Locale pullback via dcpos Dr Christopher Townsend (Open University)

Main Idea THESIS: When changing base it is only really the directed joins that need to be modelled/worried about. All the rest of the finitary data takes care of itself. DIRECTEDStudy locales only, i.e. frames. I.e. the data is finite meets and arbitrary joins. Equivalently: finite meets, finite joins and DIRECTED joins. The definition of geometric morphism (suggests atleast) that the finitary structure is preserved.

Technical Aims Given f:E -> E’, a geometric morphism. Then the direct image functor preserves dcpo structure. I.e. f * :dcpo E -> dcpo E’ is well defined. Further f * :dcpo E -> dcpo E’ has a left adjoint f #. In any topos Dlat(dcpo)=Frames. Where we look at order- internal distributive lattices in the order enriched category dcpo. The left adjoint f # restricts to a functor Dlat(dcpo E’ )-> Dlat(dcpo E ) left adjoint to f * :Fr E -> Fr E’ That is: locale pullback.

What is known already? This trick has been done by Joyal and Tierney already with suplattices: - dcpo E dcpo E’ Fr E sup E sup E’ Fr E’ NEW! Frames as order- internal dlats in dcpo Joyal and Tierney ‘84 f#f# f*f* f*f* f#f# Frames as ring objects in sup f#f# f*f*

Is f * :dcpo E -> dcpo E’ well defined? YES. Use external definition of dcpo. As with sup lattices: - Internal Definition of dcpo External Definition There exists V:IdlA->A left adjoint to A->IdlA Then unravel the adjunction of the geometric morphism with the external definition to prove f * is well defined. This works as fiber directedness is stable under the inverse image. (Known?) For every fiber directed x:I->J, the map x*:Pos E (J,A)-> Pos E (I,A) has a left adjoint (and Beck-Chevalley holds). x fiber directed iff x -1 (j) directed for all j. (I,J posets.)

Defining f # :dcpo E’ -> dcpo E e1e2 f * e1 f * e1 Note: e1 and e2 are suitably geometric and so f * e1 and f * e1 well defined TRICK: Use presentations. For every dcpo A, there exists posets G and R and dcpo maps e1 and e2 such that: Idl(R) e1 e2 Idl(G) A is a coequalizer f#f# Idl( f * R) f * e1 f * e2 Idl( f * G) f# Af# Af# Af# A defines f # A. Dcpo coequalizer well defined? Folklore, or adapt Johnstone & Vickers 91

Frames as Order-Internal DLats DEFINITION: For any order enriched category C (with lax products), an object X is an order-internal meet semilattice iff !:X- >1 and Δ:X>->XxX have right adjoints. … in other words, iff finite complete wrt to the order enrichment Define order-internal distributive lattice in the standard way from this. Then: - THEOREM: Fr=Dlat(dcpo) Proof: A in Dlat(dcpo), then  A a dcpo /\:AxA->A a dcpo hom. A a DLat A finite joins/meets + finite dist. dcpo A directed cocomplete Finite meet distr. over directed joins since AxA=A(x)A in dcpo

Defining f # :Fr E’ -> Fr E (Certainly f * :Fr E -> Fr E’ since the direct image preserves finite products) So to prove that f # :dcpo E’ -> dcpo E restricts to f # :Fr E’ -> Fr E it must be verified that (a) f # (A)x f # (A)= f # (AxA) and (b) f # preserves the order enrichment. PROOF:PROOF: (b) is immediate from construction since f* certainly preserves the order enrichment. (a) follows since product in dcpo is tensor and so is a coequalizer construction preserved by left adjoint.  Detailed construction available

Applications (Locale Pullback) If f:X->Y is locale map then f * :Fr SX -> Fr SY is equivalent to Σ f :Loc/X->Loc/Y (by the [JT84] equivalence Loc SX =Loc/X). A left adjoint to f * is therefore right adjoint to Σ f i.e. pullback. (Triquotient Assignments) If p:Z->Y is a locale map then a triquotient assignment for p is a dcpo map p # :ΩZ-> ΩY, satisfying a mixed Frobenius/coFrobenius condition with Ωp: p # [c/\(d\/Ωp(e)]=(p # c/\e)\/p # (c/\d). Using the dcpo description of pullback it can be shown that maps with triquotient assignment are pullback stable. This implies the well known pullback stability results for both proper and open maps.

Applications: Double Power Locale P (Double Power Locale) If Z is a locale map then the double power locale on Z, denoted PZ is define by P Ω PZ=Fr. PP The points of PZ at stage Y (i.e. the locale maps Y-> PZ) are therefore exactly dcpo maps ΩZ-> ΩY. Using the dcpo description of change of base it can be shown (Townsend/Vickers 03) that dcpo(ΩZ, ΩY)=Nat(Loc(_xZ,$),Loc(_xY,$)) where Loc(_xZ,$):Loc op ->Set is the presheaf, Nat(_) the collection of natural transformations and $ the Sierpiński locale. This gives a universal description of the double power locale.

Further Work Beck-Chevalley for f # :dcpo E’ -> dcpo E ? (Following on from pullback stability of triquotient assignments): Decent for Triquotient Surjections? Topos theoretic version. (With toposes taking the place of locales.) Using pretopos sites to describe filtered cocontinuous maps between toposes.

Summary An external description of dcpos is available showing the dcpo structure is preserved by the direct image of a geometric morphism. Since dcpo presentations are models of geometric theories they are preserved by the left adjoints of geometric morphisms. This defines a left adjoint to the direct image functor on dcpos. Frames are exactly order-internal distributive lattices in the category of dcpos since dcpo tensor is given by set-theoretic product. The left adjoint f # :dcpo E’ -> dcpo E preserves order and tensor and so preserves order internal distributive lattices. It therefore defines a left adjoint to f * :Fr E -> Fr E’ which must be equivalent to locale pullback. Well known pullback stability results (open/proper maps) can be reproved by a single appeal to the pullback stability of maps with triquotient assignment.