nLab
A Categorical Manifesto

A Categorical Manifesto is an article written by Joseph Goguen, encouraging the use of category theory in computer science. To quote from his introduction:

> This paper tries to explain why category theory is useful in computing science. The > basic answer is that computing science is a young field that is growing rapidly, is > poorly organised, and needs all the help it can get, and that category theory can > provide help with at least the following: > > * Formulating definitions and theories. In computing science, it is often > more > difficult to formulate concepts and results than to give a proof. The > seven guidelines of this paper can help with formulation; the guidelines > can also be > used to measure the elegance and coherence of existing formulations. > > * Carrying out proofs. Once basic concepts have been correctly formulated > in a > categorical language, it often seems that proofs “just happen”: at each > step, > there is a “natural” thing to try, and it works. Diagram chasing provides > many examples of this. It could almost be said that the purpose > of category theory is to reduce all proofs to such simple calculations. > > * Discovering and exploiting relations with other fields. Sufficiently > abstract > formulations can reveal surprising connections. For example, an analogy > between Petri nets and the >5;-calculus might suggest looking for a > closed category > structure on the category of Petri nets. > > * Dealing with abstraction and representation independence. In computing > science, more abstract viewpoints are often more useful, because of the > need to > achieve independence from the often overwhelmingly complex details of how > things are represented or implemented. A corollary of the first guideline > is that two objects are “abstractly the same” if they are isomorphic. > Moreover, universal constructions (i.e., adjoints) define > their results uniquely up to isomorphism, i.e., “abstractly” in just this > sense. > > * Formulating conjectures and research directions. Connections with other > fields > can suggest new questions in your own field. Also the seven guidelines > can help > to guide research. For example, if you have found an interesting functor, > then > you might be well advised to investigate its adjoints. > > * Unification. Computing science is very fragmented, with many different > sub-disciplines having many different schools within them. Hence, we > badly need > the kind of conceptual unification that category theory can provide.

Abstract

> This paper tries to explain why and how category theory is useful in > computing science, by giving guidelines for applying seven basic > categorical concepts: category, functor, natural transformation, limit, > adjoint, colimit and comma category. Some examples, intuition, and > references are given for each concept, but completeness is not attempted. > Some additional categorical concepts and some suggestions for further > research are also mentioned. The paper concludes with some philosophical > discussion.

Reference

Joseph A. Goguen, A Categorical Manifesto. In Mathematical Structures in Computer Science, Vol. 1, No. 1. (1991), pp. 49-67 pdf ps

A Categorical Manifesto. CiteSeerX page.

category: computer science

Revised on April 3, 2013 17:55:19 by Zoran Škoda (161.53.130.104)