ora-0123

9.12.1 Horn fillers as repair spaces

Let \(p:E\to X\) be a simplicial family of decorated decisions. Suppose \(e_\Lambda :\Lambda _i^n\to E\) is a partial decorated history and \(\sigma :\Delta ^n\to X\) is a proposed underlying history satisfying \(p e_\Lambda =\sigma |_{\Lambda _i^n}\). The derived space of repairs is the homotopy fiber

\begin{equation} \mathsf{Fill}_p(e_\Lambda ,\sigma ) = \operatorname {hofib}_{(e_\Lambda ,\sigma )}\! \left[ \operatorname {Map}(\Delta ^n,E) \longrightarrow \operatorname {Map}(\Lambda _i^n,E) \mathop{\times }_{\operatorname {Map}(\Lambda _i^n,X)} \operatorname {Map}(\Delta ^n,X) \right]. \end{equation}

9.6

Definition 9.15 Homotopical repair profile

The homotopical repair profile of a registered missing face is the weak homotopy type of \(\mathsf{Fill}_p(e_\Lambda ,\sigma )\). An empty repair space records incompatibility, distinct connected components record qualitatively different repair classes, and a contractible repair space records a repair canonical up to coherent homotopy.

A two-component inner-horn example.

Consider the unresolved inner horn \(\Lambda ^2_1\), consisting of two observed one-step decisions

\[ x_0\xrightarrow {a}x_1\xrightarrow {b}x_2 \]

but no registered composite edge \(x_0\to x_2\). In a strict categorical nerve the missing edge must be \(b\circ a\), with a unique \(2\)-simplex. A learned decision object need not yet identify a strict composite. Suppose its left-stage extension instead generates two candidate fillers:

\[ (c_{\mathrm{s}},\alpha _{\mathrm{s}}) \qquad \text{and}\qquad (c_{\mathrm{r}},\alpha _{\mathrm{r}}), \]

where \(c_{\mathrm{s}}\) is a safe composite, \(c_{\mathrm{r}}\) is a risky composite, and \(\alpha _{\mathrm{s}},\alpha _{\mathrm{r}}\) are coherent witnesses that each candidate completes the observed horn. Let \(W_{\mathrm{s}}\) and \(W_{\mathrm{r}}\) denote their respective witness spaces.

Proposition 9.16 Two-component inner-horn repair

Assume \(W_{\mathrm{s}}\) and \(W_{\mathrm{r}}\) are nonempty and contractible, with no path between their components. Before a distinguishing constraint is revealed, the repair space is

\[ \mathsf{Fill}_1 \simeq W_{\mathrm{s}}\sqcup W_{\mathrm{r}} \simeq S^0. \]

Let later evidence supply a consistency map

\[ q:\mathsf{Fill}_1\longrightarrow \{ 0,1\} , \qquad q(W_{\mathrm{s}})=1, \quad q(W_{\mathrm{r}})=0, \]

and require the value \(1\). Then the right-stage consistency construction is the homotopy fiber

\[ \mathsf{Fill}_2 =\operatorname {hofib}_{1}(q) \simeq W_{\mathrm{s}} \simeq *, \]

and the risky component is removed.

Proof ▼

The two candidate composites determine two disjoint components of the lifting space. Contractibility of their witness spaces identifies their coproduct up to weak equivalence with the discrete two-point space \(S^0\). The homotopy fiber of \(q\) over \(1\) contains precisely \(W_{\mathrm{s}}\), which is contractible by assumption.

This example realizes the UDL factorization directly:

\[ \Lambda ^2_1\text{-decoration} \xrightarrow {\ \operatorname {Lan}^h\ } W_{\mathrm{s}}\sqcup W_{\mathrm{r}} \xrightarrow {\ \operatorname {Ran}^h\ } \operatorname {hofib}_{1}(q). \]

Candidate generation does not prematurely choose between the two composites; consistency removes the component contradicted by later evidence. The natural refinement map points backward,

\[ \mathsf{Fill}_2\hookrightarrow \mathsf{Fill}_1, \]

just as the online Kan quotients form an inverse system. A forward persistent update therefore requires a registered repair span or a declared collapse; it is not an invertible transport of the earlier semantics.

The example also separates structural ambiguity from numerical regret. An observer may assign \(c_{\mathrm{s}}\) and \(c_{\mathrm{r}}\) the same currently observed loss, making them numerically indistinguishable while \(\pi _0(\mathsf{Fill}_1)\) still has two elements. After the new constraint, the change \(S^0\leadsto *\) deletes a component. No infinitesimal tangent path crosses between the two components, so this update is a repair event, not an ordinary gradient or tangent correction within a fixed decision branch.

Thus repair is not forced into a binary success/failure flag or a scalar penalty. Inner horns describe missing composition; outer horns can encode attempts to reverse a decision and need not be fillable. The bar and cobar constructions provide calculational models for homotopy-coherent colimit and limit behavior under the appropriate enriched hypotheses [ Riehl , 2014 ] . This suggests a concrete computational division: bar-like resolution for left-stage candidate generation and cobar-like resolution for right-stage consistency, without asserting that either resolution is unique before the enrichment is declared.