Schemas as Toposes

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Schemas as Toposes
Steven Vickers Department of Pure
Mathematics Open University
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Z schemas
e.g.
meaning function sets ? sets X ??
R???X?X? ? R?R ? R transitive relations on
X
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First-order theory
e.g.
sort X binary predicate R(x,y) ?x,y,zX.
(R(x,y) ? R(y,z) ? R(x,z))
Meaning Set of all logical consequences of
axioms amongst well-formed formulae using
vocabulary.
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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
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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
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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 cant do this.
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Geometric logic

• First order, many sorted.
• Two levels of axiom formation
• Formulas built using ?, V, ?, ?, true, false
• Axioms ?xX, yY, (?(x,y,) ? ?(x,y,))

formulas
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e.g. groups
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Types out of logic
e.g. forcing Z ? XY (disjoint union)
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Moral
either can eliminate type constructor XY by
introducing new sort with 1st order structure and
axioms or
can harmlessly extend geometric logic with as
type constructor
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Using infinite disjunctions
e.g. forcing Y ? F(X) (finite power set)
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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))
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Topology e.g. real line
R L, R ? Q true ? ?q Q. L(q) ? ?q' Q. R(q') ?q,
q' Q. (q lt q' ? L(q') ? L(q)) ?q Q. (L(q) ?
?q' Q. (q lt q' ? L(q')) ?q, q' Q. (q gt q' ?
R(q') ? R(q)) ?q Q. (R(q) ? ?q' Q. (q gt q' ?
R(q')) ?q Q. (L(q) ? R(q) ? false) ?q, q' Q. (q
lt q' ? L(q) ? R(q'))
Each model is a real number (Dedekind section)
L(q) q lt x R(q) x lt q
Topology is intrinsic each proposition is an
open set
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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

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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)
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Challenge
Can the mathematics be made less abstruse for the
sake of specificational practice? (And to the
benefit of the mathematics too!)
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Topology-free spacesSynthetic topology
Idea Treat spaces like sets forget
topology For functions Use constraints on mode
of definition to ensure definable ? continuous
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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.

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Newer example

• (Escardo) ?-calculus
• ?-definable functions between topological spaces
  are automatically continuous
• even if some of the function spaces dont
  properly exist!
• Simple proofs of topological results
  (compactness, closedness, )
• express logical essence of proof
• hide topological housekeeping (continuity proofs
  etc.)

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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 (propositio
nal theories) toposes (predicate theories)
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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

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(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. - ?
- SFP2 ? 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
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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

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