nLab model stack

The notion of model stack is an adaptation of model categories to stacks. Nikolai Durov (arXiv) has studied them in passing and noted that a more flexible notion for that context is the weaker notion of pseudomodel stack which he introduced there.

A stack $C$ over a site $S$ is a model stack if in every fiber three local classes of morphisms are distinguished: fibrations, cofibrations and weak equivalences (with the terminology that an acyclic (co)fibration is a (co)fibration which is also a weak equivalence) and

(MS1) all fiber categories are finitely complete and finitely cocomplete and all cartesian morphisms in $C$ admit left and right adjoints,

(MS2) each distinguished class is local and stable under global retracts,

(MS3) (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,

(MS4) any cofibration has the local left lifting property? with respect to any acyclic fibration, and any acyclic fibration has the local left lifting property with respect to any fibration,

(MS5) (factorization) Any morphism can be globally factored as a cofibration followed by a fibration, where one can choose either one of the two acyclic.