ora-0118

9.5 Prefix-indexed UDL and non-anticipation

Fix functors

\[ J:\mathcal{O}\longrightarrow \mathcal C, \qquad K:\mathcal C\longrightarrow \mathcal Q, \]

and a target category \(\mathcal D\) with the Kan extensions below. Ordinary UDL sends local data \(F:\mathcal{O}\to \mathcal D\) to

\[ \mathsf U(F)=\operatorname {Ran}_K\operatorname {Lan}_JF. \]

An evidence filtration is a functor

\[ \mathbf F:\mathbb T\longrightarrow [\mathcal{O},\mathcal D], \qquad t\longmapsto F_t, \]

whose transition maps register newly revealed evidence without using future components. Its online UDL semantics is the time-indexed family

\begin{equation} \widehat{\mathbf F}_t = \mathsf U(F_t) = \operatorname {Ran}_K\operatorname {Lan}_JF_t. \end{equation}

9.1

When observations are presented as a complete history \(h\), \(F_t\) means the restriction determined by \(h|_{\downarrow t}\).

Definition 9.2 Online UDL

An online UDL is a prefix-indexed UDL family \(\{ \widehat{\mathbf F}_t\} _{t\in \mathbb T}\) whose action readout at every time \(t\) factors through the prefix restriction

\[ \rho _t:h\longmapsto h|_{\downarrow t}. \]

Equivalently, it satisfies non-anticipation:

\[ h|_{\downarrow t}=h'|_{\downarrow t} \quad \Longrightarrow \quad A_t(h)=A_t(h'). \]

No convexity, probability, regret, or optimization is required by this definition. “Online” changes the information semantics of UDL while leaving its left/right Kan form intact.

Definition 9.3 Persistent online UDL

A persistent online UDL is an online UDL whose synthesized semantic objects and comparison maps form a functor

\[ \widehat{\mathbf F}:\mathbb T\longrightarrow [\mathcal Q,\mathcal D]. \]

Thus for \(s\leq t\) there is a registered transport

\[ u_{s,t}:\widehat{\mathbf F}_s\longrightarrow \widehat{\mathbf F}_t, \qquad u_{t,r}\circ u_{s,t}=u_{s,r}. \]

The current action is a readout from \(\widehat{\mathbf F}_t\), not the persistent object itself.

The transport may preserve an earlier structure, enrich it, or expose a repair obligation. Requiring literal inclusion would rule out legitimate revision, so preservation claims must identify the particular registered subobject that survives.