nLab pseudomodel stack

A notion of pseudomodel stack was introduced by Nikolai Durov (arXiv) as a generalization of model stacks, one which seems to be a more appropriate internalization of model categories in the setting of topoi.

A stack $C$ over a site $S$ is a pseudomodel stack if it is a flat stack? and in every fiber five local classes of morphisms are distinguished: fibrations, acyclic fibrations, cofibrations, acyclic cofibrations and weak equivalences: and

(PM1) all fiber categories are finitely complete and finitely cocomplete,

(PM2) each distinguished class is local and stable under composition and global retracts,

(PM3) (2-out-of-3) If $f$, $g$, $g\circ f$ are defined morphisms in a fixed fiber, then if any two of the three are weak equivalences then so is the third,

(PM5) (factorization) Any morphism can be globally factored as a cofibration followed by a fibration, where one can make either one chosen of the two acyclic,

(PM8) any acyclic fibration is both a fibration and weak equivalence, and any acyclic cofibration is both a cofibration and a weak equivalence (but not necessarily other ways around); any weak equivalence can be factored as an acyclic cofibration folowed by an acyclic fibration.

There is no lifting axiom! But the axioms are still self-dual.

Each fiber of a pseudomodel stack is a pseudomodel category? (in another way, it is a pseudomodel stack over the point).