Remark [efr-0045]

Markov categories generally do not have pullbacks, for the same reason that they usually don't have products. This issue generally hinders the construction of fibrations, in the ordinary sense, of Markov categories. However, we can go part of the way. The idea of the following definitions is that given a pullback in the deterministic category, say A \times _Y X, a map P \to A \times _Y X where the X-coordinate is deterministic should be uniquely determined by a choice of (deterministic) map P \to X and (stochastic) P \to A such that the square commutes---as we claimed above (and will see below), this holds for the Markov category of discrete probability Kl(\Delta ). The analogous statement for products---that a map P \to A \otimes X with deterministic X-component is uniquely determined by the projections (or marginals) P \to X, P \to A---is a consequence of positivity (see Proposition [efr-OYB6]), and hence holds in most Markov categories of interest.