sec-decision-nerve

9.9 Decision nerves and compositional depth

The principal-past semantics uses an external clock. A complementary description makes the length of a composite decision history intrinsic. Let \(\mathcal H\) be a small decision-history category: its objects are information states, its morphisms are admissible one-step decisions or transitions, and categorical composition is sequential execution. Its nerve

\[ X=N\mathcal H \]

is the simplicial set with

\[ X_n=\operatorname {Fun}([n],\mathcal H). \]

This is the ordinary categorical nerve [ Riehl , 2016 ] . Thus an \(n\)-simplex is a composable history

\[ h_0\xrightarrow {a_1}h_1\xrightarrow {a_2}\cdots \xrightarrow {a_n}h_n. \]
Proposition 9.8 Decision nerve and skeletal filtration

For the nerve \(X=N\mathcal H\):

  1. the terminal face deletes the most recent step, the initial face deletes the first step, and each internal face composes two adjacent morphisms;

  2. degeneracy maps insert identity decisions; and

  3. the simplicial set is the filtered colimit of its skeleta,

    \[ X\cong \operatorname *{colim}_{t\geq 0}\operatorname {sk}_tX. \]
Proof ▼

The face and degeneracy operators are induced by the coface and codegeneracy maps in \(\Delta \). Functoriality turns an internal coface into composition and a codegeneracy into an identity. Every simplex has finite dimension, so it belongs to some skeleton; the skeletal inclusions therefore have colimit \(X\).

Histories available through length \(t\) first form the truncation \(\operatorname {tr}_{\leq t}X\). The smallest simplicial object generated by that truncation is \(\operatorname {sk}_tX\). This yields the distinction The notation \(\downarrow t\) records information revealed by external time \(t\), whereas \(\operatorname {sk}_nX\) records information represented through compositional depth \(n\). One decision per round permits the diagonal identification \(n=t\). In general, however, a decision model is naturally bifiltered by \((t,n)\): an agent may acquire new observations without increasing compositional depth, or construct a longer internal plan before receiving new external evidence. The flow-reasoning example in 9.16 supplies a concrete second realization of this distinction: continuous flow time advances the generative state, while recurrent depth counts repeated repairs at a fixed flow state.