ora-0122

9.10 Simplicial online UDL

Let \(\Delta /X\) be the category of simplices of \(X\), and let

\[ \mathsf{Simp}_{\leq t}(X) = \Delta _{\leq t}/X \xhookrightarrow {j_t} \Delta /X \]

be its full subcategory on simplices of dimension at most \(t\). A decoration functor records the evidence, losses, actions, or warrants attached to the observed simplices.

Definition 9.9 Simplicial online UDL

Given compatible decorations \(F_t:\mathsf{Simp}_{\leq t}(X)\to \mathcal D\), the depth-\(t\) simplicial online UDL semantics is

\[ \mathsf U_t^{\Delta }(F_t) = \operatorname {Ran}_{K_t}\operatorname {Lan}_{J_t}F_t, \]

where \(J_t\) and \(K_t\) declare candidate extension and consistency over the truncated simplex category. Persistent simplicial UDL additionally registers coherent comparison maps along \(\mathsf{Simp}_{\leq s}(X)\hookrightarrow \mathsf{Simp}_{\leq t}(X)\) for \(s\leq t\).

If the entire category \(\mathcal H\) and its nerve are already known, the compositional fillers carry no learning problem. The nontrivial cases begin with a partial nerve, an unknown decision category, or a known nerve whose observations and decision values are only partially decorated.

Proposition 9.10 Inner horns encode decision composition

The nerve of every category has a unique filler for every inner horn. It is a Kan complex, with fillers for all horns, if and only if the underlying category is a groupoid.

Proof ▼

An inner horn specifies a composable configuration with one composite face missing; associativity gives its unique filler. Outer horn fillers provide left and right inverses for morphisms. Hence all outer horns are fillable exactly when every morphism is invertible.

This result separates two uses of Kan’s ideas. A Kan extension universally extends a diagram; a simplicial Kan condition fills horns. They interact in the present construction but are not synonyms. Requiring a full Kan complex would also impose reversibility, which is inappropriate for irreversible actions such as resource consumption or pushing an object into an unrecoverable state. Inner-horn or quasi-categorical structure is the more natural default.

At a schematic level, partial simplicial information supplies the UDL factorization

\[ \text{boundary or horn data} \xrightarrow {\operatorname {Lan}} \text{candidate completions} \xrightarrow {\operatorname {Ran}} \text{compatible fillers}. \]
Making this slogan a theorem requires declaring the precise horn category, the decoration, and the target enrichment; unique inner fillers in the underlying nerve alone do not supply learned values or policies.

The crossword construction in 9.15 makes this distinction concrete. Clue agents generate local candidate words, crossing letters define compatibility faces, and a partial filling may leave a completion problem unresolved. A strict filler records an exactly compatible extension; a homotopy-coherent filler can additionally retain alternative meanings and witnesses of compatibility.