Consider the walking reflexive pair category W, which consists of two objects 0 and 1 and three generating morphisms f: 0→1, g: 0→1, and h: 1→0 satisfying the relation fh=gh=id₁.
Consider the category of diagrams indexed by W valued in some model category C (which can be assumed to be combinatorial, left proper, etc.). Colimits of such diagrams are known as reflexive coequalizers.
I am interested in sufficient conditions for such a diagram to be projectively cofibrant that are weaker than the usual criterion for projective cofibrancy of a nonreflexive coequalizer diagram, i.e., a diagram f,g: A⇉B is projectively cofibrant if and only if A is cofibrant and the morphism f⊔g: A⊔A→B is a cofibration.
Motivation for this question comes from a desire to establish a criterion for levelwise cofibrancy of a coproduct of two modules A⊔O over some operad O such that A is a cofibrant left O-module (even an O-algebra) and the second term O denotes O taken as a left O-module over itself. Such a coproduct can be expressed as a reflexive coequalizer and reflexive coequalizers are preserved by the forgetful functor from modules to the underlying category of symmetric sequences, so it is sufficient to establish the projective cofibrancy of the underlying reflexive coequalizer diagram. Unfortunately, the condition for cofibrancy of a nonreflexive coequalizer diagram is far too restrictive to be satisfied here. One could hope that the reflexivity property might allow for a criterion that can be satisfied more easily. Of course, one could also extend this reflexive coequalizer diagram to a full simplicial diagram and try to establish cofibrancy in this setting, but I would like first to see whether it is sufficient to look at the 1-truncation. Note also that the question becomes trivial if the unit in the underlying monoidal model category is cofibrant, because then O, hence also A⊔O will be cofibrant O-modules, in particular, they will be levelwise cofibrant.
