Canonical Minimal Source Completion: The Coarsest Readout Extension on which a Nominated Family of Source Relations Becomes Well Defined

Completion determined by a family of target answers

Jeremy RodgersSection 9 of 16Original PDF ↗Publication DOI ↗

Completion determined by a family of target answers

Paper 2 [2] works with a single target $Q$ and its correct-answer map $a\colon S\to A_Q$ , and constructs the coarsest extension of the readout that determines that answer; we will denote that single-target object by $E_Q$ , following Paper 2. We now treat a family of targets, and we choose notation that cannot collide with the single-target object.

Let $\mathcal Q$ be a set of targets, each equipped with a correct-answer map $a_Q\colon S\longrightarrow A_Q,\qquad Q\in\mathcal Q ,\tag{8.1}$ and define $j_{\mathcal Q}\colon S\to T\times\prod_{Q\in\mathcal Q}A_Q, \qquad j_{\mathcal Q}(s)=\Bigl(p(s),\ \bigl(a_Q(s)\bigr)_{Q\in\mathcal Q}\Bigr), \qquad E_{\mathcal Q}:=\mathop{\mathrm{Im}}\bigl(j_{\mathcal Q}\bigr),\tag{8.2}$ with induced surjection $e_{\mathcal Q}\colon S\twoheadrightarrow E_{\mathcal Q}$ , projection $u_{\mathcal Q}\colon E_{\mathcal Q}\to T$ , and structure maps $\sigma_Q\colon E_{\mathcal Q}\to A_Q$ . Write $\sim_{\mathcal Q}$ for the induced equivalence relation: $s\sim_{\mathcal Q}s'$ iff $p(s)=p(s')$ and $a_Q(s)=a_Q(s')$ for every $Q\in\mathcal Q$ .

Proposition 8.1 (Target-family completion).

The data $\bigl(E_{\mathcal Q},e_{\mathcal Q},u_{\mathcal Q},(\sigma_Q)_{Q\in\mathcal Q}\bigr)$ is the terminal object of the category $\mathbf{Suff}\bigl(p;\{a_Q\}_{Q\in\mathcal Q}\bigr)$ of surjective extensions of the readout on which every answer map $a_Q$ becomes well defined. In particular $E_{\mathcal Q}\cong S/\!\sim_{\mathcal Q}$ , and $E_{\mathcal Q}$ is the coarsest extension of the readout determining every answer in the family.

Proof. The answer maps $a_Q\colon S\to A_Q$ form a set-indexed family of functions on $S$ , exactly as the $r_\alpha$ do. Definition 3.1, Definition 4.1, Definition 4.2 and Proposition 3.2, Proposition 3.4, Theorem 5.1, Corollary 5.3 were stated and proved for an arbitrary such family and use no property of the $r_\alpha$ beyond their being functions on $S$ . Substituting $\mathcal Q$ for $A$ , $a_Q$ for $r_\alpha$ , and $A_Q$ for $V_\alpha$ throughout yields the assertions verbatim. ◻

Mediation and the canonical surjection

Now suppose the nominated relation family is sufficient for the targets, in the precise sense that each answer is computable from the readout together with the relation values.

Definition 8.2 (Mediation).

The family $\{r_\alpha\}_{\alpha\in A}$ mediates $\mathcal Q$ if for every $Q\in\mathcal Q$ there is a map $\Phi_Q\colon T\times V_A\longrightarrow A_Q \qquad\text{with}\qquad a_Q(s)=\Phi_Q\bigl(p(s),\ R_A(s)\bigr)\quad\text{for all }s\in S .\tag{8.3}$

Theorem 8.3 (Relation completion versus target completion).

Suppose $\{r_\alpha\}_{\alpha\in A}$ mediates $\mathcal Q$ . Then there is a unique map $\chi\colon E_A\longrightarrow E_{\mathcal Q} \qquad\text{with}\qquad e_{\mathcal Q}=\chi\circ e_A ,\tag{8.4}$ and $\chi$ is surjective; explicitly, $\chi(x)=\bigl(u_A(x),(\Phi_Q(x))_{Q\in\mathcal Q}\bigr)$ for $x\in E_A\subseteq T\times V_A$ . Moreover, $\chi$ is a bijection if and only if $\sim_{\mathcal Q}\ =\ \sim_A ,\tag{8.5}$ that is, if and only if equality of the readout together with all target answers induces exactly the same equivalence relation on $S$ as equality of the readout together with all selected relation values. In general one has only $\sim_A\ \subseteq\ \sim_{\mathcal Q} ,\tag{8.6}$ so the target completion may be strictly coarser than the relation completion.

Proof. $E_A$ is sufficient for $\mathcal Q$ . Define $\sigma_Q^{A}:=\Phi_Q|_{E_A}\colon E_A\to A_Q$ (the restriction of $\Phi_Q$ to $E_A\subseteq T\times V_A$ ). For $s\in S$ , $\sigma_Q^{A}\bigl(e_A(s)\bigr)=\Phi_Q\bigl(p(s),R_A(s)\bigr)=a_Q(s)$ by (8.3). Together with the identity $p=u_A\circ e_A$ and the surjectivity of the map $e_A$ , this exhibits the tuple $\bigl(E_A,e_A,u_A,(\sigma_Q^A)_{Q\in\mathcal Q}\bigr)$ as an object of the category $\mathbf{Suff}(p;\{a_Q\})$ .

Existence, uniqueness, surjectivity of $\chi$ . By Proposition 8.1, $E_{\mathcal Q}$ is terminal in that category, so Theorem 5.1 supplies a unique morphism $\chi\colon E_A\to E_{\mathcal Q}$ , necessarily satisfying $e_{\mathcal Q}=\chi\circ e_A$ and necessarily surjective; and by the explicit formula (5.3) it is given on $x\in E_A$ by $\chi(x)=\bigl(u_A(x),(\sigma_Q^A(x))_{Q\in\mathcal Q}\bigr) =\bigl(u_A(x),(\Phi_Q(x))_{Q\in\mathcal Q}\bigr)$ .

The ordering (8.6). Suppose $s\sim_A s'$ . Then $p(s)=p(s')$ and $R_A(s)=R_A(s')$ , so by (8.3), $a_Q(s)=\Phi_Q(p(s),R_A(s))=\Phi_Q(p(s'),R_A(s'))=a_Q(s')$ for every $Q$ . Hence $s\sim_{\mathcal Q}s'$ . (This is Corollary 5.3 applied to the sufficient extension $E_A$ of the $\mathcal Q$ -problem.)

Injectivity criterion. By Proposition 3.4 applied to both families, $\chi$ is identified with the canonical map $S/\!\sim_A\ \to\ S/\!\sim_{\mathcal Q}$ , $[s]_A\mapsto[s]_{\mathcal Q}$ , which is well defined by (8.6). Such a map is injective iff $\sim_{\mathcal Q}\ \subseteq\ \sim_A$ ; combined with (8.6) this is exactly (8.5). Since $\chi$ is always surjective, it is a bijection precisely under (8.5). ◻

Remark 31 (Two different things are being retained).

The distinction encoded in Theorem 8.3 is substantive and is easy to lose. The relation completion $E_A$ retains the nominated relation values themselves. The target completion $E_{\mathcal Q}$ retains only those distinctions among relation values that affect the nominated answers. If two source states carry different relation signatures but the mediating maps $\Phi_Q$ send both signatures to the same answers for every $Q\in\mathcal Q$ , then $E_A$ separates them and $E_{\mathcal Q}$ does not. Hence the strict-coarsening possibility in (8.6) is not an artifact; Section 11 exhibits it concretely, where the relation completion retains an orientation that no answer in the nominated spectral family can detect.