Model structures on exact categories

Abstract

We define model structures on exact categories, which we call exact model structures. We look at the relationship between these model structures and cotorsion pairs on the exact category. In particular, when the underlying category is weakly idempotent complete, we get Hovey’s one-to-one correspondence between model structures and complete cotorsion pairs. We classify the right and the left homotopy relation in terms of the cotorsion pairs and look at examples of exact model structures. In particular, we see that given any hereditary abelian model category, the full subcategories of cofibrant, fibrant and cofibrant–fibrant subobjects each has natural exact model structures equivalent to the original model structure. These model structures each has interesting characteristics. For example, the cofibrant–fibrant subobjects form a Frobenius category, whose stable category is the same as the homotopy category of its model structure.