Partial Continuous Functions and Admissible Domain Representations
Partial Continuous Functions and Admissible Domain Representations Fredrik Dahlgren ([email protected]) 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.
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.
