I introduce and develop the theory of
model ∞-categories -- that is, of model structures on ∞-categories -- which provides a robust theory of resolutions that is entirely native to the ∞-categorical context (papers 3, 6, 7, 8, 9, and 10). Using this, I generalize
Goerss--Hopkins obstruction theory to an arbitrary (presentably symmetric monoidal stable) ∞-category (paper 14). As a sample application, I use this generalized obstruction theory to construct E
∞ structures on the motivic Morava E-theory spectra (paper 15); just as in the classical case, these E
∞ structures turn out to be essentially unique, and their automorphism groups turn out to be essentially discrete (though they will generally be strictly larger than the usual Morava stabilizer group).
I have split out the first section of the introductory chapter into an essay called The zen of ∞-categories, which you can see here. This is an introduction to abstract homotopy theory. In the interest of accessibility to a broad mathematical audience, it is centered around the classical theory of abelian categories, chain complexes, derived categories, and derived functors.
I've given a number of talks about this material, most of which have been entitled Every love story is a GHOsT story: Goerss--Hopkins obstruction theory for ∞-categories (a reference to one of my preferred authors).
Among these, I first shared my work in a two-hour talk at the Harvard Thursday seminar at the end of the fall 2013 semester. The slides for that talk are here. As I explained the Blanc--Dwyer--Goerss obstruction theory I drew an accompanying diagram, a dramatic reenactment of which you can find here. Moreover, there were a lot of things that I wanted to say that I didn't actually put into the slides themselves (which as you can see are nevertheless still quite overloaded), and many of those things are collected here.
I also gave a 30-minute talk on this work at the 2014 Young Topologists Meeting in Copenhagen, in which of course I had to get to the point much more quickly. The slides for that talk are here. I think these provide a much better and cleaner introduction to the theory, and as a bonus they also contain some new and (to my mind) compelling results about model ∞-categories.
You can also see a movie adaptation of my thesis here.
I last updated my thesis on 4/8/2019. I am happy to receive any comments, errata, typos, etc.
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