Model categories with simple homotopy categories

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In the present article, we describe constructions of model structures on general bicomplete categories. We are motivated by the following question: given a category C with a subcategory wC closed under retracts, when is there a model structure on C with wC as the subcategory of weak equivalences? We begin exploring this question in the case where wC = F⁻¹(iso D) for some functor F:C→ D. We also prove properness of our constructions under minor assumptions and examine an application to the category of infinite graphs.