Ordnung muss sein

TL;DR

This paper characterizes length categories that are equivalent to categories of modules over sheaves of division rings on finite T0-spaces, establishing necessary and sufficient rules for a partial order on simple objects.

Contribution

It provides a complete set of rules to identify when a length category corresponds to modules over a sheaf of division rings on finite T0-spaces.

Findings

Established necessary and sufficient conditions for a partial order on simple objects.

Characterized length categories as categories of modules over sheaves of division rings.

Connected length categories with topological structures on finite T0-spaces.

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.