Download Partial Continuous Functions and Admissible Domain Representations

Partial Continuous Functions and Admissible
Domain Representations
Fredrik Dahlgren (fredrik.dahlgren@math.uu.se)
Department of mathematics at Uppsala university
CiE 2006, 30 June – 5 July
Representing topological spaces
A domain representation D of a space X is a triple (D, DR , δ ) where
Representing topological spaces
A domain representation D of a space X is a triple (D, DR , δ ) where
X
Representing topological spaces
A domain representation D of a space X is a triple (D, DR , δ ) where
X
• D is a domain.
D
Representing topological spaces
A domain representation D of a space X is a triple (D, DR , δ ) where
X
• D is a domain.
• DR is a subset of D.
D
Representing topological spaces
A domain representation D of a space X is a triple (D, DR , δ ) where
X
• D is a domain.
• DR is a subset of D.
• δ : DR → X is continuous and
onto.
δ
D
Representing continuous functions
Suppose that we have domain representations of two topological
spaces X and Y.
Representing continuous functions
Suppose that we have domain representations of two topological
spaces X and Y.
X
Y
δ
D
ε
E
Representing continuous functions
Suppose that we have domain representations of two topological
spaces X and Y.
X
Y
δ
D
ε
E
Which continuous functions from X to Y lift to continuous functions
on the domain representations?
Representing continuous functions
Suppose that we have domain representations of two topological
spaces X and Y.
f
X
δ
D
Y
ε
E
Which continuous functions from X to Y lift to continuous functions
on the domain representations?
Representing continuous functions
Suppose that we have domain representations of two topological
spaces X and Y.
f
X
Y
ε
δ
D
E
fˉ
Which continuous functions from X to Y lift to continuous functions
on the domain representations?
Admissible domain representations
A domain representation E of Y is admissible if
Y
ε
E
Admissible domain representations
A domain representation E of Y is admissible if for each domain D,
and each continuous map f : DR → Y where DR is dense in D,
Y
f
ε
D
E
Admissible domain representations
A domain representation E of Y is admissible if for each domain D,
and each continuous map f : DR → Y where DR is dense in D,
Y
f
ε
D
E
fˉ
then f factors through ε.
Admissible domain representations are interesting for the following
reason:
Theorem
If D is a dense domain representation of X and E is an admissible
domain representation of Y, then every sequentially continuous
function f : X → Y lifts to a continuous function f : D → E.
f
X
Y
ε
δ
D
E
fˉ
Not every domain representation is dense
What goes wrong if DR is not dense in D?
Y
f
ε
D
E
Not every domain representation is dense
What goes wrong if DR is not dense in D?
Y
f
ε
D
E
fˉ
If f : X → Y is continuous then f lifts to the closure of DR .
Not every domain representation is dense
What goes wrong if DR is not dense in D?
Y
f
ε
D
E
fˉ
If f : X → Y is continuous then f lifts to the closure of DR .
Thus, an alternative is to view f as a partial continuous function from
D to E.
Partial continuous functions
A partial continuous function from D to E is a pair (S, f ) where
Partial continuous functions
A partial continuous function from D to E is a pair (S, f ) where
• S ⊆ D is closed.
Partial continuous functions
A partial continuous function from D to E is a pair (S, f ) where
• S ⊆ D is closed.
• f : S → E is continuous.
We may now show
Theorem
E is admissible ⇐⇒ for each domain D and each continuous map
f : DR → Y where DR ⊆ D,
Y
f
ε
D
E
We may now show
Theorem
E is admissible ⇐⇒ for each domain D and each continuous map
f : DR → Y where DR ⊆ D,
Y
f
ε
D
E
fˉ
f factors through ε via some partial continuous function f .
This suggests representing maps from X to Y by partial continuous
functions from D to E:
This suggests representing maps from X to Y by partial continuous
functions from D to E:
We say that f : D * E represents f : X → Y if the diagram
This suggests representing maps from X to Y by partial continuous
functions from D to E:
We say that f : D * E represents f : X → Y if the diagram
f
X
ε
δ
D
E
fˉ
commutes.
Y
If E is admissible, then
Theorem
Every sequentially continuous function from X to Y lifts to a partial
continuous function from D to E.
If E is admissible, then
Theorem
Every sequentially continuous function from X to Y lifts to a partial
continuous function from D to E.
If both D and E are admissible then
Theorem
f : X → Y lifts to a continuous function from D to E if and only if f is
sequentially continuous.
The domain of partial continuous functions
We let
• [D * E] = the set of partial continuous functions from D to E
The domain of partial continuous functions
We let
• [D * E] = the set of partial continuous functions from D to E
ordered by
• f v g ⇐⇒ dom(f ) ⊆ dom(g) and f (x) v g(x) for all
x ∈ dom(f ).
The domain of partial continuous functions
We let
• [D * E] = the set of partial continuous functions from D to E
ordered by
• f v g ⇐⇒ dom(f ) ⊆ dom(g) and f (x) v g(x) for all
x ∈ dom(f ).
Theorem
[D * E] is an domain and [D * E] is effective if D and E are effective.
Representing the space of sequentially
continuous functions
Suppose E is admissible. We let
• [X →ω Y] = the space of sequentially continuous functions from
X to Y.
Representing the space of sequentially
continuous functions
Suppose E is admissible. We let
• [X →ω Y] = the space of sequentially continuous functions from
X to Y.
To construct a domain representation of [X →ω Y] over [D * E], let
Representing the space of sequentially
continuous functions
Suppose E is admissible. We let
• [X →ω Y] = the space of sequentially continuous functions from
X to Y.
To construct a domain representation of [X →ω Y] over [D * E], let
• [D * E]R = {partial continuous functions which represent
sequentially continuous maps from X to Y}.
Representing the space of sequentially
continuous functions
Suppose E is admissible. We let
• [X →ω Y] = the space of sequentially continuous functions from
X to Y.
To construct a domain representation of [X →ω Y] over [D * E], let
• [D * E]R = {partial continuous functions which represent
sequentially continuous maps from X to Y}.
We define a map [δ * ε] : [D * E]R → [X →ω Y] by
Representing the space of sequentially
continuous functions
Suppose E is admissible. We let
• [X →ω Y] = the space of sequentially continuous functions from
X to Y.
To construct a domain representation of [X →ω Y] over [D * E], let
• [D * E]R = {partial continuous functions which represent
sequentially continuous maps from X to Y}.
We define a map [δ * ε] : [D * E]R → [X →ω Y] by
• [δ * ε](f ) = f ⇐⇒ f represents f .
Theorem
[D * E] is a domain representation of [X →ω Y].
Theorem
[D * E] is a domain representation of [X →ω Y].
Moreover,
[D * E] is effective/admissible if D and E are effective/admissible.
If we let ADM be the category with objects admissible domain
representations
X
δ
D
If we let ADM be the category with objects admissible domain
representations
f
X
ε
δ
D
E
fˉ
and morphisms representable maps, then
Theorem
ADM is Cartesian closed.
Y
Effectivity and Cartesian closure
A partial continuous function f : D * E is effective if we can
enumerate the set of compact approximations to f (a) for each
compact a ∈ dom(f ).
Effectivity and Cartesian closure
A partial continuous function f : D * E is effective if we can
enumerate the set of compact approximations to f (a) for each
compact a ∈ dom(f ).
We let EADM be the effective counterpart of ADM.
Effectivity and Cartesian closure
A partial continuous function f : D * E is effective if we can
enumerate the set of compact approximations to f (a) for each
compact a ∈ dom(f ).
We let EADM be the effective counterpart of ADM.
All the constructions on ADM preserve effectivity, except for
currying.
Effectivity and Cartesian closure
A partial continuous function f : D * E is effective if we can
enumerate the set of compact approximations to f (a) for each
compact a ∈ dom(f ).
We let EADM be the effective counterpart of ADM.
All the constructions on ADM preserve effectivity, except for
currying.
The map curry : [D × E * F] * [D * [E * F]] is not effective in
general.
The partial continuous function curry from [D × E * F] to
[D * [E * F]] is effective in many interesting cases:
The partial continuous function curry from [D × E * F] to
[D * [E * F]] is effective in many interesting cases:
Theorem
curry is effective if the relation “a ∈ the closure of ER ” is
semidecidable for compact a ∈ E.
Y
ε
E
?
a∈
Thank you.