Schemas as Toposes

1. Z schemas – specification 1st order theories – logic geometric theories toposes – as generalized topological spaces Schemas as Toposes Steven Vickers Department of Pure Mathematics Open University

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

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

4. 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

5. can translate geometric logic Schemas as Theories Relational calculus Predicate calculus RR  Rx,y,z. (R(x,y)  R(y,z)  R(x,z)) Generics as parameters Sorts as carriers Higher-order First-order

6. Types in Z Z  integers power set cartesian product 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.

7. 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

8. Group [G] e: G -1: G  G •: GG  G x,y,z: G (true  x•(y•z) = (x•y) •z) x: G (true  x•e = x  e•x = x) x: G (true  x•x-1 = e  x-1•x = e) e.g. groups

9. 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) Types out of logic e.g. forcing Z  X+Y (disjoint union)

10. 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

11. 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  Vnnat x1, …, xn. y = {x1}  …{xn})  x1, …, xm, x'1, …, x'n: X. ({x1}  …{xm} = {x'1}  …{x'n}  1imV1jn xi = x'j  1jnV1im x'j = xi) Using infinite disjunctions e.g. forcing Y  F(X) (finite power set)

12. 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 • Vnnat x1, …, xn. (S = {x1}  …{xn}  1in (xi))

13. 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) L(q): q < x R(q): x < q Topology is intrinsic: each proposition is an open set

14. 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

15. 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)

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

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

18. Old examples • polynomial functions p: R R are automatically continuous • p(x) = anxn + … + a1x + a0 • 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.

19. 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.)

20. 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)

21. topos geometric morphism 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]

22. (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()  F2()  F3()  … – take colimit

23. Mathematical payofffrom 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

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

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

26. 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