  Random variables $X^i$, $i=1,2$ are 'probabilistically equivalent' if they
have the same law. Moreover, in any class of equivalent random variables it is
easy to select canonical representatives.
  The corresponding questions are more involved for processes $X^i$ on filtered
stochastic bases $(\Omega^i, \mathcal F^i, \mathbb P^i, (\mathcal F^i_t)_{t\in
[0,1]})$. Here equivalence in law does not capture relevant properties of
processes such as the solutions to stochastic control or multistage decision
problems. This motivates Aldous to introduce the stronger notion of synonymity
based on prediction processes. Stronger still, Hoover--Keisler formalize what
it means that $X^i$, $i=1,2$ have the same probabilistic properties. We
establish that canonical representatives of the Hoover--Keisler equivalence
classes are given precisely by the set of all Markov-martingale laws on a
specific nested path space $\mathsf M_\infty$. As a consequence we obtain that,
modulo Hoover--Keisler equivalence, the class of stochastic processes forms a
Polish space.
  On this space, processes are topologically close iff they model similar
probabilistic phenomena. In particular this means that their laws as well as
the information encoded in the respective filtrations are similar. Importantly,
compact sets of processes admit a Prohorov-type characterization. We also
obtain that for every stochastic process, defined on some abstract basis, there
exists a process with identical probabilistic properties which is defined on a
standard Borel space.
