The Gene Ontology Categorizer, developed jointly by the Los Alamos National Laboratory and Procter & Gamble Corp., provides a capability for the categorization task in the Gene Ontology (GO): given a list of genes of interest, what are the best nodes of the GO to summarize or categorize that list? The motivating question is from a drug discovery process, where after some gene expression analysis experiment, we wish to understand the overall effect of some cell treatment or condition by identifying where in the GO the differentially expressed genes fall: clustered together in one place? in two places? uniformly spread throughout the GO? high, or low? In order to address this need, we view bio-ontologies more as combinatorially structured databases than facilities for logical inference, and draw on the discrete mathematics of finite partially ordered sets (posets) to develop data representation and algorithms appropriate for the GO. In doing so, we have laid the foundations for a general set of methods to address not just the categorization task, but also other tasks (e.g. distances in ontologies and ontology merger and exchange) in both the GO and other bio-ontologies (such as the Enzyme Commission database or the MEdical Subject Headings) cast as hierarchically structured taxonomic knowledge systems.