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Model categories with simple homotopy categories (1312.4245v3)
Published 16 Dec 2013 in math.AT, math.CO, and math.CT
Abstract: 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.