Category theory and applications to computer science

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Category theory is the language of many parts of modern mathematics and computer science. One of the tools often used in developing category theory is the distributive law, a form of rewriting rule. An example of this rewriting is familiar from arithmetic where every student learns that a sum of products with a common factor has a shorter expression as the common factor times a sum. When a high level distributive law holds for an ordered structure there are three related comparisons of the structure with its completion. The first comparison is a very canonical inclusion that is essentially one of the first structures found in algebra and the others express completeness and distributivity. This sort of distributivity can be considered for categories instead of just ordered structures. In that case some rather subtle arguments are needed to describe the distributive laws and their relations to the completion. Parallel and sequential computing processes are modeled very fruitfully using category theory. Combining these processes with composition is done using (generalized) products and sums, and once again the idea of distributive law provides guidance. When the language of category theory is used to express database system design it allows a clear criterion for solving an important question known as the the View Update Problem. Similar questions arise elsewhere in computer science and category theoretic criterion works there as well.