Schemas as Toposes Steven Vickers Department of Pure Mathematics Open University Z schemas – specification1st order theories – logic geometric theories toposes – as generalized topological spaces

Z schemas e.g. Trans [X] R: X  X R  R  R schema name generic set declaration predicate meaning: “function”: {sets}  {sets} X  {R  X  X   R  R  R} = {transitive relations on X}

First-order theory e.g. sortX binary predicateR(x,y)  x,y,z:X. (R(x,y)  R(y,z)  R(x,z)) vocabulary axiom Meaning: Set of all logical consequences of axioms amongst well-formed formulae using vocabulary.

Models – of a first-order theory Interpret vocabulary as actual sets, relations, etc. … in such a way that the axioms are all true e.g. for Trans A model of Trans is a pair (X,R) with X a set and R a transitive binary relation on it. Guiding principle: The purpose of a schema or theory is to delineate a class of models

SchemasasTheories Relational calculusPredicate calculus R  R  R  x,y,z. (R(x,y)  R(y,z)  R(x,z)) Generics as parametersSorts as carriers Higher-orderFirst-order can translate geometric logic

Types in Z ZZ integers cartesian product power set Presence of  means can have variables and terms for sets, not just for elements. e.g.  S:  X. … Higher order! 1st order logic can’t do this.

Geometric logic First order, many sorted. Two levels of axiom formation: Formulas: built using , V, , , true, false Axioms:  x:X, y:Y, … (  (x,y,…)   (x,y,…)) formulas

e.g. groups Group [G] e: G -1 : G  G : G  G  G  x,y,z: G (true  x(yz) = (xy) z)  x: G (true  xe = x  ex = x)  x: G (true  xx -1 = e  x -1 x = e)

Types out of logic Th [X, Y, Z] i: X  Z j: Y  Z  x, x': X. (i(x) = i(x')  x = x')  y, y': Y. (j(y) = j(y')  y = y')  z: Z. (true   x: X. z = i(x)   y: Y. z = j(y))  x: X, y: Y. (i(x) = j(y)  false) e.g. forcing Z  X+Y (disjoint union)

Moral either: can eliminate “type constructor” X+Y by introducing new sort with 1st order structure and axioms or: can harmlessly extend geometric logic with + as type constructor

Using infinite disjunctions e.g. forcing Y  F(X) (finite power set) Th2 [X, Y]  : Y {–}: X  Y  : Y  Y  Y  y,y',y": Y. (true  (y  y')  y" = y  (y'  y"))  y,y': Y. (true  y  y' = y'  y)  y: Y. (true  y  = y   y = y)  y: Y. (true  V n  nat  x 1, …, x n. y = {x 1 }  …  {x n })  x 1, …, x m, x' 1, …, x' n : X. ({x 1 }  …  {x m } = {x' 1 }  …  {x' n }   1  i  m V 1  j  n x i = x' j   1  j  n V 1  i  m x' j = x i )

Weak 2nd order Geometric logic has 2nd order capabilities for finite sets. e.g.  S: F(X). (…)– in formulas  S: F(X). (…)– in axioms Also, if S finite and  a formula then  x  S.  (x) definable as a formula V n  nat  x 1, …, x n. (S = {x 1 }  …  {x n }   1  i  n  (x i ))

Topology – e.g. real line R L, R  Q true   q: Q. L(q)   q': Q. R(q')  q, q': Q. (q < q'  L(q')  L(q))  q: Q. (L(q)   q': Q. (q < q'  L(q'))  q, q': Q. (q > q'  R(q')  R(q))  q: Q. (R(q)   q': Q. (q > q'  R(q'))  q: Q. (L(q)  R(q)  false)  q, q': Q. (q < q'  L(q)  R(q')) Each model is a real number (Dedekind section) Topology is intrinsic: each proposition is an open set L(q): q < x R(q): x < q

GeoZ – geometric logic as specification language Take Z-style calculus Modify type system and logic to be geometric Type constructors: , +, equalizers, coequalizers, N, Z, Q, F, free algebras But not:  (power set),  (function set), R Constrains the language … but practical expressive power seems comparable with Z

Geometric logic – summary of features Advantages (simplicity) of 1st order logic … but can emulate higher order features (e.g. weak 2nd order) Natural picture: schema specifies “space of implementations” Good structure on each class of models – categorical, topological Natural to consider maps that are functorial, continuous Full mathematical answer is abstruse! – geometric morphisms between classifying toposes (topos as generalized topological space)

Challenge Can the mathematics be made less abstruse for the sake of specificational practice? (And to the benefit of the mathematics too!)

Topology-free spaces Synthetic topology Idea: Treat spaces like sets – forget topology For functions: Use constraints on mode of definition to ensure definable  continuous

Old examples polynomial functions p: R  R are automatically continuous p(x) = a n x n + … + a 1 x + a 0 denotational semantics of programming languages Given: a functional programming language, and a denotational semantics for it. Each function written in that language denotes a continuous map between two topological spaces, “semantic domains”. Continuity guaranteed by general semantic result.

Newer example (Escardo) -calculus -definable functions between topological spaces are automatically continuous – even if some of the function spaces don’t properly exist! Simple proofs of topological results (compactness, closedness, …) express logical essence of proof hide topological housekeeping (continuity proofs etc.)

Geometric reasoning Describe points of space = models of geometric theory  intrinsic topology Describe function using geometric constructions  automatic continuity Geometrically constructivist mathematics  “topology-free spaces” Logical approach  locales / formal topologies(propositional theories) toposes(predicate theories)

Topical Categories of Domains (Vickers) Apply methods to denotational semantics. [SFP] = “space” of SFP domains Solving recursive domain equations X  F(X): – any continuous map F: [SFP]  [SFP] – has initial algebra X – and its structure map  : F(X)  X is an isomorphism (a fixed point) – copes with problems like F(X) = [X  X] topos geometri c morphis m

(Topical Categories of Domains) Task: define basic domain constructions ( , +, function spaces, power domains, …) geometrically (geometric constructivism). Then e.g. function space construction is a geometric morphism. [-  -]: [SFP] 2  [SFP] Constructive reasoning  geometric morphism  generalized continuity required for fixed points as limits If E any local topos (e.g. [SFP]), F: E  E any geometric morphism, then F has an initial fixed point.   F(  )  F 2 (  )  F 3 (  )  …– take colimit

Mathematical payoff from geometric constructivism Focus on essence of mathematics Ignore topological housekeeping (e.g. continuity proofs) Includes generalization from topology to toposes e.g. free access to fixed point results (e.g. domain equations) [SFP] a presheaf topos – without examining category structure of topos “Spatial” proofs in locale theory

Current work (+ Townsend + Escardo) Make -calculus methods work with locales (propositional geometric theories) Combine with geometric logic e.g.P U (P L (X))  $ ($ X ) upper and lower powerlocales (cf. powerdomains) Definable in terms of geometric theories function spaces – so can use -calculus

Specificational aim Mathematical “logic of continuity” (topology-free spaces)  Formal GeoZ specification language  Can test more fully in application

Conclusions Computer science  has big influence on Pure mathematics The maths it leads to is worth investigating even for its own sake … BUT it retains links with computer science: motivation potential applications