For any positive integer $n$, let $A_n$ be a linearly oriented quiver of type $A$ with $n$ vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories ${ℳ}_{n+1}$ and ${ℱ}_n$, where ${ℳ}_{n+1}$ and ${ℱ}_n$ are the two extriangulated categories corresponding to the representation category of $A_{n+1}$ and the morphism category of projective representations of $A_n$, respectively. As a by-product,...

Let be an abelian category, or more generally a weakly idempotent complete exact category, and suppose we have two complete hereditary cotorsion pairs $(,\tilde{})$ and $(\tilde{},)$ in satisfying $\tilde{}\subseteq$ and $\cap \tilde{}=\tilde{}\cap$. We show how to construct a (necessarily unique) abelian model structure on with (resp. $\tilde{}$) as the class of cofibrant (resp. trivially cofibrant) objects, and (resp. $\tilde{}$) as the class of fibrant (resp. trivially fibrant) objects.