CONCEPT ANALYSIS IN CATEGORIES

Abstract

Network computation requires concept mining and analysis to extract semantics shared between nodes. The impor- tance of this has grown immensely with the advent of the web, and the distribution of content through channels relies on concept analysis. While the methods of concept extraction vary widely, many of them appear to lend themselves to a categorical description in which the basis is an enriched matrix. The development of a categorical toolkit for concept mining and analysis is currently in its infancy. We further the development of this toolkit by focusing on set-valued matrices, constructing a category of concepts, and defining a formal concept in this new setting. Understanding this concept category is an exercise in understanding the structures of categories themselves. However, categories were introduced as a tool to understand structure without being concerned with the inner details of objects, focusing on morphisms and their composition. With the current presentation, we must carry objects along the path to the concept category, without ever appealing to their inner structure. To provide a less cumbersome description of the situation, we forget the objects of a category, and then state precisely when the objects can be recovered. This establishes an equiv- alence between certain categories and partial semigroups, similar to the theory of locales in pointless topology. This equivalence is intended to provide a supplemental approach to categories that is suitable for situations when objects prove to be unnecessary or clumsy, in particular, the concept category.