Exercise. [dt-005O]

Liam O'Connor

Show that, in contrast to $\mathcal {F}(X) = X\twoheadrightarrow \mathbb {Z}_\bot$ from Problem [dt-0053], the mapping $\mathcal {G}(X) = \mathbb {Z}_\bot \twoheadrightarrow X$ can be extended to continuous functions giving a covariant endofunctor on $\textbf {Cpo}$ .

Solution.

Liam O'Connor

Given a continuous function $f : A \twoheadrightarrow B$ , we must produce a continuous function $\mathcal {F}(f) : (\mathbb {Z}_\bot \twoheadrightarrow A) \twoheadrightarrow (\mathbb {Z}_\bot \twoheadrightarrow B)$ , defined as follows: $\mathcal {F}(f)(g)(x) = f \circ g(x)$ Composition of continuous functions preserves continuity. The functor laws can be shown straightforwardly.

Exercise. [dt-005O]

Solution.

Problem. Contravariance in the function arrow [dt-0053]

Exercise. [dt-005O]

Solution.

Definition. Retraction pair [dt-0054]

Example. Examples and counterexamples of retraction pairs [dt-0055]

Definition. Composition of retraction pairs [dt-0056]

Corollary. Derived products of retraction pairs [dt-0057]

Definition. The category $\textbf {Cpo}^\textbf {R}$ [dt-0058]

Definition. Information ordering in $\textbf {Cpo}^\textbf {R}$ [dt-0059]

Generalising to $\textbf {Cpo}^\textbf {R}$ [dt-005A]

Definition. $\omega$ -chains of cpos in the category $\mathbf {Cpo}^\textbf {R}$ [dt-005C]

Definition. Endofunctors on $\textbf {Cpo}^\textbf {R}$ [dt-005D]

Definition. Upper bounds in the category $\mathbf {Cpo}^\mathbf {R}$ [dt-005G]

Definition. Colimits in the category $\mathbf {Cpo}^\mathbf {R}$ [dt-005F]

Definition. Cocontinuous endofunctors on $\textbf {Cpo}^\textbf {R}$ [dt-005E]

Extending cpo Mappings to endofunctors on $\textbf {Cpo}^\mathbf {R}$ [dt-005I]

Definition. $\times$ operation for retraction pairs [dt-005J]

Definition. $\twoheadrightarrow$ operation for retraction pairs [dt-005K]

Theorem. Fixed point theorem for endofunctors on $\textbf {Cpo}^\textbf {R}$ [dt-005B]

Construction. The colimit $D^\infty$ [dt-005H]

Proof.


( $y \mapsto b \mapsto z$ but $z \not \sqsubseteq y$ )	( $b \mapsto x \mapsto a$ but $a \neq b$ )