Target-Relative Necessity of Completion: When Readout Loss Obstructs, and What a Sufficient Extension Must Retain

Readout extensions and target sufficiency

Jeremy RodgersSection 7 of 16Original PDF ↗Publication DOI ↗

Readout extensions and target sufficiency

If the readout does not determine the answer, one may enlarge the readout. The following definition says what an enlargement is, and the crucial point is that it need not be an enlargement all the way back to the source.

Definition 6.1 (Readout extension).

A readout extension (or enrichment) of $p\colon S\to T$ is a surjective map $e\colon S\longrightarrow E \qquad\text{together with a map}\qquad u\colon E\longrightarrow T \qquad\text{such that}\qquad p=u\circ e .\tag{15}$

Lemma 6.2 (Basic properties of extensions).

Let $(e,E,u)$ be a readout extension of $p$ . Then:

$u$ is uniquely determined by $e$ and $p$ ;
$u$ is surjective;
the partition of $S$ into fibers of $e$ refines the partition into fibers of $p$ ; equivalently $e(s)=e(s')\implies p(s)=p(s')$ .

Proof. (1) If $u,u'$ both satisfy $p=u\circ e=u'\circ e$ , let $x\in E$ . By surjectivity of $e$ choose $s$ with $e(s)=x$ ; then $u(x)=u(e(s))=p(s)=u'(e(s))=u'(x)$ .

(2) Let $t\in T$ . By surjectivity of $p$ choose $s$ with $p(s)=t$ ; then $u(e(s))=p(s)=t$ , so $t\in\operatorname{Im}(u)$ .

(3) If $e(s)=e(s')$ then $p(s)=u(e(s))=u(e(s'))=p(s')$ . ◻

Lemma 6.2 (3) is the content of the word “extension”: $E$ is at least as informative as $T$ , because anything determined by $E$ determines the $T$ -value. Equivalently, the equivalence relation $\sim_e$ induced by $e$ is contained in $\sim_p$ . But $E$ need not be $S$ , and $\sim_e$ need not be equality: $E$ sits somewhere between the readout and the source, an intermediate quotient of $S$ , or, in partition language, a refinement of the readout partition that need not be the discrete partition. The two extreme cases are $E=T$ (with $e=p$ , $u=\operatorname{id}$ ), which adds nothing, and $E=S$ (with $e=\operatorname{id}$ , $u=p$ ), which retains everything. The interesting extensions are strictly between, and the point of the next two sections is that for a given target there is a canonical one, generally far closer to $T$ than to $S$ .

Definition 6.3 (Sufficiency for a target).

A readout extension $e\colon S\to E$ is sufficient for $Q$ , or $Q$ -sufficient, if there exists a map $b\colon E\longrightarrow A_Q \qquad\text{such that}\qquad a_Q=b\circ e .\tag{16}$ We also call a $Q$ -sufficient readout extension a completion of $p$ for $Q$ .

Remark 6.4 (What “completion” means here).

The word “completion” is used in Definition 6.3 in a sense that must be separated from several established and entirely different uses. It is not metric completion, not topological or uniform completion, not Cauchy completion, not the completion of a local ring, and not the adjunction of arbitrary hidden variables posited to restore determinism. A completion in the present sense is an enrichment of the readout that suffices for a nominated target: a surjection $e$ through which both $p$ and $a_Q$ factor. It is target-relative by construction; an extension may complete $p$ for one target while remaining insufficient for another. And it is constrained: Theorem 7.1 shows that a completion cannot be arbitrary, and Theorem 8.3 shows that there is a least one.

Lemma 6.5 (Sufficiency criterion).

A readout extension $e\colon S\to E$ is $Q$ -sufficient if and only if $a_Q$ is constant on every fiber of $e$ ; and then $b$ in (16) is unique.

Proof. Apply Theorem 3.1 with $p$ replaced by the surjection $e$ and $T$ by $E$ . ◻

Note the immediate sanity check: $e=\operatorname{id}_S$ (with $E=S$ , $u=p$ , $b=a_Q$ ) is always $Q$ -sufficient, so completions exist trivially. The theorems that follow are therefore not existence theorems. They are theorems about how little a completion can get away with, and what it must contain no matter what.