There has been a lot of news recently on government surveillance of its citizens. The biggest two that have pervaded my news feeds are the protests in Turkey, which in particular have resulted in particular oppression of social media users, and the recent light on the US National Security Agency’s widespread “backdoor” in industry databases at Google, Verizon, Facebook, and others.

I’ve been spending a little less time on my blog recently then I’d like to, but for good reason: I’ve been attending two weeks of research conferences, I’m getting ready for a summer internship in cybersecurity, and I’ve finally chosen an advisor. Visions, STOC, and CCC I’ve been taking a break from the Midwest for the last two weeks to attend some of this year’s seminal computer science theory conferences.

Last time we defined and gave some examples of rings. Recapping, a ring is a special kind of group with an additional multiplication operation that “plays nicely” with addition. The important thing to remember is that a ring is intended to remind us arithmetic with integers (though not too much: multiplication in a ring need not be commutative). We proved some basic properties, like zero being unique and negation being well-behaved.

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.

This post is mainly mathematical. We left it out of our introduction to categories for brevity, but we should lay these definitions down and some examples before continuing on to universal properties and doing more computation. The reader should feel free to skip this post and return to it later when the words “isomorphism,” “monomorphism,” and “epimorphism” come up again.

Pablo Picasso Simplicity and the Artist Some of my favorite of Pablo Picasso’s works are his line drawings. He did a number of them about animals: an owl, a camel, a butterfly, etc. This piece called “Dog” is on my wall: Dachshund-Picasso-Sketch (Jump to interactive demo where we recreate “Dog” using the math in this post) These paintings are extremely simple but somehow strike the viewer as deeply profound.

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. The reader unfamiliar with the ML programming language should consult our earlier primer.

Previously on this blog, we’ve covered two major kinds of algebraic objects: the vector space and the group. There are at least two more fundamental algebraic objects every mathematician should something know about. The first, and the focus of this primer, is the ring. The second, which we’ve mentioned briefly in passing on this blog, is the field.

For a list of all the posts on Category Theory, see the Main Content page. 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.

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.

Probabilistic arguments are a key tool for the analysis of algorithms in machine learning theory and probability theory.