Axioms for a category of spaces

1. Axioms for a category of spaces Dr Christopher Townsend (Open University)

2. Main Idea • THESIS: Just as axioms exist for the category of Sets, axioms can also be found for the category Top of topological spaces. • Categorical axioms give the structure of the class of all spaces and so are ‘higher order’ • These axioms will model the category of LOCALES (constructive topological spaces) • Such axiomatizations are powerful unifying tools, e.g. toposes

3. Outline Objectives • Define Category, Functor, Natural Transformation. • Describe the axiomatization of regular categories • Discuss the difficulties with axiomatizing spaces • Introduce locales as the correct category for constructive topology • Give the axiomatization of spaces • Describe compact Hausdorff and discrete spaces • Discuss proper/open duality

4. Basic Definitions • A category is a set of objects and morphisms between objects. E.g. the category of Sets (with functions), topological spaces (with cts functions.) • For any category C there is a category Cop it has morphisms in the opposite direction. • For any category C there is a functor category [Cop ,Set]. Its objects are functors, F • And morphisms natural transformations, a:F->G, which are indexed maps • aA :FA->GA • for every object A satisfying a naturality condition. F Set Cop A F(A) f F(f) B F(B)

5. Regular Categories • Regular categories are those good enough for basic subset manipulations. • Formally defined as those categories such that • (a) pullbacks exist • (b) equalizers exist • (c) pullback stable image factorizations exists. AxBC C A B f E B C A B • E.g. category of sets and of compact Hausdorff spaces is regular. • Posets definable in any reg. cat. A Im(f) B

6. Order Enriched Categories • A category is order enriched if for any objects A,B the collection of morphisms from A to B is a poset (and function composition preserves order). • E.g. the category of posets is order enriched since the set of monotone functions is itself a poset. In any category with products you can define distributive lattices. These are objects X with maps /\:XxX->X, \/:XxX->X satisfying the usual axioms. In the order-enriched case these maps MUST agree with background order.

7. What’s the Problem? Q: What’s so difficult about axiomatizing a category that looks like topological spaces? (After all, you’ve said you can axiomatize Set.) A: It must be close enough to set theory so as to contain set-like parts (the regular subcategories of compact Hausdorff and discrete spaces), but is NOT a set theory in structure (principally there are no function spaces). It is somewhere in between.

8. Locales as the category Top • The axioms to follow cover the category of locales. This is a category which is well known to be good enough for topological theory. E.g. Stone-Cech compactification, Stone duality, Priestley duality, Pontryagin duality etc. • The category of Locales is defined as the dual of the category of frames (complete Heyting algebras). • Frame homomorphism are particular directed join preserving maps between frames. • The categorical structure of locales is slice stable. I.e. anything true of the category Loc is also true of the slice Loc/Y for any locale Y. (This is not true of topological spaces.) JT84 g X1 X2 The objects of C/Y are maps f:X->Y and the morphisms are arrows g such that: - f2 f1 Y

9. Axioms I: The Sierpinski Space Some of the less central axioms omitted. E.g. pullbacks exists, coequalizers exists. ALSO: The category of spaces is axiomatized to be ORDER ENRICHED The Sierpinski space (classically) is the 2 point topological space such that the singleton of one of the points is open. E.g. $={0,1} Opens of $={ Empty, $, {1}}.Points are TRUTH VALUES. Axiom I: There exists an internal distributive lattice $ such that for every space X the pullbacks: a 1 ¬a 1 …uniquely define every open (resp. closed) subspace of X 1 0 a a X $ X $

10. Application of Sierpinski Axiom I • The existence of $ makes the following very simple set-theoretic fact true of spaces: - THEOREM: If i, j are subsets of {*}=1, then (i subset of j) if and only if (* in i ) implies (* in j). Proof: (classically) the subsets of {*} is the poset {0,1} • Recall that $={0,1} classically, and so the theorem gives us a ‘trick’ for arguing when two ‘truth values’ are less than or equal to each other. • Of course, Loc has a Sierpinski space.

11. Power Spaces background • Just as set theory is axiomatized via power sets, spaces are to be axiomatized via a power locale. • The correct notion of power locale for spaces is the double power locale, P. It is the combination of two more well known power locales, and its points can be visualised as sets of subsets of points. • Let C be a category, then recall the category [Cop ,Set]. For any X, there is a functor C(_xX,$): Cop ->Set, which takes any Y to the set of morphisms C(YxX,$). It is denoted $X (it is the exponential in [Cop ,Set] ) • Recent result (with Vickers) shows that Loc(X, PY)= Nat[$Y, $X] where Nat[_] is the collection of natural transformations... hence the double power space axiom...

12. Double Power Space, Axiom II • Since Loc(X, PY)=Nat[$Y, $X] for locales, we have Axiom II: For any space Y there exists another space PY such that (for all X), C(X, PY)=Nat[$Y, $X] • In contrast to set theory (the points of the power set are subsets), the points of the double power space are natural transformations. • The axiom forces PY=$^($Y) (double exponentiation) since: • C(X, PY)=Nat[$Y, $X]=Nat[Xx$Y, $]=Nat[X, $^($Y)]

13. Coverage Axiom III • (Called ‘coverage’ since it mimics a technical result in locale theory; Johnstone’s coverage theorem). Axiom III (coverage): If e:E->X is an equalizer in C then $e :$X -> $E is (like a) coequalizer in [Cop ,Set] . Compare to set theory: If i:B>->A a subset inclusion then i-1:PA->PB is a surjection (i.e. coequalizer). • This axiom is more technical. (And we hope to remove it, following experience from the monadicity result in topos theory.)

14. Applications: Open maps. • Recall from topology that a map f:X->Y is open iff the direct image of an open subset is open. • For a category of spaces, f:X->Y is open iff there exists a map f #:$X -> $Y in [Cop ,Set] , which satisfies a Frobenius condition (plus others omitted). • This is the correct notion for C=Loc. • It can be shown that a space is discrete iff X>->XxX and !:X->1 are open maps. Therefore we can define discrete spaces in this abstract context: X is discrete iff X>->XxX and !:X->1 are open maps. Theorem: The category of discrete spaces is regular. (In other words good enough for basic theory of subsets, e.g. posets etc)

15. Applications: Proper maps. • Recall from topology that a map f:X->Y is proper iff the direct image of a closed subset is closed and every fibre is compact. • For a category of spaces, f:X->Y is proper iff there exists a map f #:$X -> $Y in [Cop ,Set] , which satisfies a coFrobenius condition (plus others omitted). • This is the correct notion for C=Loc. • It can be shown (Vermeulen) that a space is compact Hausdorff iff X>->XxX and !:X->1 are proper maps. Therefore we can define compact Hausdorff spaces in this abstract context: X is compact Hausdorff iff X>->XxX and !:X->1 are proper maps. Theorem: The category of compact Haus. spaces is regular. (In other words good enough for basic theory of subsets, e.g. posets etc)

16. Proper/Open Duality • The theories of proper and open maps are dual. Recall that C is order enriched, simply turn the order around and everything compact Hausdorff is discrete etc. (E.g. $ is a distributive lattice and so its opposite is again a distributive lattice.) • The theory of ‘sets’ and the theory of ‘compact Hausdorff spaces’ therefore have equal status in this setting. • This does not mean that all facts about set theory are true of compact Hausdorff spaces; you must restrict yourself to working within that part of set theory determined by the axioms for spaces.

17. Summary • It is possible to axiomatize classes of mathematical investigation categorically. E.g. set theory via toposes, or regular categories for a basic theory with subsets/relational composition. • This work attempts to do the same for spaces. • The key appears to be axiomatizing the notion of power space. • It has been observed that the points of the (double) power space are natural transformations, i.e. morphisms of [Locop ,Set], and this is then taken as an axiom. • A Sierpinski space is introduced which classifies open and closed subspaces and so allows basic results about truth values to go through. • Proper and open maps can be defined leading to regular categories of discrete and compact Hausdorff spaces. • These two theories are identical under open/proper duality.