Dominant and codominant dimensions for quiver representations

Let M be a module category and Q a rooted quiver. In this paper, we study the dominant (resp. codominant) dimension of the category Rep(Q,M) of M-valued representations of Q. To do this, we first study injective envelopes and projective covers that play important roles in homological algebra and give explicit formulas for them in the category Rep(Q,M), whose origins go back to the classical representation theory of a finite quiver over a field. Then, by using such descriptions, we compute the dominant (resp. codominant) dimension of Rep(Q,M). We show that the dominant dimension of Rep(Q,M) is at most one for every nonzero module category M and any right rooted quiver with at least one arrow.