1.7 Functor categories and online information

For categories \(\mathbb T\) and \(\mathcal D\), the functor category \([\mathbb T,\mathcal D]\) has time-indexed diagrams as objects and natural transformations as morphisms. This is the minimal setting for persistence.

Let \(\mathbb T\) be a poset of times. The information available at \(t\) is its principal past

An action at \(t\) is online when it factors through restriction to \(\downarrow t\). Equivalently, two complete histories with identical prefixes through \(t\) must induce the same action at \(t\).

Persistence is stronger than repeated computation. A collection of unrelated objects, one for each time, does not say what survived. A functor \(\mathbb T\to \mathcal D\) registers the transport maps and their composition law. Later chapters allow those transports to be coherent only up to higher homotopy.