Pre-abelian clan categories. (Q1415021)

Let \(k\) be a field. The author considers Krull-Schmidt \(k\)-categories \(\mathcal C\) with \(\text{ind}\,\mathcal C\) finite such that \(\hom_{\mathcal C}(I,J)\) is at most one-dimensional for \(I,J\in\text{ind}\,\mathcal C\) and the composition of two non-isomorphisms in \(\text{ind}\,\mathcal C\) is zero. Thus \(\mathcal C\) is essentially given by a quiver \(Q\) with vertex set \(\text{Ob}\,\mathcal C\). The main result states that \(\mathcal C\) is preabelian if and only if \(Q\) has no path of length \(> 2\). Moreover, it is shown that \(\mathcal C\) is abelian if and only if, in addition, every edge of \(Q\) belongs to a path of length two. The author shows that such categories \(\mathcal C\) arise as categories of representations of certain clans without special loops in the sense of \textit{W. W. Crawley-Boevey} [J. Lond. Math. Soc., II. Ser. 40, No. 1, 9--30 (1989; Zbl 0725.16012)].