### Categorical Homotopy Theory

#### Posted by Emily Riehl

In my first year at Harvard, I had an opportunity to teach a graduate-level topics course entitled “Categorical Homotopy Theory.” Its aim was to highlight areas in which category theoretic abstractions provide a particularly valuable insight into classical homotopy theoretic constructions. Over the course of the semester I gave lectures that focused on homotopy limits and colimits, enriched category theory, model categories, and quasi-categories.

In hopes that attendees would be able to drop in and out without feeling totally lost, I decided to write lecture notes. And now they have just been published by Cambridge University Press as an actual physical book and also as an ebook (or so I’m told).

One of the wonderful things about working with CUP is that they have given me permission to host a free PDF copy of the book on my website. At the moment, this is the pre-copyedited version. There is an extra section missing from chapter 14 and various minor changes made throughout. In a few years time, I’ll be able to post the actual published version.

So what’s in the book? Part I tells a story I learned from a really fantastic paper written by Mike Shulman. It introduces a particular model for the homotopy limit and colimit functors associated to diagrams of any shape with which it is easy to prove their global universal property (as point-set level derived functors) and their local universal property (representing “homotopy coherent cones”). The proof makes use of an independently useful observation of Dwyer-Kan-Hirschhorn-Smith that full model structures are not necessary to define derived functors. In this case, this means that we don’t need to establish model structures on the diagram categories.

Our particular model for homotopy colimits is first defined via the two-sided bar construction, but it is later re-expressed as a weighted colimit, from which viewpoint it is recognizable as the Bousfield-Kan formula. This emphasis might be slightly unusual — a homotopy limit or colimit is something that is weakly equivalent to a particular ur-model — but I think it can be valuable. A number of homotopical theorems have an up-to-isomorphism component, which can be easier to understand. (For instance, the left adjoint of a simplicial Quillen adjunction preserves homotopy colimits, as weighted colimits.)

Part II continues with the study of enriched homotopy theory. We show that the total derived functors of simplicially enriched functors between simplicial model categories are enriched over the homotopy category of spaces. I like to think of this derived enrichment as a proxy for the “homotopical correctness” of the functors. There is also a chapter giving a fairly detailed introduction to weighted limits and colimits, which (unsurprisingly) turn out to be the key categorical tool used in proofs throughout the manuscript.

Part III finally turns focus to Quillen’s model categories, which are black boxed in the first half of the book as good settings in which to implement the Dwyer-Kan-Hirschhorn-Smith axiomatization. Given the wealth of excellent textbooks and surveys on the topic, this section isn’t meant to serve as a comprehensive introduction to model categories. For instance, I say very little about the construction of the homotopy category of a model category. Instead, I develop the theory of weak factorization systems leading up to André Joyal’s definition of a model category: a model category is a homotopical category equipped with classes of cofibrations and fibrations that combine with the weak equivalences to define a pair of weak factorization systems.

As won’t surprise anyone familiar with my thesis work, I spend a fair amount of time discussing the small object argument, both in its original form and in its modern algebraic variant, due to Richard Garner. I then segue into the enriched small object argument and its accompanying enriched weak factorization systems. This definition, which is the obvious enrichment of the usual notion, isn’t well-known, but I think it is interesting. The weak factorization systems in any simplicial model category are automatically enriched. (This is true more generally for any $V$-model category in which tensoring with an object of $V$ defines a left Quillen functor.) Equally interesting is when this does not hold: for instance, for Quillen’s model structure on the category of chain complexes over a ring admitting non-projective modules. In this case it is most productive to think about enrichment over the category of modules (not thought of as a model category). Surprisingly, the usual generating cofibrations and trivial cofibrations for the Quillen model structure also generate the Hurewicz model structure, when we interpret “cofibrant generation” in a non-enriched sense. Some of the details can be found here.

Part III closes with a chapter of Reedy categories that describes a small part of a joint paper with Dominic Verity, connects these ideas to the Bousfield-Kan approach to localizations and completions of spaces, and closes up some loose ends from earlier in the book

The final part is about quasi-categories. My aim here, given the location of the course, was to overlap as little as possible with *Higher Topos Theory* and so avoid boring Jacob’s students. The first chapter focuses on the construction of and comparison between various models of mapping spaces between vertices in a quasi-category, explaining why quasi-categories are $(\infty,1)$-categories. A second chapter discusses simplicial categories, which provide an important source of examples of quasi-categories, and homotopy coherence.

I then study *isomorphisms* in quasi-categories, by which I mean 1-simplices that become invertible in the homotopy category of a quasi-category. These are usually called equivalences, but I think this terminology is better. There’s no possibility of confusing with any stricter notion, and it allows for weaker notions of equivalence, which might be of interest for constructing localizations or the like. The final chapter is a very glancing preview of joint with work Dom on the 2-category theory of quasi-categories and its sequels.

For those who are curious, here is the table of contents. Should you happen to read any of this, I hope you enjoy it!

Part I. Derived functors and homotopy (co)limits

Chapter 1. All concepts are Kan extensions

Chapter 2. Derived functors via deformations

Chapter 3. Basic concepts of enriched category theory

Chapter 4. The unreasonably effective (co)bar construction

Chapter 5. Homotopy limits and colimits: the theory

Chapter 6. Homotopy limits and colimits: the practice

Part II. Enriched homotopy theory

Chapter 7. Weighted limits and colimits

Chapter 8. Categorical tools for homotopy (co)limit computations

Chapter 9. Weighted homotopy limits and colimits

Chapter 10. Derived enrichment

Part III. Model categories and weak factorization systems

Chapter 11. Weak factorization systems in model categories

Chapter 12. Algebraic perspectives on the small object argument

Chapter 13. Enriched factorizations and enriched lifting properties

Chapter 14. A brief tour of Reedy category theory

Part IV. Quasi-categories

Chapter 15. Preliminaries on quasi-categories

Chapter 16. Simplicial categories and homotopy coherence

Chapter 17. Isomorphisms in quasi-categories

Chapter 18. A sampling of 2-categorical aspects of quasi-category theory

## Re: Categorical Homotopy Theory

Congratulations! What a superb feeling to be able to hold the book in your hand. I assume that hand is yours — the reflection in the cover isn’t clear enough to tell, and the phone’s blocking the view of whoever-it-is’s face.

And congratulations too on arranging with CUP to make the book freely available. Anyone reading this who’s thinking of writing a book should strongly consider CUP; they have an excellent track record of allowing free online versions (perhaps most famously with Allen Hatcher’s topology text).