Computational Category Theory (404 as of 21 August 2015)
Published in 1990 by D.E. Rydeheard and R.M. Burstall, Computational Category Theory uses ML to illustrate the relevance of category theory to CS.
Updated link: Computational Category Theory by D.E. Rydeheard and R.M. Burstall.
BTW, ML code is available as well.
Enjoy!
Introduction to Categories and Categorical Logic by Samson Abramsky and Nikos Tzevelekos.
Category theory is important for theoretical CS and I suspect should skilled explanations come along, CS practice as well.
From the preface:
The aim of these notes is to provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The notes are based on a lecture course given at Oxford over the past few years. They contain numerous exercises, and hopefully will prove useful for self-study by those seeking a first introduction to the subject, with fairly minimal prerequisites. The coverage is by no means comprehensive, but should provide a good basis for further study; a guide to further reading is included.
The main prerequisite is a basic familiarity with the elements of discrete mathematics: sets, relations and functions. An Appendix contains a summary of what we will need, and it may be useful to review this first. In addition, some prior exposure to abstract algebra—vector spaces and linear maps, or groups and group homomorphisms—would be helpful.
Under further reading, F. Lawvere and S. Schanuel, Conceptual Mathematics: A First Introduction to Categories, Cambridge University Press, 1997, is described by the authors as “idiosyncratic.” Perhaps so but I found it to be a useful introduction.
Applicatives are generalized functors
A continuation of Heiko Seeberger’s coverage of Scala and category theory.
Highly recommended.
I ran across Alfred Korzybski’s dictum “…the map is not the territory…” the other day.
I’ve repeated it and have heard others repeat it.
Not to mention it being quoted in any number of books on mapping and mapping technologies.
It’s a natural distinction, between the artifact of a map and the territory it is mapping.
But it is important to note that Korzbski did not say “…a map cannot be a territory….”
Like the wave/particle duality in quantum mechanics, maps can be maps or they can be territories.
Depends upon the purpose with which we are viewing them.
A rather wicked observer effect that changes the formal properties of a map vis-a-vis a territory to being the properties of a territory vis-a-vis a map.
Maps (that is syntaxes/data models) try to avoid that observer effect by proclaiming themselves to be the best possible of all possible maps in the traditional of Dr. Pangloss.
They may be the best map for some situation, but they remain subject to being viewed as a territory, should the occasion arise.
(If that sounds like category theory to you, give yourself a gold star.)
The map-as-territory principle is what enables the viewing of subject representatives in different maps as representatives of the same subjects.
Otherwise, we must await the arrival of the universal mapping strategy.
It is due to arrive on the same train as the universal subject identifier for all subjects, for all situations and time periods.
From the Preface (in the book):
This book should be helpful to computer scientists wishing to understand the computational significance of theorems in category theory and the constructions carried out in their proofs. Specialists in programming languages should be interested in the use of a functional programming language in this novel domain of application, particularly in the way in which the structure of programs is inherited from that of the mathematics. It should also be of interest to mathematicians familiar with category theory – they may not be aware of the computational significance of the constructions arising in categorical proofs.
In general, we are engaged in a bridge-building exercise between category theory and computer programming. Our efforts are a first attempt at connecting the abstract mathematics with concrete programs, whereas others have applied categorical ideas to the theory of computation.
Website has a pdf of a book by the same title and source code for programs.
This may not be useful for the “meatball” semantics of everyday practice, developing tools for use in the design information systems is another matter.
Questions:
Introduction to Category Theory in Scala.
Jack Park and I have been bouncing posts about category theory resources off of each other for years.
This one looks like a keeper.
It may be the sort of series that acts as a bridge between an abstraction (category theory) and the real world of programming.
They are related you know.
Powered by WordPress