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

The canonical minimal-completion theorem

Jeremy RodgersSection 6 of 16Original PDF ↗Publication DOI ↗

The canonical minimal-completion theorem

Theorem 5.1 (Canonical minimal completion).

The tuple $\mathcal E_A=\bigl(E_A,\ e_A,\ u_A,\ (\rho_\alpha)_{\alpha\in A}\bigr)\tag{5.1}$ of Definition 3.1 is an object of $\mathbf{Suff}(p;\{r_\alpha\})$ , and it is terminal: for every relation-sufficient extension $\mathcal E=(E,e,u,(\rho_\alpha^E))$ there exists a unique morphism $\phi_{\mathcal E}\colon\ \mathcal E\longrightarrow\mathcal E_A, \qquad\text{that is,}\qquad \phi_{\mathcal E}\colon E\longrightarrow E_A .\tag{5.2}$ It is given by the explicit formula $\phi_{\mathcal E}(x)=\Bigl(u(x),\ \bigl(\rho_\alpha^E(x)\bigr)_{\alpha\in A}\Bigr), \qquad x\in E,\tag{5.3}$ and $\phi_{\mathcal E}$ is surjective.

Before the proof, we emphasize the direction, since reversing it inverts the meaning of the theorem. The universal map runs from an arbitrary sufficient extension to $E_A$ , not from $E_A$ to it. $E_A$ is therefore the terminal, the coarsest, sufficient extension: every sufficient extension maps onto it, and so retains at least as much information as it does. $E_A$ is not an initial object, not a universal cover, and not the finest extension; the finest sufficient extension is $S$ itself (with $e=\mathop{\mathrm{id}}_S$ ), which is precisely the object one is trying not to be forced into.

Proof. $\mathcal E_A$ is an object. $e_A$ is surjective and the factorizations (3.4) hold, by Proposition 3.2.

Now fix a relation-sufficient extension $\mathcal E=(E,e,u,(\rho_\alpha^E))$ and define $\phi_{\mathcal E}$ by the formula (5.3). We establish the required properties in turn.

(i) The formula is well defined as a function $E\to T\times V_A$ . For each $x\in E$ the value $u(x)\in T$ is determined, and for each $\alpha\in A$ the value $\rho_\alpha^E(x)\in V_\alpha$ is determined; the assignment $\alpha\mapsto\rho_\alpha^E(x)$ is therefore an element of the set-theoretic product $V_A=\prod_{\alpha\in A}V_\alpha$ . Hence (5.3) assigns to each $x\in E$ exactly one element of $T\times V_A$ . No choice is involved, and no representative of $x$ is used: the formula depends only on $x$ through the given structure maps of $\mathcal E$ .

(ii) The image lies in $E_A$ , and $e_A=\phi_{\mathcal E}\circ e$ . Let $x\in E$ . By surjectivity of $e$ there is $s\in S$ with $x=e(s)$ . Then, using (4.2), $\phi_{\mathcal E}(x) =\phi_{\mathcal E}\bigl(e(s)\bigr) =\Bigl(u\bigl(e(s)\bigr),\ \bigl(\rho_\alpha^E(e(s))\bigr)_{\alpha\in A}\Bigr) =\Bigl(p(s),\ \bigl(r_\alpha(s)\bigr)_{\alpha\in A}\Bigr) =j_A(s) =e_A(s).\tag{5.4}$ The right-hand side lies in $\mathop{\mathrm{Im}}(j_A)=E_A$ by Definition 3.1. Since $x\in E$ was arbitrary, $\phi_{\mathcal E}(E)\subseteq E_A$ , so $\phi_{\mathcal E}$ may be regarded as a map $E\to E_A$ ; and reading (5.4) from the ends, $\phi_{\mathcal E}(e(s))=e_A(s)$ for every $s\in S$ , i.e. $e_A=\phi_{\mathcal E}\circ e .\tag{5.5}$ Note that the computation (5.4) is legitimate for every choice of preimage $s$ of $x$ , and its left-hand side does not depend on $s$ ; so no consistency question arises.

(iii) It is a morphism, and preserves all structure maps. Equation (5.5) is precisely condition (4.3) of Definition 4.2, so $\phi_{\mathcal E}$ is a morphism $\mathcal E\to\mathcal E_A$ . Preservation of the structure maps then follows from Lemma 4.3, but in the present case it is also visible directly from (5.3): applying the coordinate projections $u_A$ and $\rho_\alpha$ to $\phi_{\mathcal E}(x)=\bigl(u(x),(\rho_\beta^E(x))_\beta\bigr)$ gives $u_A\circ\phi_{\mathcal E}=u, \qquad \rho_\alpha\circ\phi_{\mathcal E}=\rho_\alpha^E \qquad(\alpha\in A).\tag{5.6}$

(iv) Surjectivity. Let $y\in E_A$ . By Proposition 3.2 the map $e_A$ is surjective, so $y=e_A(s)$ for some $s\in S$ . By (5.5), $y=\phi_{\mathcal E}(e(s))$ , and $e(s)\in E$ . Hence $\phi_{\mathcal E}$ is onto $E_A$ .

(v) Uniqueness. Suppose $\psi\colon\mathcal E\to\mathcal E_A$ is any morphism, i.e. any function $\psi\colon E\to E_A$ with $e_A=\psi\circ e$ . Let $x\in E$ and choose $s\in S$ with $x=e(s)$ . Then $\psi(x)=\psi\bigl(e(s)\bigr)=e_A(s)=\phi_{\mathcal E}\bigl(e(s)\bigr)=\phi_{\mathcal E}(x),$ using $e_A=\psi\circ e$ for the second equality and (5.5) for the third. Since $x$ was arbitrary, $\psi=\phi_{\mathcal E}$ . (This is exactly where surjectivity of $e$ is consumed: it guarantees that the condition $e_A=\psi\circ e$ pins down $\psi$ at every point of $E$ , not merely on a subset.) ◻

Corollary 5.2 (Uniqueness up to unique isomorphism).

Any two terminal objects of the category $\mathbf{Suff}(p;\{r_\alpha\})$ are isomorphic by a unique isomorphism, and this isomorphism commutes with the completion maps from $S$ and with all of the structure maps. In particular, any relation-sufficient extension that satisfies the universal property of Theorem 5.1 is canonically isomorphic to $\mathcal E_A$ .

Proof. Let $\mathcal E,\mathcal E'$ be terminal. Terminality of $\mathcal E'$ gives a unique morphism $f\colon\mathcal E\to\mathcal E'$ ; terminality of $\mathcal E$ gives a unique morphism $g\colon\mathcal E'\to\mathcal E$ . Then $g\circ f\colon\mathcal E\to\mathcal E$ is a morphism, as is $\mathop{\mathrm{id}}_{\mathcal E}$ ; by terminality of $\mathcal E$ there is only one morphism $\mathcal E\to\mathcal E$ , so $g\circ f=\mathop{\mathrm{id}}_E$ . Symmetrically $f\circ g=\mathop{\mathrm{id}}_{E'}$ . Hence $f$ is an isomorphism, and it is unique because morphisms into a terminal object are unique. Compatibility with the completion maps is (4.3); compatibility with the structure maps is Lemma 4.3. ◻

Corollary 5.3 (Coarsest sufficiency, elementary form).

Let $\mathcal E=(E,e,u,(\rho_\alpha^E))$ be any relation-sufficient extension, and define its induced equivalence relation by $s\sim_e s'\quad:\Longleftrightarrow\quad e(s)=e(s').\tag{5.7}$ Then $\sim_e\ \subseteq\ \sim_A .\tag{5.8}$ Equivalently: every sufficient extension separates at least every pair that $E_A$ separates, while $E_A$ separates no pair that sufficiency does not force it to separate.

Proof. Suppose $e(s)=e(s')$ . Applying $u$ and using $p=u\circ e$ , $p(s)=u\bigl(e(s)\bigr)=u\bigl(e(s')\bigr)=p(s') .$ Applying $\rho_\alpha^E$ and using $r_\alpha=\rho_\alpha^E\circ e$ , for each $\alpha\in A$ , $r_\alpha(s)=\rho_\alpha^E\bigl(e(s)\bigr)=\rho_\alpha^E\bigl(e(s')\bigr)=r_\alpha(s') .$ By (3.5) this is precisely $s\sim_A s'$ . ◻

Corollary 5.3 and Theorem 5.1 are two faces of one statement. Terminality is the categorical form; the inclusion (5.8) is the elementary, checkable form. The inclusion also settles the direction question once and for all: since $\sim_e$ is contained in $\sim_A$ , the partition induced by $e$ refines the partition induced by $e_A$ , so there is a well-defined map from the finer quotient onto the coarser one, from $E$ onto $E_A$ , and not back.

Remark 18 (The sense in which the completion is a construction).

Theorem 5.1 constructs the coarsest relation-sufficient extension in the paper’s category $\mathbf{Suff}(p;\{r_\alpha\})$ of set-theoretic source quotients: the object $E_A$ exists, is terminal, and is unique up to unique isomorphism, and those claims are unconditional. What the theorem builds it builds from the nominated relations themselves. The joint map $j_A=(p,R_A)$ takes the $r_\alpha$ as given input, so the construction presupposes the relation data rather than producing it; in particular it furnishes no procedure for recovering the $r_\alpha$ from the readout $p$ when they fail to descend, if such a procedure existed the relations would already be functions of $p$ , and the completion would be $T$ itself. Nor does terminality establish that $E_A$ is experimentally accessible, measurable, smooth, geometric, or realized as a physical state space independent of its defining coordinates; $E_A$ is, as constructed, a quotient of $S$ recording exactly the nominated values. A concrete realization must independently supply and justify the relation data, and any additional geometric or physical structure carried by $E_A$ , on its own terms. That realization question is not settled here; it is deferred to the companion manuscript on geometric realization discussed in Section 13.

Remark 19 (What “minimal” means, and what it does not).

Minimality here means: least retained distinction among sufficient extensions. It is a statement about the ordering of equivalence relations by inclusion, equivalently about the ordering of partitions by refinement. It does not mean minimum cardinality selected without regard to structure, the one-point set is smaller than $E_A$ and is not, in general, sufficient, and it does not mean that $E_A$ is a small or economical description of $S$ in any other sense. Above all, minimal completion is not reconstruction of $S$ : by Proposition 3.5 the map $e_A$ need not be injective, and Section 12 determines exactly when it is.