1. Scott is Natural between Frames Christopher Townsend, Open University UK. Reporting on joint work with Vickers, Birmingham University, UK Approx. 20 minutes.

2. Overview Certain locale maps = Natural Transformations Scott continuous maps between frames Why? Categorical logic approach to proper maps in topology. = Today’s Talk Lattice TheoreticCategorical

3. Definitions Frame = complete lattice satisfying distributivety law a /\ \/ T= \/ {a /\ t, tεT} Example: Opens(X) any top. space X. Always denote frames ΩX. Frame homomorphism = preserves \/ and /\ Example: f -1 :Opens(Y)  Opens(X) any cts f : X  Y Always denote frames homs Ωf: ΩY  ΩX (more general) q : ΩY  ΩX Scott continuous = preserves directed joins (therefore definable between arb. dcpos). Example: Computer Science.

4. Relationship between Scott continuous and Frames Proper map in topology. If f : X  Y is proper then –there exists Scott continuous f * :Opens(X)  Opens(Y) with f * right adjoint to f -1 and f * (U\/f -1 V)=f * U\/V for U, V open in X,Y resp. Converse true provided Y is T D (i.e. every point is open in its closure). Broadly speaking: Topological notion of properness definable using interaction of Scott continuous maps and frames.

5. Natural Transformations Must define functor for every frame ΩX: Generalising ΩY+ΩX can be constructed as a dcpo, and so any Scott continuous Λ ΩX : Frm  Set ΩY ¦  ΩY+ΩX Hence every Ωg: ΩX  ΩW gives rise to nat. trans. by + is frame coproduct. ΩY+ΩX ΩY+ΩW 1+Ωg q: ΩX  ΩW gives rise to dcpo maps ΩY+ΩX  ΩY+ΩW for every ΩY. Scott  Nat. trans.

6. Scott maps from Nat Trans. UL is frame of upper closed subsets of any poset L. Lemma: for any poset L and frame ΩW Scott(idl(L),ΩW)=UL+ ΩW. Infact: Lemma is natural w.r.t Scott maps idl(R)  idl(L), and so this argument extends to any Scott continuous map since they are maps Case ΩX=idl(L) follows since, given a:Λ ΩX  Λ ΩW IdεScott(idl(L), ΩX)= UL+ ΩX UL+ ΩW=Scott(idl(L),ΩW) a UL q : idl(L)  ΩW such that qe 1 =qe 2 where e 1,e 2 : idl(R)idl(L) is the data for a dcpo presentation of ΩX.

7. Locale Theoretic Interpretation We can give an interpretation of Λ ΩX using locale theory; it is an exponential. Loc=Frm op This is the exponential $ X in [Loc op,Set]. $ and X embed into [Loc op,Set] via Yoneda. $ defined to behave like Sierpiński top. space Λ ΩX : Frm  Set ΩY ¦  ΩY+ΩX $ X : Loc op  Set Y ¦  Loc(Y  X,$) =

8. Application: Proper Maps Locale map f : X  Y is proper if there exists Scott continuous f * : ΩX  ΩY with f * right adjoint to Ωf and f * (a\/ Ωf b)=f * (a)\/b for aεΩX and bεΩY. Same as topological definition, providing examples. Equivalent definition: f : X  Y is proper if there exists natural trans. f * : $ X  $ Y satisfying coFrobenius equation. No set theory! I.e. totally categorical account of a topological notion (properness).

9. Further Applications Open maps Points of double power locale: $ X exists in [Loc op,Set] for every X, but exists in Loc only if X is locally compact. Remarkably, $^($ X ) always exists as a locale. Its construction is related to power domain constructions used in theoretical computer science.

10. Summary Scott continuous maps between frames can be represented by certain natural transformations. This provides an entirely categorical account for a well known class of functions. A categorical axiomatization of the topological notion of properness is available.