Model categories of quiver representations

Gillespie's Theorem gives a systematic way to construct model category structures on C(M), the category of chain complexes over an abelian category M. We can view C(M) as the category of representations of the quiver ⋯→2→1→0→−1→−2→⋯ with the relations that two consecutive arrows compose to 0. This is a self-injective quiver with relations, and we generalise Gillespie's Theorem to other such quivers with relations. There is a large family of these, and following Iyama and Minamoto, their representations can be viewed as generalised chain complexes. Our result gives a systematic way to construct model category structures on many categories. This includes the category of N-periodic chain complexes, the category of N-complexes where ∂N=0, and the category of representations of the repetitive quiver ZAn with mesh relations.