Welcome! My name is Tasmin Chu, and I am a U4 student at McGill. I am teaching a mini-course in category theory in Fall 2023. This is my second time teaching this. **When and where:** Burnside 1105E, on Wednesdays, from 12pm to 1pm. The room is one of the big glassed-in rooms near the back. If you cannot get into the 1105 study space, knock on the door and someone should let you in. **GitHub**: Exercises can be found [here](https://github.com/zhaoshenzhai/CatTheory-F23). I invite you to choose an exercise and typeset a full solution to it, then upload it to the linked GitHub repository. **Join my email list**: Interested (even tentatively)? You can always just show up, but if you fill out [this survey](https://forms.gle/u715cHSnXeToX3BV9 "https://forms.gle/u715cHSnXeToX3BV9") it would be great. **Overview:** Always wanted to learn category theory? Are you a fan of abstract nonsense? We'll begin by covering categorical fundamentals: the Yoneda lemma, colimits, limits, and adjunctions. Then, time permitting, we'll hold a series of lectures on special topics: - the Stone representation theorem (which formalizes the fact that totally disconnected, compact Hausdorff spaces are "the same" as Boolean algebras) - Stone-Cech compactification, which gives us a natural way to make a generic topological space $(X, \mathcal{T})$ compact and Hausdorff - ultrafilters, filters, and the $\text{Spec}$ functor - fruitful intersections with analytic logic and model theory (see [here](https://rynchn.github.io/math/math595-2019fa.pdf) for an idea) - probability monads and the Giry monad Likely we will not be able to cover all of these special topics, so we will collectively decide on the topics most interest the class. **Why category theory?** > Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial. (Peter Freyd) Category theory arose in the study of homological algebra, but it quickly grew beyond that. Today, category theory has fruitful intersections with logic (topos theory, homotopy type theory, model theory), topology and algebraic topology, and computer science (functional programming, semantics). It is the basic language of algebraic geometry and topology. Higher category theory is also a field in its own right, and can be very fun. **Who is this for?** Anyone who wants to learn some category theory should come—undergraduate or graduate. Category theory is relatively self-contained, although examples from group theory and topology will be used frequently. Please let me know if you need additional resources in these areas. I believe anyone interested in math, whether they are majoring or not, could get something out of this course. **Where else can I learn this?** There will be some overlap in content between this course and Higher Algebra I, as well as Geometry and Topology 1. However, most of the content cannot be learned at any course in McGill. Someone deeply comfortable with 1-cat theory should probably only attend the second half of this course. **Textbook:** The first section of the course, on category-theoretic fundamentals, will mostly pull from [Emily Riehl's Category Theory in Context](https://math.jhu.edu/~eriehl/context.pdf). The second section of the course, on special topics, will pull from Tai-Danae Bradley, Bryson, and Terilla's [Topology: A Categorical Approach ](https://topology.mitpress.mit.edu/), these [notes](https://personalpages.manchester.ac.uk/staff/Marcus.Tressl/papers/StoneDualityBooleanAlgebras.pdf) on Stone duality, and Ruiyuan Chen's notes on [categorical logic](https://rynchn.github.io/math/math595-2019fa.pdf). **Who:** I will be the main instructor. My friend Owen Rodgers may co-teach if we cover logic and model theory. **What I want you to take away from this:** My main goal is that you open up [nLab](https://ncatlab.org/nlab/show/HomePage) and find it fun, rather than completely intimidating. (This may be too ambitious, as it still occasionally intimidates me...) More to the point, category theory will teach you a basic way of looking at mathematical objects that is highly structural and general. It will make you precise about quantifying the relationships between distinct classes of mathematical objects (e.g. groups vs topological spaces). Also, it is really nice to be comfortable with many of the examples and objects encountered in category theory (posets, ultrafilters, filters, fibrations, fundamental groups). For example, ultraproducts and ultrafilters are useful in non-standard analysis and geometric group theory.