Equilogical spaces A Chu-like Extension of Topological Spaces There are several constructions that embed the traditional category of topological spaces and continuous functions in a cartesian closed category. Paul Taylor (All topological spaces will be sober.) Honorary Research Fellow The best known was proposed by Dana Scott, University of Birmingham with details by Andrej Bauer and Lars Birkedal: An equilogical space is a topological space X together with an equivalence relation ∼ on its set of points. Dagstuhl Seminar 15441 A morphism f : ( X , ∼ X ) → ( Y , ∼ Y ) of equilogical spaces is a Monday 25 October 2015 continuous function f : X → Y that respects the equivalence relations: www.Paul Taylor.EU/ASD/equideductive/ x 1 ∼ X x 2 = ⇒ fx 1 ∼ Y fx 2 . www.Paul Taylor.EU/slides/15-Domains-Cork.pdf The category of equilogical spaces is locally cartesian closed. Funded by: my late parents. Presheaves and exact completions My objections To be rather more categorical, This is all excellent work, but it is not to my taste. the process of formally adding quotients of equivalence ◮ The equivalence relation defining an equilogical space relations to a category is called its exact completion. need not respect the topological structure in any way. Two authors in particular studied this: So the points and not the topology are carrying the structure. (Unfortunately, we will not be able to replace Giuseppe Rosolini surveyed several cartesian closed extensions topological spaces with locales in this work.) of topology, showing how they are reflective subcategories ◮ Toposes, locally cartesian closed categories and exact of categories of presheaves (or functor categories). completions are tools for set theory — the study of discrete Jiˇ ri Rosický showed that the exact completion structures, not continuous ones. of a category is cartesian closed You wouldn’t do linear algebra in a topos, whenever the original category is weakly cartesian closed. so why try to do topology in one? This means that λ -abstraction or Currying X → E ◮ The logical formulae for the equivalence relations get more exists for any map f : X × Y → Z but need not be unique. and more complicated, even for iterated powers of Σ . Moreover, the category of sober topological spaces is weakly cartesian closed. Can we approach the problem in another way?
A metaphor due to Martín Escardó Developing the metaphor complex numbers have negatives and square roots real numbers have negatives positive numbers have square roots equilogical spaces have equalisers and exponentials The category of equilogical spaces topological spaces have equalisers is like the field of complex numbers: algebraic lattices have exponentials all problems (equations, exponentials) can be solved in it. The construction R [ i ] gives an algebraically closed field Ordinary topological spaces play the role of the real numbers. in essentially only the case R ≡ R : If R ⊂ C is a proper finite extension of fields It’s convenient to consider third, smaller, systems too, with C algebraically closed then say algebraic lattices and positive real numbers. R must be a real closed field (Artin–Schreier theorem). C gets all the credit for being algebraically closed, but R has done the heavy lifting of solving algebraic equations — all those of odd degree — √ leaving just − 1 to be added. What are the special properties of sober spaces? Nice cartesian closed extensions Equideductive categories This property with equalisers and exponentials can be stated within the smaller category: When does a category lie nicely in its cartesian closed ✲ ✲ extension? E . . . . . e . i . . ✻ . . . . . . . . ✲ We are adding equalisers and exponentials of Σ . . . . . a . . ✲ Γ A Mixing them gives diagrams of the form ✻ ✻ π 0 f ✲ ✲ π 0 E × Y ✲ ✲ ✲ X e × Y . ✲ Σ Y . . . E . . i × Y . . . . . . ✲ . α ✲ . ❄ . . . g . a × Y ✲ Γ × Y A × Y ✲ Σ β Evaluating the diagram in equilogical spaces, Then we would like to write if X is a sober topological space then so is E . E ≡ { x ∈ A | ∀ y . α xy = β xy } Reinhold Heckmann’s cartesian closed extension of the category of locales does not have this property: and when Y is another object of this form, there are more sub-equi-locales than sub-locales. � � x ∈ A | ∀ y ∈ B . ( ∀ z . γ yz = δ yz ) ⇒ α xy = β xy .
Equideductive logic The Chu construction The metaphor of the complex numbers suggests pairs of objects. The use of ∀ and ⇒ (together) suggested by the diagram An idea of Michael Barr and his student Po-Hsaing Chu. does satisfy the proof-theoretic rules that we would expect, Given a symmetric monoidal closed category V Γ , x : A , p ( x ) ⊢ α x = β x and a chosen object Σ ∈ V , we define: = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = A Chu object consists of two objects X , A ∈ V with a map Γ ⊢ ∀ x : A . p ( x ) = ⇒ α x = β x σ : X ⊗ A → Σ . so long as every variable that occurs freely on the left of ⇒ is A morphism ( f , g ) : ( X , A , σ ) → ( Y , B , τ ) consists of bound by ∀ . f : X → Y and g : B → A (NB reverse!) in V such that The formulae for (partial) equivalence relations in equilogical X ⊗ B f ⊗ B ✲ Y ⊗ B spaces satisfy this restriction. We also need a (restricted) λ -calculus for the terms (of type Σ ) X ⊗ g τ in the equations. ❄ ❄ σ ✲ Σ X ⊗ A This is set out in Equideductive Categories and their Logic . commutes. Any equideductive category is weakly cartesian closed in the Switching the objects defines a contravariant involution sense of Rosický. (square root of the identity) on the Chu category. Chu with Stone duality Square root of the monad? Now let ( T , η, µ, ζ ) be a strong monad with initial algebra ( Σ , κ ). An ambispace consists of an object X ∈ R , a T -algebra ( A , α ) and a map σ : X × A → Σ that is an X -indexed homomorphism from We have A to Σ . $$( X , A , α, σ ) = ( TX , TA , µ A , $$ σ ) A morphism ( f , F ) : ( X , A , α, σ ) → ( Y , B , β, τ ) consists of a map f : X → Y in R and a homomorphism F : B → A making but to justify saying that this is (the square root of) the given monad T we need to define an embedding R ֒ → C . X × B f × B ✲ Y × B To make this (and later constructions) work for an equideductive category R we rely on its injectives. X × F τ For example, any sober topological space X is a subspace of ❄ ❄ some algebraic lattice X 0 . σ ✲ Σ X × A The embedding should take X to ( X , Σ X 0 , µ X 0 , ev ). commute. Instead of just switching the objects, we apply the monad: $( X , A , α, σ ) ≡ ( A , TX , µ X , $ σ ) , where $ σ ≡ ζ A , X ; T σ ; κ .
The emdedding The duality again Our monad is the extension of Σ Σ ( − ) and the embedding should The contravariant endofunctor $ is self-adjoint: take X to the real ambispace ( X , Σ X 0 , µ X 0 , ev ). there is a bijection between morphisms To make this a well defined (full and faithful) functor, we need to modify the morphisms between ambispaces: ( X , A , α, σ ) → $( Y , B , β, τ ) and ( Y , B , β, τ ) → $( X , A , α, σ ) Pairs ( f , F ) and ( g , G ) are equivalent (define the same because both sides correspond to pairs of maps morphism) if they have the same diagonal: f × B f : Y → A and g : X → B ✲ X × B ✲ Y × B . . . . . . . . . . g × B . (NB we’ve mixed objects with algebras!) such that . . . . . . . . . . . . . X × F X × G . τ . . . . . . . X × f ✲ X × A . . . . . . . ❄ ❄ . ❄ . ✲ . . . X × Y . σ ✲ Σ X × A Doing the same thing in the original Chu construction is g × Y σ equivalent to restricting to biextensional Chu objects, i.e. those for which both transposes of σ are monos: ❄ ❄ τ ✲ Σ B × Y X ֒ → ( A ⊸ Σ ) and A ֒ → ( X ⊸ Σ ) . commutes. The Chu tensor product Chu tensor with weak exponentials What makes the Chu construction interesting is not the duality alone but that we can define a new tensor and internal hom. In a symmetric monoidal closed category V with pullbacks, we may form We could repeat the same construction ✲ X ⊸ B for the monoidal Chu category, P replacing internal homs by exponentials, if the basic category were cartesian closed with pullbacks. X ⊸ ˜ τ ❄ ❄ But an equideductive category is only weakly cartesian closed. σ ✲ ( X ⊗ Y ) ⊸ Σ Y ⊸ ˜ Y ⊸ A Is this good enough? Then the morphisms We rely on the plentiful supply of injectives, but since these are not assigned functorially, ( X , A ) → $( Y , B ) and ( Y , B ) → $( X , A ) it gets rather messy in categorical notation. also correspond to [ I is the tensor unit] ( X , A ) ⊗ ( Y , B ) ≡ ( X ⊗ Y , P ) −→ Σ ≡ ( Σ , I ) . This defines the tensor product of the Chu category, which is then symmetric monoidal.
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