . This rule is important in the construction. Later, we tighten it to ensure that all spaces are definable using exponentials and equalisers. LHS is a partial equivalence relation. A morphism M : X ≡ ( A , p , q ) → Y ≡ ( B , r , s ) is an realiser M : ( A → Σ ) → B → Σ such that φ : Σ A , p ( φ ) ⊢ r ( M φ ) φ, ψ : Σ A , p ( φ ) , ∀ a . q ( a ) = ⊲ φ a = ψ a ⊢ ∀ b . s ( b ) = ⊲ M φ b = M ψ b , where M 1 = M 2 if φ : Σ A , p ( φ ) ⊢ ∀ b : B . s ( b ) = ⊲ M 1 φ b = M 2 φ b .
Categorical structure 1 ≡ ( 0 , ⊤ , ⊤ ), Σ ≡ ( 1 , ⊤ , ⊤ ). � � The product is ( A , p , q ) × ( B , r , s ) ≡ A + B , ( p · π 0 & r · π 1 ) , [ q , s ] . The equaliser is M I > E ≡ ( A , t , q ) > > ( A , p , q ) > ( B , r , s ) N t ( φ ) ≡ p ( φ ) & ∀ b : B . s ( b ) = ⊲ M φ b = N φ b , The exponential of X ≡ ( A , p , q ) is Σ X ≡ ( Σ A , q p , p ), where q p ( F ) ≡ ∀ φ, ψ : Σ A . p ( φ ) & ( ∀ a : A . q ( a ) = ⊲ φ a = ψ a ) = ⊲ F φ = F ψ. (The modulation p ( φ )& · · · is the source of many di ffi culties.)
All objects are definable If q is defined using ⊤ , equations, & and ∀ = ⊲ then q ( a ) ⊣⊢ q ⊤ ( λφ. φ a ). ( A , p , ⊤ ) � ( Σ Σ A , p ⊤ & prime , ⊤ ) ( A , ⊤ , q ) � ( Σ Σ A , ⊤ , q ⊤ & prime ) � Σ ( Σ A , q ⊤ & prime , ⊤ ) . F �→ λ F . F F > ( Σ A , prime , ⊤ ) >> ( Σ A , ⊤ , ⊤ ) > ( Σ 3 A , ⊤ , ⊤ ) � � F �→ λ F . F λ a . F ( λφ. φ a ) Σ 2 M > > ( B , ⊤ , r ) � Σ ( Σ B , r ⊤ & prime , ⊤ ) ( Σ A , p ⊤ & prime , ⊤ ) >> ( Σ A , prime , ⊤ ) Σ 2 N Σ 2 M > > � Σ { B | r } { A | p } > > { A | ⊤} Σ 2 N
An exactness property i Z ≡ { Σ A | p } ≡ ( A , p , ⊤ ) > > Σ A ≡ ( A , ⊤ , ⊤ ) A ∧ Σ j j ∨ ∨ ∨ ∨ ∧ X ≡ { Σ { A | q } | p } ≡ ( A , p , q ) > Σ Y ≡ Σ { A | q } ≡ ( A , ⊤ , q ) > Y ≡ { A | q } W ≡ ( A , q p , ⊤ ) > > Σ 2 A ≡ ( Σ A , ⊤ , ⊤ ) Σ A ∧ Σ j i ∨ ∨ ∨ ∨ ∧ Σ X ≡ ( Σ A , q p , p ) > Σ Z ≡ ( Σ A , ⊤ , p ) Z ≡ { Σ A | p } >
Exactness property Let L be the full subcategory of objects ( A , p , ⊤ ). (In the case of equilogical spaces, L consists of sober Bourbaki ( = textbook) spaces.) L is closed under × , regular monos and Σ Σ ( − ) . Σ is injective wrt regular monos in L . Given regular mono ( A , p , ⊤ ) ( A , ⊤ , ⊤ ), Σ ( − ) takes it to a regular epi, the pullback of this along any regular mono is still regular epi. Set obeys similar (but stronger) properties.
A Chu-like construction We can represent any equideductive space ( A , p , q ) by two L -objects ( A , p , ⊤ ) and ( Σ A , q p , ⊤ ). Similarly any morphism ( A , p , q ) → ( B , r , s ) is given by ( A , p , ⊤ ) → ( B , r , ⊤ ) and ( Σ A , q p , ⊤ ) ← ( Σ B , s r , ⊤ ). ( Σ A , q p , ⊤ ) ← ( Σ B , s r , ⊤ ) is a homomorphism of Σ 2 -algebras. Like the real and imaginary parts of a complex number. So equideductive spaces have a topological part and an algebraic one, cf. Stone duality. However, ( A , p , ⊤ ) is not the reflection of ( A , p , q ) in L , and indeed does not depend functorially on it.
What kind of theory Should generalised topology be ◮ bipartite, with a topological (“real”) part and an algebraic (“imaginary” one), or ◮ unitary, where the same (exactness) properties apply to all objects? (In “free” equideductive logic, the exactness property only holds when the basic object is ( A , ⊤ , ⊤ ), essentially a locally compact space.)
What kind of theory Should generalised topology be ◮ bipartite, with a topological (“real”) part and an algebraic (“imaginary” one), or ◮ unitary, where the same (exactness) properties apply to all objects? (In “free” equideductive logic, the exactness property only holds when the basic object is ( A , ⊤ , ⊤ ), essentially a locally compact space.) An analogy from the history of Science: ◮ Aristotle had a bipartite theory, with rectilinear motion on Earth and circular motion for the planets. ◮ Galileo and Newton unified them. √ Similarly, whilst C adds − 1 to R , it otherwise obeys the same laws of algebra.
A critical example B ≡ N N is not locally compact, so i : B ≡ N N R (where R ≡ Σ N × N or N N ⊥ ) is not Σ -split, i.e. there is no I : Σ B → Σ R with Σ i · I = id . Hence there is no diagonal fill-in i × id B × Σ B > > R × Σ B ............................................... ev ∨ < Σ so Σ i × id is not surjective. (( − ) × Σ B is crucial to this counterexample.)
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