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 over a site 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
(PM2) each distinguished class is local and stable under composition and global retracts,
(PM3) (2-out-of-3) If , , 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.