Problem: Prove that for vectors $ v, w$ in an inner product space, the inequality $$\displaystyle |\left \langle v, w \right \rangle | \leq \| v \| \| w \|$$ Solution: There is an elementary proof of the Cauchy-Schwarz inequality (see the Wikipedia article), and this proof is essentially the same. What makes this proof stand out is its insightful technique, which I first read about on Terry Tao’s blog.

Last time we worked through some basic examples of universal properties, specifically singling out quotients, products, and coproducts. There are many many more universal properties that we will mention as we encounter them, but there is one crucial topic in category theory that we have only hinted at: functoriality. As we’ve repeatedly stressed, the meat of category theory is in the morphisms.

Problem: Given a data stream of unknown size $ n$, pick an entry uniformly at random. That is, each entry has a $ 1/n$ chance of being chosen. Solution: (in Python) import random def reservoirSample(stream): for k,x in enumerate(stream, start=1): if random.random() < 1.0 / k: chosen = x return chosen Discussion: This is one of many techniques used to solve a problem called reservoir sampling.

Guest Post: Torus-Knotted Baklava at Baking And Math, a blog by my friend and colleague, Yen.

Problem: Determine if a number is prime, with an acceptably small error rate.

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 The Simons Institute at UC Berkeley 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.