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Scott is Natural between Frames. Christopher Townsend , Open University UK. Reporting on joint work with Vickers, Birmingham University, UK Approx. 20 minutes. Overview. Certain locale maps. Today’s Talk. =. Scott continuous maps between frames. Natural Transformations. =.
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Scott is Natural between Frames Christopher Townsend, Open University UK. Reporting on joint work with Vickers, Birmingham University, UK Approx. 20 minutes.
Overview Certain locale maps Today’s Talk = Scott continuous maps between frames Natural Transformations = Lattice Theoretic Categorical Why? Categorical logic approach to proper maps in topology.
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.
Relationship between Scott continuous and Frames • Proper map in topology. If f : XY is proper then • there exists Scott continuous f* :Opens(X) Opens(Y) with f* right adjoint to f-1 and f* (U\/f-1V)=f* U\/V • for U, V open in X,Y resp. • Converse true provided Y is TD (i.e. every point is open in its closure). Broadly speaking: Topological notion of properness definable using interaction of Scott continuous maps and frames.
Must define functor for every frame ΩX: Generalising ΩY+ΩX can be constructed as a dcpo, and so any Scott continuous Natural Transformations ΛΩX: Frm Set ΩY ¦ ΩY+ΩX q: ΩX ΩW + is frame coproduct. • Hence every Ωg: ΩX ΩW gives rise to nat. trans. by gives rise to dcpo maps ΩY+ΩX ΩY+ΩW for every ΩY. 1+Ωg ΩY+ΩW ΩY+ΩX Scott Nat. trans.
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. Case ΩX=idl(L) follows since, given a:ΛΩX ΛΩW IdεScott(idl(L), ΩX)= UL+ ΩX UL+ ΩW=Scott(idl(L),ΩW) aUL 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 q : idl(L) ΩW such that qe1 =qe2 where e1 ,e2 : idl(R) idl(L) is the data for a dcpo presentation of ΩX.
Locale Theoretic Interpretation • We can give an interpretation of ΛΩX using locale theory; it is an exponential. Loc=Frmop $X: Locop Set Y ¦ Loc(YX ,$) ΛΩX: Frm Set ΩY ¦ ΩY+ΩX = $ defined to behave like Sierpiński top. space This is the exponential $X in [Locop,Set]. $ and X embed into [Locop,Set] via Yoneda.
Application: Proper Maps • Locale map f : XY is proper if there exists Scott continuous f* : ΩX ΩY with f* right adjoint to Ωfand f* (a\/ Ωf b)=f*(a)\/b • for aεΩX and bεΩY. • Same as topological definition, providing examples. Equivalent definition: f : XY 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).
Further Applications • Open maps • Points of double power locale: $X exists in [Locop,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.
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.
