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Stream: community: general

Topic: Course on infinity-categories in Amsterdam

Simon Pepin Lehalleur (May 17 2020 at 11:23):

TL;DR: I am preparing a course on infinity-categories for next semester at the University of Amsterdam, and I would be grateful for suggestions and feedback.

Freeze frame So you are probably wondering how I ended up in that situation... Before Covid-19 came into our lives, I committed to give a master-level course at UvA on $\infty$-categories (i.e., $(\infty,1)$-categories). That seemed like a good idea at the time. Now it looks very likely that the course will be online, which complicates things.

Here are the design criteria I am gravitating towards:

• The course is aimed primarily at master students. The prerequisites are a class in algebraic topology and some exposure to ordinary category theory.

• It will be in a (14 weeks * 2 hours) = 28 hours format, which is not a lot. A mitigating factor is that in the Dutch system, it seems that it is customary to skip over quite a lot of details and that the students are used to check and fill in the blanks.

• I still have to decide on the format of online course. I do not have any fancy tablet and I am not sure whether to buy one for this. Otherwise, it should be possible to either record the course in an empty classroom or to prepare slides.

• The main goal is to help students “graduate” from usual (1-)category theory to $\infty$-categories, mostly considered as a tool for other parts of math. Ideally, at the end, the students should be able to start reading papers written in the language. My background is in algebraic geometry and algebraic topology and this will certainly guide my choice of examples and possible applications.

• The focus is on generalising ideas and theorems from categories to $\infty$-categories,
  and explaining what goes through directly and what requires additional work.

• The model of choice is quasicategories à la Joyal-Lurie, but I want to gradually move the focus from the simplicial minutiae to the properties of quasicategories which are invariant under categorical equivalences and to "model-independent" reasoning.

• This is an overview course. This means that I will not go over many of the more technical proofs,
  I try to avoid model category techniques for as long as possible, although the ideas underlying model categories will be introduced along the way (lifting properties, factorisation systems, etc.).

• Towards the end, I will try to give a flavour of more advanced topics like presentable $\infty$-categories, monoidal $\infty$-categories and stable $\infty$-categories (but almost certainly not $\infty$-topoi...). This last part will be tailored depending on the students' interests and motivations for learning all this stuff.

• Some references I like: Lurie's Higher Topos Theory and Higher Algebra (and Kerodon!), Cisinski's Higher Categories and homotopical algebra, Riehl's Categorical Homotopy theory, Rezk's online lecture notes, Groth's Short course on $\infty$-categories.

How reasonable does this seem so far? Does anyone have some experience with teaching this material? Have I missed your favourite reference on the subject?

Morgan Rogers (he/him) (May 17 2020 at 12:11):

Simon Pepin Lehalleur said:

• The main goal is to help students “graduate” from usual (1-)category theory to $\infty$-categories, mostly considered as a tool for other parts of math. Ideally, at the end, the students should be able to start reading papers written in the language. My background is in algebraic geometry and algebraic topology and this will certainly guide my choice of examples and possible applications.

I would love to experience this kind of graduation! Many results about toposes are presented in an infinity-form (by Lurie, say), and I find this rather intimidating. Even if you're not covering $\infty$-toposes, this could be useful for me, so...

Now it looks very likely that the course will be online, which complicates things.

... does this mean it will be possible for students not enrolled at the university to participate?

Reuben Stern (they/them) (May 18 2020 at 03:55):

This seems like a perfectly doable course! I do think that introducing the complete Segal space model is important, because that's how to reason about $\infty$-categories "internal to the world of $\infty$-categories".

Simon Pepin Lehalleur (May 18 2020 at 07:29):

Morgan Rogers said:

I would love to experience this kind of graduation! Many results about toposes are presented in an infinity-form (by Lurie, say), and I find this rather intimidating. Even if you're not covering $\infty$-toposes, this could be useful for me, so...

... does this mean it will be possible for students not enrolled at the university to participate?

I don't plan to say anything about $\infty$-toposes (except maybe the definition and some handwaving motivation) because I think they are really difficult to appreciate without a good grounding on classical topoi and/or sheaf cohomology on general Grothendieck sites, and most students of the course will not have that background.

My favourite introduction to $\infty$-toposes, probably easier than HTT Chapter 6 to start with, are Sections 2-7 of Rezk's survey:

https://faculty.math.illinois.edu/~rezk/sag-chapter-web.pdf

Rezk also has some related lecture notes, with more details on descent in $\infty$-toposes:

https://faculty.math.illinois.edu/~rezk/leeds-lectures-2019.pdf

I don't know yet if it will be possible for people who are not students at UvA to attend. At any rate I will post all the teaching materials on my website.

Simon Pepin Lehalleur (May 18 2020 at 07:34):

Reuben Stern said:

This seems like a perfectly doable course! I do think that introducing the complete Segal space model is important, because that's how to reason about $\infty$-categories "internal to the world of $\infty$-categories".

Thanks for the suggestion! I am not so familiar with complete Segal spaces. What would you recommend as sources to learn about them? I know of (as in, have not read!) the Joyal-Tierney paper and Lurie's Goodwillie calculus paper.

Simon Pepin Lehalleur (May 18 2020 at 07:48):

I found a survey on complete Segal spaces:

https://arxiv.org/abs/1805.03131

which looks good.

Reuben Stern (they/them) (May 18 2020 at 15:42):

Simon Pepin Lehalleur said:

Reuben Stern said:

This seems like a perfectly doable course! I do think that introducing the complete Segal space model is important, because that's how to reason about $\infty$-categories "internal to the world of $\infty$-categories".

Thanks for the suggestion! I am not so familiar with complete Segal spaces. What would you recommend as sources to learn about them? I know of (as in, have not read!) the Joyal-Tierney paper and Lurie's Goodwillie calculus paper.

To be honest, I'm not sure about sources. I think complete Segal spaces should be presented internal to the world of $\infty$-categories, in the sense that a CSS is a functor $\Delta^{\sf op} \to {\sf Spaces}$ to the $\infty$-category of spaces that satisfies the Segal condition using homotopy pullbacks (i.e. pullbacks in the $\infty$-category of spaces) and the completeness condition. This makes it easy to generalize to complete Segal objects in any $\infty$-category with pullbacks.

John Baez (May 18 2020 at 21:10):

It sounds like a great course. I like how you've already recognized that 28 hours is very little time; in this amount of time it seems that giving definitions, precise theorem statements, general philosophy and examples is much more important than giving proofs.

John Baez (May 18 2020 at 21:13):

I emphasize "general philosophy" and "examples" because people in a rush often (foolishly) economize on these. I find that with most math, except for proofs of famously hard theorems, I can read the proofs and learn them myself once I know "what's going on" - what the theorem says, what it means in some examples, and how it fits into the structure of the subject.

Arthur Parzygnat (May 19 2020 at 04:57):

John Baez said:

I emphasize "general philosophy" and "examples" because people in a rush often (foolishly) economize on these. I find that with most math, except for proofs of famously hard theorems, I can read the proofs and learn them myself once I know "what's going on" - what the theorem says, what it means in some examples, and how it fits into the structure of the subject.

I can't agree with this enough. It took me an embarrassingly long time to figure out that this is the main reason I found it difficult to follow most of my math classes when I was in the earlier stages or learning. However, I do recommend providing a sketch if the proof introduces a technique that is incredibly useful/interesting (but again, only after sufficiently many examples are given and intuition is gained since otherwise these techniques may be unappreciated).

Simon Pepin Lehalleur (May 19 2020 at 06:59):

John Baez said:

I emphasize "general philosophy" and "examples" because people in a rush often (foolishly) economize on these. I find that with most math, except for proofs of famously hard theorems, I can read the proofs and learn them myself once I know "what's going on" - what the theorem says, what it means in some examples, and how it fits into the structure of the subject.

I mostly agree with this. However, even at the master level, some students may not have the necessary mindset and intellectual discipline to go through a lot of proofs on their own. Also, I imagine that many students who take the course will not necessarily go on to work with $\infty$-categories everyday, but the proof techniques (simplicial combinatorics, lifting arguments, reasoning with slice categories, ...) are more widely applicable; one of my aims is that students who forget everything about quasicategories afterwards will still take those techniques home, and for this I need to do a number of "representative" proofs.

Simon Pepin Lehalleur (May 19 2020 at 07:02):

Reuben Stern said:

To be honest, I'm not sure about sources. I think complete Segal spaces should be presented internal to the world of $\infty$-categories, in the sense that a CSS is a functor $\Delta^{\sf op} \to {\sf Spaces}$ to the $\infty$-category of spaces that satisfies the Segal condition using homotopy pullbacks (i.e. pullbacks in the $\infty$-category of spaces) and the completeness condition. This makes it easy to generalize to complete Segal objects in any $\infty$-category with pullbacks.

This is certainly interesting if one wants to go in the direction of $(\infty,2)$-categories or internal $\infty$-topos theory and univalence, but does it belong in an introductory course?

Reuben Stern (they/them) (May 19 2020 at 13:50):

Simon Pepin Lehalleur said:

This is certainly interesting if one wants to go in the direction of $(\infty,2)$-categories or internal $\infty$-topos theory and univalence, but does it belong in an introductory course?

I suggested it because that is how I think about $\infty$-categories in practice, but it requires a little bit of putting up with circular reasoning if you don't want to really muck about with the formal proofs of things.

Simon Pepin Lehalleur (May 19 2020 at 15:14):

Reuben Stern said:

Simon Pepin Lehalleur said:

This is certainly interesting if one wants to go in the direction of $(\infty,2)$-categories or internal $\infty$-topos theory and univalence, but does it belong in an introductory course?

I suggested it because that is how I think about $\infty$-categories in practice, but it requires a little bit of putting up with circular reasoning if you don't want to really muck about with the formal proofs of things.

I see. I tend to think about $\infty$-categories informally in a slightly less precise way, as "categories coherently enriched in spaces", but your perspective may be better! It certainly meshes well with the nerve functor from 1-categories.

John Baez (May 19 2020 at 18:39):

Simon wrote:

I mostly agree with this. However, even at the master level, some students may not have the necessary mindset and intellectual discipline to go through a lot of proofs on their own. Also, I imagine that many students who take the course will not necessarily go on to work with ∞-categories everyday, but the proof techniques (simplicial combinatorics, lifting arguments, reasoning with slice categories, ...) are more widely applicable; one of my aims is that students who forget everything about quasicategories afterwards will still take those techniques home, and for this I need to do a number of "representative" proofs.

Yes, I definitely think it's good to do some carefully chosen proofs to show the techniques, and to convince people a subject is not black magic. What I hate is when people leave out the "soft" stuff - the general philosophy, and sometimes even the examples - because they feel short on time and think proofs are the essence of mathematics.

It sounds like you're thinking about the actual students, which is great!

Mike Shulman (May 20 2020 at 05:04):

In case you aren't aware of it already, let me mention http://www.math.jhu.edu/~eriehl/elements.pdf.