The past $\sigma$-algebra is a single pullback along the restriction

For the coordinate process $Y_i(\omega) = \omega_i$ on path space, the $\sigma$-algebra generated by the coordinates up to time $k$ coincides with the pullback of the product $\sigma$-algebra on $\prod_{i \le k}\mathsf{X}$ along the restriction map $\omega \mapsto (\omega_i)_{i \le k}$:

Companion to the future-$\sigma$-algebra identification. Mixing coefficients compare the past $\sigma(Y_0,\dots,Y_k)$ with the future $\sigma(Y_i : i \ge k+n)$; both appear in their definitions as suprema of single-coordinate pullbacks. This theorem handles the past exactly as its companion handles the future: every past event is $\{\omega : \mathrm{fr}_k\omega \in A_0\}$ for a single measurable $A_0 \subseteq \prod_{i\le k}\mathsf{X}$.

This is what allows the Ionescu–Tulcea decomposition $\mathrm{traj}_0 = \mathrm{traj}_k \circ \mathrm{partialTraj}(0,k)$ to be applied: that decomposition conditions on the first $k+1$ coordinates, and the statement that a past event is determined by those coordinates has to be available in the form "$A$ is a pullback along $\mathrm{fr}_k$", not as membership in an abstract supremum. Together with the future identification, a mixing estimate becomes a genuine computation: condition on $\mathrm{fr}_k$, use the restart property on the shifted future, and compare with stationarity.

Proof. The product $\sigma$-algebra on $\prod_{i \le k}\mathsf{X}$ is the supremum of the pullbacks along its own coordinate evaluations; pullback commutes with suprema and composition, so the right-hand side is the supremum over $i \in \mathrm{Iic}\,k$ of the pullbacks along $\omega \mapsto \omega_i$. Matching this with $\bigsqcup_{i \le k}$ is the translation between the subtype of the finite set $\mathrm{Iic}\,k$ and the condition $i \le k$, in both directions.