An Introduction to Category Theory

Going beyond the partial functions and multifunctions already considered, one might invent other useful notions of the input/output function from X to Y. In addition to the need to considerX, Y as “data structures,” there are theoretical approaches to semantics in which all X, Y must carry further structure. Rather than embark on the misguided task of presenting an exhaustive list of present and future possibilities, we introducecategories as a framework for semantics which possess so little structure that most models of semantics can be represented this way. Surprisingly, what structure remains can be extensively developed and there is a great deal to say.