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