Definition Weak conditionals [efr-2MH5]

Let p: \mathcal {D} \to \mathcal {C} be a Markov prefibration. We say p (or, abusing notation, \mathcal {D}) admits weak conditionals if, given a Cartesian map \bar {Y} \to \bar {Z} and any map \bar {X} \to \bar {Z}, the existence part of the Cartesian condition holds---that is, every factorization p(\bar {X}) \to p(\bar {Y}) admits a lift, although not necessarily a unique one.

We say a Markov category \mathcal {C} admits weak conditionals if its codomain functor \mathcal {C}^\to \to \mathcal {C} is a prefibration which admits weak conditionals---this is equivalent to requiring that it is pullback-positive, and that all deterministic pullbacks are carried to weak pullbacks by the inclusion \mathcal {C}_\mathrm {det} \to \mathcal {C} (in other words, that they satisfy the existence part of the universal property even for pairs of nondeterministic maps).