There is a (necessarily unique closed) mixed model structure on in which the fibrations are the -fibrations, but the weak equivalences are the -equivalences.
An object is cofibrant in the mixed model structure if and only if it has the -homotopy type of a -cofibrant object.
From the Quillen and Strøm model structures on topological spaces, we obtain a mixed model structure in which the weak equivalences are the weak homotopy equivalences, the fibrations are the Hurewicz fibrations, and the cofibrant objects are those of the homotopy type of a CW complex. (This example influences the notation above; the -model structure is the Quillen one, the -model structure is the “Hurewicz” (or “homotopy”) one.)
Similarly, we can mix the - and - model structures on chain complexes.
Let be an accessible strict 2-monad on a locally finitely presentable strict 2-category . Then by a theorem of Lack, the category of strict -algebras admits a transferred model structure from the 2-trivial model structure on , where the weak equivalences and fibrations are the morphisms which become internal equivalences and internal isofibrations in .
On the other hand, has its own 2-trivial model structure. Since the forgetful functor preserves equivalences and isofibrations (the latter since it has a strict left 2-adjoint), we can mix these two model structures to obtain one whose weak equivalences are the equivalences in (which are also the equivalences in the category of -algebras and pseudo morphisms), but whose fibrations are the isofibrations in .
The original paper is
There is also an exposition in