ora-0117

9.4 Time, revelation, and the least available past

Let \(\mathbb T\) be a small poset of times, written \(s\leq t\) when \(s\) is no later than \(t\). The information available at \(t\) is the principal down-set

\[ \downarrow t=\{ s\in \mathbb T\mid s\leq t\} . \]

Equivalently, it is the representable presheaf \(y(t)=\mathbb T(-,t)\). Since \(\mathbb T\) is a poset, \(y(t)\) is subterminal and records the truth value “revealed by time \(t\).”

Proposition 9.1 Least causally closed information region

The principal past \(\downarrow t\) is the least downward-closed subobject of \(\mathbb T\) containing \(t\).

Proof ▼

Every downward-closed subset containing \(t\) must contain every \(s\leq t\), hence contains \(\downarrow t\). The principal past is itself downward closed and contains \(t\).

Thus \(\{ t\} \) is generally too small because it omits history, while all of \(\mathbb T\) is too large because it admits future information.