Another Word For It Patrick Durusau on Topic Maps and Semantic Diversity

Category Theory References

Ten (10) pages of category theory citations that I bookmarked recently.

The citations are not annotated so they are of limited utility but it looked worth passing along.

Are there any ongoing annotated lists of references for category theory?

The American Mathematical Association (AMA) indexing scheme for 18 Category theory; homological algebra isn’t detailed enough to substitute for an annotated listing. (Be aware that category theory appears under other classifications so use the search function for 2010 Mathematics Subject Classification if you want to find all appearances of category theory.)

Enjoy!

Category Theory Using String Diagrams by Dan Marsden.

Abstract:

In work of Fokkinga and Meertens a calculational approach to category theory is developed. The scheme has many merits, but sacrifices useful type information in the move to an equational style of reasoning. By contrast, traditional proofs by diagram pasting retain the vital type information, but poorly express the reasoning and development of categorical proofs. In order to combine the strengths of these two perspectives, we propose the use of string diagrams, common folklore in the category theory community, allowing us to retain the type information whilst pursuing a calculational form of proof. These graphical representations provide a topological perspective on categorical proofs, and silently handle functoriality and naturality conditions that require awkward bookkeeping in more traditional notation.

Our approach is to proceed primarily by example, systematically applying graphical techniques to many aspects of category theory. We develop string diagrammatic formulations of many common notions, including adjunctions, monads, Kan extensions, limits and colimits. We describe representable functors graphically, and exploit these as a uniform source of graphical calculation rules for many category theoretic concepts. We then use these graphical tools to explicitly prove many standard results in our proposed string diagram based style of proof.

This form of visualization does seem to be easier on the eyes. 😉

Whether it is sufficient or not for some particular purpose, remains to be seen.

Notes on Category Theory by Robert L. Knighten.

From the preface:

There are many fine articles, notes, and books on category theory, so what is the excuse for publishing yet another tome on the subject? My initial excuse was altruistic, a student asked for help in learning the subject and none of the available sources was quite appropriate. But ultimately I recognized the personal and selfish desire to produce my own exposition of the subject. Despite that I have some hope that other students of the subject will find these notes useful.

The other generous explanation for “another tome” is to completely master a subject.

Either teach it or write a master tome on it.

I first saw this in a tweet by OGE Search.

Category Theory for Scientists by David Spivak.

Abstract:

There are many books designed to introduce category theory to either a mathematical audience or a computer science audience. In this book, our audience is the broader scientific community. We attempt to show that category theory can be applied throughout the sciences as a framework for modeling phenomena and communicating results. In order to target the scientific audience, this book is example-based rather than proof-based. For example, monoids are framed in terms of agents acting on objects, sheaves are introduced with primary examples coming from geography, and colored operads are discussed in terms of their ability to model self-similarity.

I first saw this at: Category Theory for Scientists by John Baez.

I forwarded it to Jack Park who responded with a link to an earlier post: Spivak on Category Theory by Bruce Bartlett.

David Spivak responds to the earlier post with a link to a Google doc for posting comments:

CT4S book: Typos, comments, questions, and suggestions

The MIT course description with links to supplemental materials: 18-S996, Spring 2013: Category theory for scientists.

If you post about this book, please include a pointer to the Google doc for comments as well.

Universal Properties by Jeremy Kun.

From the post:

Previously in this series we’ve seen the definition of a category and a bunch of examples, basic properties of morphisms, and a first look at how to represent categories as types in ML. In this post we’ll expand these ideas and introduce the notion of a universal property. We’ll see examples from mathematics and write some programs which simultaneously prove certain objects have universal properties and construct the morphisms involved.

Just in time for a long holiday weekend in the U.S., Jeremy continues his series on category theory.

Enjoy!

Categories as Types by Jeremy Kun.

From the post:

In this post we’ll get a quick look at two ways to define a category as a type in ML. The first way will be completely trivial: we’ll just write it as a tuple of functions. The second will involve the terribly-named “functor” expression in ML, which allows one to give a bit more structure on data types.

Jeremy gets us closer to application of category theory to programming with an introduction to types.

Introducing Categories by Jeremy Kun.

From the post:

It is time for us to formally define what a category is, to see a wealth of examples. In our next post we’ll see how the definitions laid out here translate to programming constructs. As we’ve said in our soft motivational post on categories, the point of category theory is to organize mathematical structures across various disciplines into a unified language. As such, most of this post will be devote to laying down the definition of a category and the associated notation. We will be as clear as possible to avoid a notational barrier for newcomers, so if anything is unclear we will clarify it in the comments.

Definition of a Category

Let’s recall some examples of categories we’ve seen on this blog that serve to motivate the abstract definition of a category. We expect the reader to be comfortable with sets, and to absorb or glaze over the other examples as comfort dictates. The reader who is uncomfortable with sets and functions on sets should stop here. Instead, visit our primers on proof techniques, which doubles as a primer on set theory (or our terser primer on set theory from a two years ago).

The go-to example of a category is that of sets: sets together with functions between sets form a category. We will state exactly what this means momentarily, but first some examples of categories of “sets with structure” and “structure-preserving maps.”

Not easy but not as difficult as some introductions to category theory.

Jeremy promises that the very next post jumps into code to show the “definition of a category as a type in ML.”

Plus some pro’s and con’s on proceeding this way.

Categories, What’s the Point? by Jeremy Kun.

From the post:

Perhaps primarily due to the prominence of monads in the Haskell programming language, programmers are often curious about category theory. Proponents of Haskell and other functional languages can put category-theoretic concepts on a pedestal or in a mexican restaurant, and their benefits can seem as mysterious as they are magical. For instance, the most common use of a monad in Haskell is to simulate the mutation of immutable data. Others include suspending and backtracking computations, and even untying tangled rope.

Category theory is often mocked (or praised) as the be-all and end-all of mathematical abstraction, and as such (and for other reasons I’ve explored on this blog) many have found it difficult to digest and impossible to master. However, in truth category theory arose from a need for organizing mathematical ideas based on their shared structure. In this post, I want to give a brief overview of what purpose category theory serves within mathematics, and emphasize what direction we’ll be going with it on this blog.

We should also note (writing this after the fact), that this article is meant to be a motivation to our future series on category theory. It is very difficult to explain what category theory is without going into very specific details, but we can explain by analogy what category theory achieves for us. That is the goal of this post.

I rather like the line:

However, in truth category theory arose from a need for organizing mathematical ideas based on their shared structure.

Which is one of the reasons why I keep trying to master category theory, for the ability to organize subject identifications based on their shared structures.

Not necessary for all subject identity cases but if topic maps extend into subjects like math, physics, etc., then it may be quite useful.