I was first introduced to category theory through an excellent, though quite difficult, course taught by Peter Johnstone at Cambridge, where I was a Part III student. A year later, I followed up by reading the Saunders MacLane's Categories for a Working Mathematician cover to cover. And now I'm working my way through Francis Borceux's Handbook of Categorical Algebra. (I try to compensate for a poor memory by being thorough.) Along the way, I've learned quite a lot from informal conversations with a number of people, including Martin Hyland, Peter May, and Mike Shulman, just to name a few.
During my first year at Chicago, I conceived of a series of short expository papers covering various topics in category theory that were beyond Categories in content if not in difficulty. The idea is that the writing process would help solidify my understanding as I attempted to find my footing in new categorical vistas. The topics on my imagined list are only very loosely connected under the heading ''things I'd like to learn about.'' My hope is that by making these papers available on the web that someone someday might find one of them useful.
The first installment, written in the summer of 2008, is a very leisurely introduction to the theory of simplicial sets. When I first began to study these objects seriously, I was being very stupid about them, and consequently it took me quite a while to become truly comfortable with various elementary concepts. My hope is that this document, by explaining the basic notions in great detail, will help smooth the way for others.
A leisurely introduction to simplicial sets
The next installment is essentially lecture notes I've written for a talk given by Mike Shulman in the fall of 2008 on weighted limits and colimits, with some of the preliminary ideas about homs and tensors of bimodules expanded into their full gorey detail. I don't actually define an enriched category or functor, but otherwise the level is appropriate for someone whose knowledge of the subject is more-or-less contained in the first three pages of Max Kelly's Basic Concepts of Enriched Category Theory.
I recently rediscovered a short note I wrote sometime in the spring of 2009 for my advisor that explains a proof I learned from Andre Joyal's unpublished introduction to the theory of quasi-categories. This is NOT an introduction to model categories by any means. Rather, it's an alternative definition of a model structure on a category that an experienced user might find cute.
A concise definition of a model category
Comments or suggestions are always appreciated. Please send to eriehl at math.uchicago.edu.
Coming Soon...