The Relationship between Topology and Logic

1. The Relationship between Topology and Logic Dr Christopher Townsend (Open University)

2. Category, Locale, Topos • Three Objectives • 1) The idea of category as ‘universe of mathematical discourse’ • 2) The category of locales as a context for topological space theory • 3) The idea of a Topos as a special category that is good enough for set theory. • From (3) we will define Geometric Logic

3. Definition: Category 1 • Imagine a graph with composition of edges • Formally a category C consists of sets Ob(C) and M(C), with domain codomain, Id & associative functional composition A A g B A f fg C

4. Category: Examples • Every poset, P, is a category. • Objects = P itself • Morphism = ‘’ • Identity exists since xx • Composition exists by transitivity • The category of Sets, • The category of Abelian Groups, • The category of Topological Spaces • Therefore: a number of ‘Universes of Mathematical Discource’

5. Relating Categories: Functors • Functors are the maps between categories • Picture: F C D F(1)=1, F(fg)=F(f)F(g) A F(A) f F(f) B F(B) • For example: • Monotone map between posets • homology functor from Topological Spaces to Abelian Groups

6. Axioms on Categories • We can argue about arrows to define axioms on categories • For example (right) is exactly the data for ‘FX is the free Abelian Group on the set X’ • Second Example: The dual of any category; just reverse the arrows. i FX X !g f A (FX is an Abelian group & there exists i, such that for all functions f to Abelian group A, there exists unique g making the triangle commute)

7. Locales: A category of Spaces • Topological Space (X, ) • iUi is open for Ui open • UV is open for U and V open • (iUi )V= i (Ui V ) • f:XY continuous thenf takes opens to opens in the opposite direction • Frame is a poset A such that • iai defined for all ai in A • ab is defined for a,b in A • (iUi )V= i (Ui V ) • f:XY is a locale map if and only if it is the formal reverse of a frame homomorphism. I.e. the category of locales is defined as the dual of the category of frames A locale DOES NOT have an an underlying set of points (cf Topological Space)

8. The Point of Locales • A framework within which to reason about spaces: • Compact Locales, Hausdorff Locales, Proper Maps, Open Maps, Stone-Cech Compactification etc • Most Topological Spaces are uniquely determined their frame of opens, e.g. Compact Hausdorff. But really... …Locales are better - Topological Spaces: Xi compact  iXi compact Locales: Xi compact  iXi compact Johnstone 1981 Axioms of Choice For free!

9. Axioms on Arrows: Toposes • We add 3 axioms to the axioms of being a category to get some special categories: (a) Products (actually ‘limits’). Exists 1 with unique !:Z 1 for all Z and for X,Y Z XY 1 2 (b) Exponentials, for all X,Y there exists X with Y X Y  Y ZY X ZX (c) Subobject classifier. Exists  such that for any subobject X0 in X there exists unique f with ! 1 X0 f (a), (b) & (c ) A topos, within which all of set theory works(no AC, no Excl. Middle) X 

10. The Point of Toposes • Toposes are generalised locales: i:LocalesToposes (a functor - gives the topos of sheaves on a space) Therefore 2 mains points to investigating Toposes: • (1)The category of toposes behaves like the category of locales/spaces, e.g. separation axioms, proper/open maps, exponentiability results etc • (2) Full force of set theory is available even if the underlying objects are not sets (e.g. sheaves on a topological space).

11. Types of Logic The natural maps between toposes (called geometric morphisms, extending the notion of locale map/continuous function) give rise to a new geometric logic: Classical Intuitionistic Geometric Power Set Excluded Middle Axiom of Choice Set of Finite Subset Finite Products

12. Standard Defn Requires Choice to prove things Formally reversed arrow Prove things without choice and excluded middle requires the algebra of posets of opens Three Incarnations of the Continuous Function Topological Space Locale Map Geometric Transformation f:XY f:XY f:XY • Geometric Translation of points • Very restricted logic • Retain spatial intuitions about points • Extends to topos theory

13. Achievements so far... • Geometric Modelling of Locales • The poset algebras (frames) can be express as multi-sorted theories which are the geometric parts of the frame. • The power locales (spaces whose points are subsets of a domain space) are geometric constructions on these theories. • Localic Locales • This is the theory of topological semi-lattices re-written in a localic context. • A remarkable symmetry exists between meet () semilattices and join () semilattices that does not appear to be visable in topological space theory.

14. Aims & Objectives • To make more precise the relationship between geometric translation/construction and continuous map, • to understand how/why certain ‘geometric’ constructions on locales arise with non-geometric spatial analogies. (Ideal completion.) • to extend work on localic locales. • And finally… to find the topos of locales. • Just as for a locale X, there is X SX (a topos) we would like Locales S(Locales) a topos. Then: Topological Space theory is just a fragment of set theory.

15. Summary • A category is a graph with composition of edges. Most of mathematics is done in a category (e.g. category of Abelian Groups) • Categorical Axioms can be written as rules on arrows, e.g. you can take the dual of any category by reversing the arrows • There is a category of locales which is the dual of a category of posets with structure. This category is a good model for topological spaces. • In fact, locales are a better model since they requires less axioms of our mathematics • Toposes: By adding a few extra axioms on arrows (products, exponentials, subobject classifier) we isolate the class of toposes. • All of set theory can be done in any topos • Toposes generalise locales (and therefore spaces) and the maps between toposes, called geometric morphisms, generalise continuous functions. • The fragment of logical constructions preserved by geometric morphisms defines geometric logic. • It is very restricted since it does not have a power set. • Locale maps (and therefore continuous functions) are exactly those constructed out of geometric manipulations.