Suppose C is a complete and cocomplete category with two model structures (C0,F0,W0) and (C1,F1,W1) such that C0⊃C1, F0⊂F1, W0⊂W1. If necessary, the model structures can be assumed to be simplicial, left proper, cofibrantly generated, combinatorial, cellular, tractable, etc.
Consider now the left Bousfield localization (C2,F2,W2) of the model structure (C0,F0,W0) with respect to the class of morphisms W1. We know that W1⊂W2.
Which additional conditions (if any) are needed to ensure that W1=W2?
This question is motivated by a desire to understand better the so-called mixed model structure (not to be confused with the notion introduced by Michael Cole) on symmetric spectra, as explained, for example, by Shipley in Proposition 1.3 in “A convenient model category for commutative ring spectra”. There C is the category of G-spaces for some finite group G, (C0,F0,W0) is the equivariant model structure (weak equivalences and fibrations are defined as maps whose induced map on H-fixed points for any finite subgroup H of G is a weak equivalence respectively fibration), (C1,F1,W1) is the projective model structure, and (a posteriori) (C2,F2,W2) is the mixed model structure: C2=C0, W2=W1. Thus it is natural to ask whether the mixed model structure can be obtained as the left Bousfield localization of the equivariant structure with respect to projective weak equivalences.
