[pl-seminar] PL Seminar

[JOHN CYPHERT <jcyphert@xxxxxxxx>]

Hi all,

It's time to start up the PL seminar for this semester. I've scheduled room 4310 for a PL seminar on Monday, Sept 9 at Noon. 

For those of you who are new, the PL seminar is a semi-regular meeting time, usually Monday at noon, where usually a student gives a presentation to the rest of the group about some PL related topic. We've used this time for a variety of types of presentations: practice talks, paper discussions, tutorials, and topic lectures. It's open to anyone, and fairly informal. Feel free to join on Monday, and you are more than welcome to bring your lunch to the seminar. Also consider giving a presentation yourself about some topic you think everyone will find interesting. On Monday, I'll talk a little more about the seminar and some ideas people have had.

Towards the end of the spring semester, I was giving lectures on category theory. As an excuse to start up the seminar, I'm going to give another lecture on category theory. This time I'm going step into the realm of universal properties. Here's a brief abstract:

Arguably universal properties is the answer to what category theory is useful for. It is often stated that universal properties are found everywhere in mathematics, but we often don't call them out as universal properties. A universal property will define an object and morphisms in a category that is the "most efficient solution" to a certain problem. At a high-level a universal object is an object that satisfies a property such that all other objects that satisfy the property can be "factored" through the universal object. In other words, if you want to understand how an object satisfies a property you can construct this understanding by going through the universal object.

I'm going to introduce universal properties by defining the equaliser, first in set theory, then slowly building to a universal construction. This generalization will hopefully give us some intuition about how a universal construction works. If we have some time, I'll introduce some other basic examples of universal properties moving towards the definition of a categorical limit.

If you missed out on some of my previous lectures, don't sweat. Fortunately, for Monday all you'll need to know is the basic axioms of a category and what a commutative diagram is. You can find these axioms and definitions in this document https://pages.cs.wisc.edu/~jcyphert/categoryTheoryNotes/basics/1_Categories.pdf . 

Also for everyone, I have typeset the notes I was using for some previous lectures and I put them on my webpage https://pages.cs.wisc.edu/~jcyphert/categoryTheoryNotes/. In that directory you can find notes on categories, functors, natural transformations, and universal properties. I'm planning on adding more notes to that directory as I find time to make them.

If anyone has any questions just let me know,

John