Semi-pullbacks and bisimulation in categories of Markov processes

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 f i:S i→S (i = 1, 2), there exists an object V and morphisms π i:V→S i (i = 1, 2) such that f 1cπ 1 = f 2cπ 2. 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.