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