A categorical manifesto

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. IntroductionIn this paper we will try 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 (see section 1.2) 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 formulationscan reveal surprising connections. For example, an analogy between Petri nets and the A-calculus might suggest looking for a closed category structure on the category of Petri nets (Meseguer and Montanari, 1988).-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 (given in section 1) is that two objects are 'abstractly the same' iff they are isomorphic; see section 1.1. Moreover, universal

Discovering Empirical Theories of Modular Software Systems. An Algebraic Approach

“…Each model in this hierarchy is a model of the entire system, as in the former approach of Angius and Tamburrini (2011). We maintain that the process, conceived in the work of (Burstall and Goguen 1977;Goguen 1991;Goguen and Burstall 1992), of getting from module specifications to software specification, resembles the process of discovery of semantic theories of modular software systems. Depending on the kind of predictions and explanation one is seeking, and on the observed executions one would like to model, the modularity of the built semantic theory can be decreased by computing bigger Institutions describing a higher number of potential computations, until a non-modular semantic theory is achieved.…”

“…It was introduced by Goguen and Burstall (1992) to face the vast number of formalisms characterising common specification activities. Based on category theory (Goguen 1991), Institutions abstract from both the syntax and the semantics of a given language to focus on the satisfaction relation of models. Institutions accomplish the Tarskian satisfaction condition requiring that truth is invariant under change of notation (Tarski 1944).…”

A categorical model of model merging and weaving

“…The use of colimits (a generalisation of pushouts in category theory) to perform merging has been encouraged since the 90s by Goguen [13] and already been applied to merge models, e.g. in the TreMer tool [14], [1].…”

Automating Proofs in Category Theory

“…A more efficient way is to write tactics based on formal theorems that establish properties of the fundamental concepts. For instance, rule (14) corresponds to the theorem…”

“…Apart from the fact that rules have to be formulated as top-down sequent rules to accommodate Nuprl's goal-oriented reasoning style, the representation of the rules in the system is identical to the version on paper, which makes it easy to check its faithfulness. Rule (14), for instance, is represented by a rule object NatTransApply with the following contents.…”