  We give a simultaneous generalization of exact categories and triangulated
categories, which is suitable for considering cotorsion pairs, and which we
call extriangulated categories. Extension-closed, full subcategories of
triangulated categories are examples of extriangulated categories. We give a
bijective correspondence between some pairs of cotorsion pairs which we call
Hovey twin cotorsion pairs, and admissible model structures. As a consequence,
these model structures relate certain localizations with certain ideal
quotients, via the homotopy category which can be given a triangulated
structure. This gives a natural framework to formulate reduction and mutation
of cotorsion pairs, applicable to both exact categories and triangulated
categories. These results can be thought of as arguments towards the view that
extriangulated categories are a convenient setup for writing down proofs which
apply to both exact categories and (extension-closed subcategories of)
triangulated categories.
