Equilogical spaces A Chu-like Extension of Topological Spaces There - - PowerPoint PPT Presentation

A Chu-like Extension of Topological Spaces

Paul Taylor Honorary Research Fellow University of Birmingham Dagstuhl Seminar 15441 Monday 25 October 2015 www.Paul Taylor.EU/ASD/equideductive/ www.Paul Taylor.EU/slides/15-Domains-Cork.pdf Funded by: my late parents.

Equilogical spaces

There are several constructions that embed the traditional category of topological spaces and continuous functions in a cartesian closed category. (All topological spaces will be sober.) The best known was proposed by Dana Scott, 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. A morphism f : (X, ∼X) → (Y, ∼Y) of equilogical spaces is a continuous function f : X → Y that respects the equivalence relations: x1 ∼X x2 =⇒ fx1 ∼Y fx2. The category of equilogical spaces is locally cartesian closed.

Presheaves and exact completions

To be rather more categorical, the process of formally adding quotients of equivalence relations to a category is called its exact completion. Two authors in particular studied this: Giuseppe Rosolini surveyed several cartesian closed extensions

• f topology, showing how they are reflective subcategories
• f categories of presheaves (or functor categories).

Jiˇ ri Rosický showed that the exact completion

• f a category is cartesian closed

whenever the original category is weakly cartesian closed. This means that λ-abstraction or Currying X → E exists for any map f : X × Y → Z but need not be unique. Moreover, the category of sober topological spaces is weakly cartesian closed.

My objections

This is all excellent work, but it is not to my taste.

◮ The equivalence relation defining an equilogical space

need not respect the topological structure in any way. So the points and not the topology are carrying the

• structure. (Unfortunately, we will not be able to replace

topological spaces with locales in this work.)

◮ Toposes, locally cartesian closed categories and exact

completions are tools for set theory — the study of discrete structures, not continuous ones. You wouldn’t do linear algebra in a topos, so why try to do topology in one?

◮ The logical formulae for the equivalence relations get more

and more complicated, even for iterated powers of Σ. Can we approach the problem in another way?

A metaphor due to Martín Escardó

The category of equilogical spaces is like the field of complex numbers: all problems (equations, exponentials) can be solved in it. Ordinary topological spaces play the role of the real numbers. It’s convenient to consider third, smaller, systems too, say algebraic lattices and positive real numbers.

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 topological spaces have equalisers algebraic lattices have exponentials The construction R[i] gives an algebraically closed field in essentially only the case R ≡ R: If R ⊂ C is a proper finite extension of fields with C algebraically closed then 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

• f 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

When does a category lie nicely in its cartesian closed extension? We are adding equalisers and exponentials of Σ. Mixing them gives diagrams of the form E

✲ ✲ X

f

✲

g

✲ ΣY

Evaluating the diagram in equilogical spaces, if X is a sober topological space then so is E. Reinhold Heckmann’s cartesian closed extension

• f the category of locales does not have this property:

there are more sub-equi-locales than sub-locales.

Equideductive categories

This property with equalisers and exponentials can be stated within the smaller category: E Γ a

✲

e . . . . . . . . . . . . . . . . . . . . . .

✲ ✻

A i

✲ ✲

E × Y Γ × Y π0

✻

a × Y

✲

e × Y . . . . . . . . . . . . . . . . . .

✲

A × Y π0

✻

α ✲ β

✲

i × Y

✲ ✲

Σ

❄

Then we would like to write E ≡ {x ∈ A | ∀y. αxy = βxy} and when Y is another object of this form,

• x ∈ A | ∀y ∈ B. (∀z. γyz = δyz) ⇒ αxy = βxy
• .

Equideductive logic

The use of ∀ and ⇒ (together) suggested by the diagram does satisfy the proof-theoretic rules that we would expect, Γ, x : A, p(x ) ⊢ αx = βx = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Γ ⊢ ∀x:A. p(x ) =⇒ αx = βx so long as every variable that occurs freely on the left of ⇒ is bound by ∀. The formulae for (partial) equivalence relations in equilogical spaces satisfy this restriction. We also need a (restricted) λ-calculus for the terms (of type Σ) in the equations. This is set out in Equideductive Categories and their Logic. Any equideductive category is weakly cartesian closed in the sense of Rosický.

The Chu construction

The metaphor of the complex numbers suggests pairs of objects. An idea of Michael Barr and his student Po-Hsaing Chu. Given a symmetric monoidal closed category V and a chosen object Σ ∈ V, we define: A Chu object consists of two objects X, A ∈ V with a map σ : X ⊗ A → Σ. A morphism ( f, g) : (X, A, σ) → (Y, B, τ) consists of f : X → Y and g : B → A (NB reverse!) in V such that X ⊗ B f ⊗ B

✲ Y ⊗ B

X ⊗ A X ⊗ g

❄

σ

✲ Σ

τ

❄

commutes. Switching the objects defines a contravariant involution (square root of the identity) on the Chu category.

Chu with Stone duality

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 A to 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 X × B f × B

✲ Y × B

X × A X × F

❄

σ

✲ Σ

τ

❄

commute. Instead of just switching the objects, we apply the monad: $(X, A, α, σ) ≡ (A, TX, µX, $σ), where $σ ≡ ζA,X ; Tσ ; κ.

Square root of the monad?

We have $$(X, A, α, σ) = (TX, TA, µA, $$σ) but to justify saying that this is (the square root of) the given monad T we need to define an embedding R ֒→ C. To make this (and later constructions) work for an equideductive category R we rely on its injectives. For example, any sober topological space X is a subspace of some algebraic lattice X0. The embedding should take X to (X, ΣX0, µX0, ev).

The emdedding

Our monad is the extension of ΣΣ(−) and the embedding should take X to the real ambispace (X, ΣX0, µX0, ev). To make this a well defined (full and faithful) functor, we need to modify the morphisms between ambispaces: Pairs ( f, F) and (g, G) are equivalent (define the same morphism) if they have the same diagonal: X × B f × B

✲

g × B

✲ Y × B

X × A X × F

❄

X × G

❄

σ

✲ Σ

τ

❄

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

✲

Doing the same thing in the original Chu construction is equivalent to restricting to biextensional Chu objects, i.e. those for which both transposes of σ are monos: X ֒→ (A ⊸ Σ) and A ֒→ (X ⊸ Σ).

The duality again

The contravariant endofunctor $ is self-adjoint: there is a bijection between morphisms (X, A, α, σ) → $(Y, B, β, τ) and (Y, B, β, τ) → $(X, A, α, σ) because both sides correspond to pairs of maps f : Y → A and g : X → B (NB we’ve mixed objects with algebras!) such that X × Y X × f ✲ X × A B × Y g × Y

❄

τ

✲ Σ

σ

❄

commutes.

The Chu tensor product

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 P

✲ X ⊸ B

Y ⊸ A

❄

Y ⊸ ˜ σ ✲ (X ⊗ Y) ⊸ Σ X ⊸ ˜ τ

❄

Then the morphisms (X, A) → $(Y, B) and (Y, B) → $(X, A) 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.

Chu tensor with weak exponentials

We could repeat the same construction for the monoidal Chu category, replacing internal homs by exponentials, if the basic category were cartesian closed with pullbacks. But an equideductive category is only weakly cartesian closed. Is this good enough? We rely on the plentiful supply of injectives, but since these are not assigned functorially, it gets rather messy in categorical notation.

Chu tensor using equideductive logic

Equideductive logic was designed to handle embedding in

• injectives. We assume that the injectives form a cartesian closed

subcategory — with all exponentials YX not just ΣX — so that we may use general λ-calculus. Then

• {X | p}, {A | q}, α, σ
• ⊗
• {Y | r}, {B | s}, β, τ)
• is
• {(x, y) : X × Y | p(x) ∧ r(y)}, P ≡ {( f, g) : AY × BX | w( f, g)}, π, ρ
• where w( f, g) is the conjunction of

∀y. r(y) ⇒ q( fy), ∀x. p(x) ⇒ s(gx) ∀xy. p(x) ∧ r(y) =⇒ σ( fy, x) = τ(gx, y) and describes the pullback of weak exponentials, π : TP → P is the appropriate product structure and ρ

• ( f, g), (x, y)
• ≡ σ(gy, x).

Properties of the tensor product

First we need to check that

◮ ⊗ is a well defined functor of two variables that respects

equivalence,

◮ it satisfies the adjunction with $, ◮ it is associative, obeying the Mac Lane – Kelly equation, ◮ it is commutative, and ◮ it has a unit, (Σ, ΣΣ).

Better than this, the unit is the terminal object, essentially arising from the choice of Σ as the initial T-algebra. Hence the “product” has projections (X, A, α, σ) ←− (X × Y, P, π, ρ) −→ (Y, B, β, τ). So we could call ⊗ an affine [monoidal] structure. Does it also have diagonals, making it the categorical product?

Is ⊗ a categorical product?

The diagonal morphism (d, D) : (X, A, α, σ) → (X × X, P, π, ρ) would have to satisfy d(x) ≡ (x, x) and σ

• D( f, g), x
• = σ( fx, x) = σ(gx, x)

for all ( f, g) ∈ P. What this means is that the λ-term λx. σ( fx, x) ∈ ΣX0 must lie in the image of ˜ σ : A → ΣX0. This does at least happen for real ambispaces, those in the image of the embedding of the equideductive category in the Chu category, for which A ≡ ΣX0. We will come back to the question of which other ambispaces have diagonals at the end of the lecture. We work with ⊗ as if it were ×.

General (pseudo) function spaces

From the defining adjunction for ⊗, we already have $(X, A, α, σ) as the internal hom (X, A, α, σ) ⊸ Σ, but we can also satisfy

• {X | p}, {A | q}, α, σ
• ⊗
• {Y | r}, {B | s}, β, τ
• −→
• {Z | u}, {C | v}, γ, υ
• =

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

• {X | p}, {A | q}, α, σ
• −→
• {W | w}, D, δ, ω
• For this, {W | w} is the weak hom-object
• ( f, F) : ZY0

0 × BC0 0 | w( f, F)

• where w( f, F) says that f : Y0 → Z0 and F : C0 → B0

define a Chu morphism from (Y, B, β, τ) to (Z, C, γ, υ). The algebra D is the copower of C by Y: the universal algebra for which Y ⊗ C → D is an indexed homomorphism and ω

• ( f, F), d)
• is the lifting to it of τ(Fc, y).

Equalisers

The embedding of objects of an equideductive category in injectives and the equivalence relation on morphisms of ambispaces allow us to “cheat” when forming equalisers: (E, A, α, σ)

✲ ✲ (X, A, α, σ)

( f, F) ✲ (g, G)

✲ (Y, B, β, τ).

The algebra part of the equaliser is that of the domain of the pair. If X ≡ {X0 | p} and B ≡ {B0 | s} then E ≡ {X0 | e} where e(x) ≡ p(x) ∧ ∀b. s(b) ⇒ σ(b, fx) = σ(b, gx).

Coequalisers

Similarly, we may “cheat” when forming coequalisers: (X, C, α, σ) ✛

✛

(X, A, α, σ) ✛ ( f, F)

✛

(g, G) (Y, B, β, τ). The spatial part of the coequaliser is that of the codomain of the pair. If A ≡ {A0 | q} and Y ≡ {Y0 | r} then C ≡ {A0 | t} where t(a) ≡ q(a) ∧ ∀y. r(y) ⇒ σ(a, fy) = σ(a, gy).

Factorisation

Using the tensor product, equaliser and coequaliser, we can factorise any morphism into a regular epi and a mono. Consider in particular the morphism

• {X | p}, ΣX, µX, ev
• (id, ˜

σ) ✲ {X | p}, {A | q}, α, σ

• ,

comparing a general ambispace (on the right) with a real one. The kernel (which is also real) is

• {X × X | k}, ΣX×X, µX×X, ev
• π0 ✲

π1

✲

• {X | p}, ΣX, µX, ev
• where

k(x, y) ≡ p(x) ∧ p(y) ∧ ∀a. q(a) ⇒ σ(a, x) = σ(a, y). The coequaliser of this pair defines the factorisation,

• {X | p}, ΣX, µX, ev

✲

✲

{X | p}, {ΣX | qp}, µX, ev

• ✲✲

{X | p}, {A | q}, α, σ

• ,

where qp(φ) ≡ ∀xy. k(x, y) ⇒ φx = φy, which is ∀xy. p(x) ∧ p(y) ∧

• ∀a. q(a) ⇒ σ(a, x) = σ(a, y)
• =⇒ φx = φy.

Modulation

Even though it depends on an arbitrary choice of injectives, the factorisation kernel

✲ ✲ real ✲ ✲ modulated ✲ ✲ general,

defines a functor (modulation), in fact a coreflection. Real objects are projective and every modulated object is a coequaliser of them, so we have rediscovered the exact completion. Modulated objects have diagonals for ⊗, so this is the categorical product for them. Hence the coreflective subcategory is cartesian closed.

Conclusions

The formula for modulation is of the same kind as we obtained for exponentials of equilogical spaces. So we have at least explained these nasty formulae just using limits and colimits in the monadic Chu category. Do more ambispaces have diagonals, forming a larger cartesian closed category, without the need for modulation? Or might this affine closed category be a more convenient setting than a CCC in which to study higher recursion theory? It would provide semantics for a version of PCF in which duplication is forbidden for variables of nested exponential type.