Are there theorems (esp. computational tools) on model categories which survive and do not trivialise when its underlying category is a (quasi-)poset ? Are there tools that may help to calculate a derived functor from such a model category to a poset?
Here a quasi-poset is simply a category such that there is at most one morphism between any two objects.
The standard tools such as (co)fibration sequences, loop and suspension functors, path and cylinder objects, are either not defined because the category is not pointed or trivialise for the following basic reason: In a poset (as a category), for an object X, both product and coproduct $X+X = X\times x = X$ is X itself, and therefore for X' a path object X, the decomposition X ---> X' ---> XxX = X implies X'=X, similarly for the cylinder object. Same reasoning applies when the underlying category is a quasi-poset. Homotopy classes of maps [X,Y] and [X,Y]1 do not make much sense either, as there is a unique, if any, map from X to Y.
Motivation: a set-theoretic model category
The motivation for my question is that I am trying to understand a set-theoretic construction of a model category. The model category "talks" about set theory, e.g. a cofibrant object is a family of countable sets, and an example of a fibration is taking the union of an increasing chain of sets. The model category is rather degenerate, as the underlying category is a quasi-poset. Still, a well-known cardinal arithmetic invariant, the covering number of PCF theory, appears as a (slightly generalised) left derived functor in this formalism, and I am interested to see whether homotopy theory tools may be able to calculate that derived functor.
