Tilting and cotilting subcategories in categories of quiver representations

In this paper, we study tilting and cotilting subcategories of the category of representations of a quiver. Let M be an abelian category, Q be a rooted quiver, and Rep(Q,M) be the category of M-valued representations of Q. By using some recent results about cotorsion torsion triples (resp. torsion cotorssion triples), under certain assumptions, we show that if T is a 1-tilting (resp. 1-cotilting) subcategory of M, then the monomorphism category Φ(T) (resp. the epimorphism category Ψ(T)) is a 1-tilting (resp. 1-cotilting) subcategory of Rep(Q,M). Then, we study another types of induced subcategories in Rep(Q,M) and, by using nice descriptions of monomorphism and epimorphism categories, show that if T is a tilting (resp. cotilting) subcategory of M, then the epimorphism category Ψ(T) (resp. the monomorphism category Φ(T)) is a tilting (resp. cotilting) subcategory of Rep(Q,M) for every finite acyclic quiver Q. This result is a generalization of a lemma due to Zhang (2011) about induced cotilting modules and some recent results due to Bauer et al. (2020). We finally extend Zhang’s reciprocity of the monomorphism operator and the left perpendicular operator for cotilting modules to cotilting subcategories. The results give us a systematic method to create new tilting and cotilting subcategories.