Scott is Natural between Frames

Christopher Townsend, Open University UK.

Reporting on joint work with Vickers, Birmingham University, UK

Approx. 20 minutes.

OverviewCertain locale maps

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Natural Transformations

Scott continuous maps between frames

Why? Categorical logic approach to proper maps in topology.

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Today’s Talk

Lattice Theoretic Categorical

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-1 V)=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.

Natural Transformations• Must define functor for

every frame ΩX:• Generalising ΩY+ΩX can

be constructed as a dcpo, and so any Scott continuous ΛΩX: Frm Set

ΩY ¦ ΩY+ΩX

• Hence every Ωg: ΩX ΩW gives rise to nat. trans. by

+ is frame coproduct.

ΩY+ΩX ΩY+ΩW1+Ωg

q: ΩX ΩW

gives rise to dcpo maps

ΩY+ΩX ΩY+ΩW

for every ΩY.

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.

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:ΛΩX ΛΩW

IdεScott(idl(L), ΩX)= UL+ ΩX UL+ ΩW=Scott(idl(L),ΩW) aUL

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

This is the exponential $X in [Locop,Set]. $ and X embed into [Locop,Set] via Yoneda.

$ defined to behave like

Sierpiński top.

space

ΛΩX: Frm Set ΩY ¦ ΩY+ΩX

$X: Locop Set Y ¦ Loc(YX ,$)

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Application: Proper Maps• Locale map f : XY 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 : 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.