Categories as algebra: An essential ingredient in the theory of monoids

Abstract

The study of categories as generalized monoids is shown to be essential to the understanding of monoid decomposition theory. A new ordering for categories, called division, is introduced which allows useful comparison of monoids and categories. Associated with every morphism φ{symbol}: M → N is a category Dφ{symbol}, called the derived category of φ{symbol}, which encodes the essential information about the morphism. The derived category is shown to be a rightful generalization of the kernel of a group morphism. Collections of categories that admit direct products and division are studied. Called varieties, these collections are shown to be completely determined by the path equations their members satisfy. Several important examples are discussed. In a major application, a strong connection is established between category varieties and certain semigroup varieties associated with recognizable languages. © 1987.