A chain complex can be viewed as a representation of a certain self-injective quiver with relations, Q Q . To define Q Q , include a vertex q n q_n and an arrow q n → ∂ q n − 1 q_n \xrightarrow {\partial } q_{n-1} for each integer n n . The relations are ∂ 2 = 0 \partial ^2 = 0 . Replacing Q Q by a general self-injective quiver with relations, it turns out that some of the key properties of chain complexes generalise. Indeed, consider the representations of such a Q Q with values in A Mod {}_{A}\mspace {-1mu}\operatorname {Mod} where A A is a ring. We showed in earlier work that these representations form the objects of the Q Q -shaped derived category, D Q ( A ) \mathcal {D}_{Q}(A) , which is triangulated and generalises the classic derived category D ( A ) \mathcal {D}_{}(A) . This follows ideas of Iyama and Minamoto. While D Q ( A ) \mathcal {D}_{Q}(A) has many good properties, it can also diverge dramatically from D ( A ) \mathcal {D}_{}(A) . For instance, let Q Q be the quiver with one vertex q q , one loop ∂ \partial , and the relation ∂ 2 = 0 \partial ^2 = 0 . By analogy with perfect complexes in the classic derived category, one may expect that a representation with a finitely generated free module placed at q q is a compact object of D Q ( A ) \mathcal {D}_{Q}(A) , but we will show that this is, in general, false. The purpose of this paper, then, is to compare and contrast D Q ( A ) \mathcal {D}_{Q}(A) and D ( A ) \mathcal {D}_{}(A) by investigating several key classes of objects: Perfect and strictly perfect, compact, fibrant, and cofibrant.