  The main results of this paper comprise proofs of the following two related
facts: (i) the Feynman--Kac formula is a functor $F_*$, namely, between a
stochastic differential equation and a dynamical system on a statistical
manifold, and (ii) a statistical manifold is a sheaf generated by this functor
with a canonical gluing condition. Using a particular locality property for
$F_*$, recognised from functorial quantum field theory as a `sewing law,' we
then extend our results to the Chapman--Kolmogorov equation {\it via} a
time-dependent generalisation of the principle of maximum entropy. This yields
a partial formalisation of a variational principle which takes us beyond
Feynman--Kac measures driven by Wiener laws. Our construction offers a robust
glimpse at a deeper theory which we argue re-imagines time-dependent
statistical physics and information geometry alike.
