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We generalize Bousfield-Friedlander's Theorem and Hirschhorn's Localization Theorem to settings where the hypotheses are not satisfied at the expense of obtaining semimodel categories instead of model categories. We use such results to answer, in the world of semimodel categories, an open problem posed by May-Ponto about the existence of Bousfield localizations for Hurewicz and mixed type model structures (on spaces and chain complexes). We also provide new applications that were not available before, e.g. stabilization of non-cofibrantly generated model structures or applications to mathematical physics. [ABSTRACT FROM AUTHOR]

We give a general framework of equivariant model category theory. Our groups G, called Hopf groups, are suitably defined group objects in any well-behaved symmetric monoidal category V . For any V, a discrete group G gives a Hopf group, denoted I[G]. When V is cartesian monoidal, the Hopf groups are just the group objects in V . When V is the category of modules over a commutative ring R, I[G] is the group ring R[G] and the general Hopf groups are the cocommutative Hopf algebras over R. We show how all of the usual constructs of equivariant homotopy theory, both categorical and model theoretic, generalize to Hopf groups for any V . This opens up some quite elementary unexplored mathematical territory, while systematizing more familiar terrain. [ABSTRACT FROM AUTHOR]

We verify the existence of left Bousfield localizations and of enriched left Bousfield localizations, and we prove a collection of useful technical results characterizing certain fibrations of (enriched) left Bousfield localizations. We also use such Bousfield localizations to construct a number of new model categories, including models for the homotopy limit of right Quillen presheaves, for Postnikov towers in model categories, and for presheaves valued in a symmetric monoidal model category satisfying a homotopy-coherent descent condition. We then verify the existence of right Bousfield localizations of right model categories, and we apply this to construct a model of the homotopy limit of a left Quillen presheaf as a right model category. [ABSTRACT FROM AUTHOR]

We consider the category of presheaves of Γ-spaces, or equivalently, of Γ-objects in simplicial presheaves. Our main result is the construction of stable model structures on this category parametrised by local model structures on simplicial presheaves. If a local model structure on simplicial presheaves is monoidal, then the corresponding stable model structure on presheaves of Γ-spaces is monoidal and satisfies the monoid axiom. This allows us to lift the stable model structures to categories of algebras and modules over a monoid. [ABSTRACT FROM AUTHOR]