Summary: Associating categories with measured or observed attributes is a central challenge for discrete mathematics in life sciences. We propose a new concept to formalize this question: Given a binary matrix of objects and attributes, determine all attribute sets characterizing object sets of cardinality \(t_{1}\) that do not characterize any object set of size \(t_{2}>t_{1}\). We determine how many such attribute sets exist, give an output-sensitive quasi-polynomial time algorithm to determine them, and show that \(k\)-sum matrix decompositions known from matroid theory are compatible with the characterization.