Reading group on abstract homotopy theory
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The idea is to have a reading group to familiarise ourselves with simplicial model categories and applications such as to homotopy (co)limits, ∞-categories, and stabilisation.
Details are to be discussed in the preliminary meeting (see organisational), but meetings will be weekly, each discussing sections of the chosen references.
Organisational
- When and where: We meet on Wednesdays 10:00 - 12:00 in JCMB 5323 (next to the common room)
- Preliminary meeting: Friday, 13/01/17, 10am, JCMB 3212
- 18/01/17: Room change: We will meet in JCMB 4312 this week
Chapter 1 in [Hov], as much as you feel like ([HoT1] as an alternative, but probably quite similar),
Refreshing categorical tools (e.g. Kan extensions from [CT3], (co)ends from [CT4], although not necessary yet)
- 25/01/17: Chapter 1 in [Hov]
- 01/02/17: Sections 2.1, 2.2 and 2.3
- 08/02/17: Read sections 2.4, 3.1 and 3.2, and bring along your favourite simplicial set:-)
- 15/02/17: Sections 3.3 - 3.6, especially homotopy groups and geometric realisation
- 22/02/17: Chapter 4 - especially simplicial model categories
- 01/03/17: No meeting this week
- 09/03/17: Chapter 5 ("let's see how far we get") - Note the time change: it's Thursday this week!
- 15/03/17: Sections 5.3, 5.4, 6.1, 6.2, and bring your favourite suspension or loop space
- 22/03/17: Finish chapter 6
- 29/03/17: No meeting this week
- 05/04/17: We will discuss [HC5]
- 12/04/17: Chapter 1 in [Rie], supplemented by some [CT2] and [CT4] if you fancy it, and bring your favourite Kan extension, end, or coend
- 19/04/17: Up to and including Section 3.3 in [Rie]
- 26/04/17: Finish chapter 3
- 03/05/17: Chapter 4
- 10/05/17: Chapter 5
- 17/05/17: Chapter 6, we might review 4.4 and 5.2.6
- 24/05/17: No meeting
- 31/05/17: Brainstorming session
Literature
I have compiled a list of literature that might be interesting.
The key references will be Hovey and, if we make it through that book, Riehl.
The other references are supplementary, or things we might find interesting in relation with the reading group.
- Core references:
- [Hov] Mark Hovey - Model categories: Basis for the (first part of the) reading group
- [Rie] Emily Riehl - Categorical homotopy theory: Second part of the reading group, great treatment of more modern developments and techniques in homotopy theory, leading on to ∞-categories
- More on homotopy theory:
- [HoT1] William G. Dwyer, Jan Spalinski - Homotopy theories and model categories: Very good and self-contained introduction to model categories (no simplicial techniques though)
- [HoT2] Philip S. Hirschhorn - Model categories and their localisations: technical but complete reference for advanced topics regarding model categories (hocolims, localisations,...)
- [HoT5] Paul G. Goerss, John F. Jardine - Simplicial homotopy theory: A detailed treatment of simplicial sets and simplicial model categories, features Dold-Kan (for which we should also keep Weibel in mind!)
- [HoT6] Emily Riehl - A leisurely introduction to simplicial sets
- Categories:
- [CT1] Tom Leinster - Basic category theory: Wonderful short but self-contained introduction to the basics of categories
- [CT2] Saunders Mac Lane - Categories for the working mathematician: more extensive introduction to categories, has more advanced topics (e.g. (co)ends,...)
- [CT4] Fosco Loregian - This is the (co)end, my only (co)friend
- Higher categories:
- [HC1] Moritz Groth - A short course on ∞-categories: seemingly good introduction/overview
- [HC2] Jacob Lurie - Higher topos theory: comprehensive standard reference on quasi-categories
- [HC3] Charles Rezk - Stuff about quasicategories
- [HC4] Emily Riehl - Quasicategories as (∞,1)-categories
- [HC5] Charles Rezk - A model for the homotopy theory of homotopy theory
- Further reading, applications: I hope I found something for everybody;-)
- [FR1] Daniel Freed, Michael Hopkins - Chern-Weil forms and abstract homotopy theory
- [FR2] Bertrand Toen, Gabriele Vezzosi - Homotopical Algebraic Geometry I: Topos theory
- [FR3] Mark Hovey, Brooke Shipley, and Jeff Smith - Symmetric spectra
- [FR4] Sharon Hollander - A homotopy theory for stacks
- [FR5] Christopher Schommer-Pries - Dualisability in low-dimensional higher category theory
- [FR6] Charles Rezk - A model category for categories
- [FR7] Daniel Dugger - Sheaves and homotopy theory
- ...