Ordnung muss sein

Abstract

For any length category, we establish a set of rules (necessary and sufficient) that ensure a partial order on the isomorphism classes of simple objects such that the category is equivalent to the category of finite dimensional representations of this partially ordered set. Equivalently, we characterise the length categories that arise as categories of modules over a sheaf of division rings on a finite $T_0$-space.

12 pages. English translation of the title: "There must be order". Thorough revision of the first version, providing generalisations in two directions: the endomorphim rings of the simple objects may vary and the posets may be infinite