We show that the category whose objects are families of Markov processes on Polish spaces,
with a given transition kernel, and whose morphisms are transition probability preserving,
surjective continuous maps has semi-pullbacks, i.e., for any pair of morphisms
fi:Si→S
(i = 1, 2), there exists an object V and morphisms
πi:V→Si
(i = 1, 2) such that f1∘π1
= f2∘π2. This property holds for various full subcategories, including that of families
of Markov processes on locally compact second countable spaces, and in the larger category
where the objects are families of Markov processes on analytic spaces and morphisms are
transition probability preserving surjective Borel maps. It also follows that the category of
probability measures on Polish spaces and measure-preserving continuous surjective maps
has semi-pullbacks. A main consequence of our result is that probabilistic bisimulation for
labelled Markov processes on all these categories, defined in terms of spans of morphisms, is
an equivalence relation.