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(Apologies in advance if this question is inappropriate for the present forum – I can pose it on MO instead in that case.)
I am currently beginning a long-term project to teach myself the foundations of modern algebraic topology and higher category theory, starting with Lurie’s HTT and eventually moving to “Higher Algebra” and derived algebraic geometry. I have some concrete goals in mind (when I asked a professor about this, I was advised to aim to learn what the Spec of a ring spectrum meant, which apparently was one of Lurie’s early achievements when developing this theory). I am currently familiar with the basic homotopy theory of simplicial sets and the formalism of model categories, but I do not have a solid background in topology. I also know basic 1-category theory (including sites, sheaves, triangulated categories, etc.), but I know nothing about higher categories. As an undergraduate, I would much prefer to go slowly if it means I can learn the material better; I am in no hurry to start doing research.
I was advised to learn everything in HTT thoroughly and in detail, which I plan to do; however, having started reading it, it’s become clear to me that the book is not organized in a linear manner, nor is it always completely detailed, the way EGA – which was my first experience (and another continuing project!) at sinking my teeth into a huge volume – is, for instance. HTT seems much more comparable in presentation to SGA, based on the limited reading I have done of each of them; which is to say, it will be difficult. As a result, it seems that to read HTT (and later HA), I’ll have to read many other papers – for instance, Bergner’s construction of the model structure on simplicial categories, possibly some of Joyal’s work on quasicategories, likely classical topos theory as in “Sheaves in Geometry and Logic” – to gain a proper facility with this material. I also am often at a loss to understand the material when it is less motivated.
Since many of the readers of this forum have read HTT in complete detail (or so it seems), I was curious if anyone had any advice on how to approach this project: what outside sources to read, what parts of HTT to skip, etc. (For instance, with EGA I was advised to ignore III.2, which was helpful and a time-saver.) I’d appreciate any thoughts; moreover, since I will be organizing a student reading group on this material in the fall, I’ll certainly pass on any thoughts here to the other students there.
Thanks!
I wrote (and continue to write) the Crossed Menagerie as an aid for exactly the sort of study you are undertaking. (In fact it was partially to do something of that form myself, that I continued it after the first 60 or so pages.) Download a copy from the nLab and it may be useful. It will not answer all your questions, especially with regard to DAG but some useful stuff is there. The present version is 830 pages long so ….! Don’t print it all out. Use it as a electronic resource. (It is not a Bible and no doubt has some errors in it. It may however help. You do not mention TQFTs and HQFTs so some of the recent pages, which are not in the version I have put on the nLab, are probably not needed. It aims to give a flavour of some of the ideas. It then directs the reader to sources. Note it is written to be dipped into so is not that ‘linear’, but has cross linking everywhere. If parts that are not there in that version are useful to you, contact me and I will send the longer file or the cutdown version that I prepared for the Lisbon meeting in February.
Hello Akhil,
Your timing is amazing! Some students at my university (University of Washington) decided to start this exact same project within the past week. When planning what to do we were feeling a little overwhelmed. We seem to have no idea how to think about the things in the first chapter mainly because the examples are so sparse. Our hope is to construct lots of examples of each concept as we go through however much of the book we do and keep them in a nice typed up and organized fashion (we’ll see how long that actually lasts). Maybe the Crossed Menagerie will help us with that, I haven’t actually checked yet.
@Akhil and Matt: Why not ask for a nLab page and post your typed up version (with comments etc.) there for the benefit of everyone. That would be better than pdf. (I think that I would have used the nLab for my Menagerie if I had had it when I started. :-(.) If I can help debug something in anyway, holler!
I would be very happy if you put your notes directly on nLab in wiki format. At one point I tried to start an “Understanding” series on the nLab. I started with “Understanding categorical constructions in Set”. Something like that. I’d give a link but I’m on my mobile now (headed into a vortex).
I think you’ll find the nLab to be perfectly suited for this kind of project. Plus the nForum and even the nCafe are perfect complements.
Reading Lurie’s work has been on my ToDo list for a long time so I’d be interested in joining in. My natural place to put any notes or whatevers would be the nLab, with discussion here.
(Tim, regarding:
I think that I would have used the nLab for my Menagerie if I had had it when I started.
would you be interested in having it converted to a series of nLab pages? Note the passive voice!)
Eventually some version of it would seem natural to put on the Lab. I have other priorities at the present (i.e. finishing off a monograph that was started in 1984) and also want to put some more stuff in to it in the early parts, namely stuff on higher generation by subgroups à là Abels and Holz. Then the first few chapters could be converted.
I was curious if anyone had any advice on how to approach this project: what outside sources to read, what parts of HTT to skip, etc.
A good brief outside source to have in reach is
Charles Rezk, Toposes and homotopy toposes (pdf)
A good deal of HTT is all about taking this model-category theoretic discussion and providing its intrinsic $\infty$-categorical analog. Many of the central ideas are in Rezk’s writeup, and part of what makes HTT long is that it has to set up all the $\infty$-categorical basics to rephrase these ideas intrinsically.
In fact, much of HTT is about setting up and then formulating intrinsically Dugger’s theorem (reviewed at combinatorial model category). In HTT this appears at the very-very end (third but last proposition in the appendix), but everything in the book on presentable $\infty$-categories sort of revolves around this.
Another main chunk that intrinsically rephrases something well-known in model category theory is the section on $\infty$-limits. Here the main statement around which everything revolves is the theorem that the intrinsically defined $\infty$-limits are indeed presented by homotopy limits. The proof of that theorem is spread out a bit through the appendix. I once tried to collect the relevant pointers at infinity-limit.
Generally: people have suggested since the early 70s (starting with Ken Brown, André Joyal) that the model structures on simplicial presheaves present $\infty$-toposes, and lots of evidence for this claim has been accumulated in the literature. To a large extent HTT is all about saying: “Yes, that suggestion is correct, we can give the following intrinsic reformulation of it all.”
On first reading of HTT I would suggest to skip chapters 2 and 3. Look at the intros and scan through the definitions and main theorems to get an idea for what’s going on there, but don’t try to work through the proofs on first reading. This is a bit a science in its own. Come back to this once you have a good grasp of section 5 and 6, because then you will also have a better idea for what the theory achieves and what it is all good for. In 6, first look at 6.1, 6.2 of course, and then maybe skip ahead to 6.5, which contains some of the crucial statements that tell you how to think of $\infty$-toposes.
Lastly, don’t ignore the appendix. There is more in there than meets the eye. Section A.2 is great to read in itself, this is about the best self-contained exposition of model category theory with the modern (otherwise still unpublished) Smith-ian theory of combinatorial model categories, which is all-important for the book. (One of the central theorems says that presentable $\infty$-categories are precisely the intrinsic incarnation of combinatorial model categories.)
And several of the central theorems of the book have proofs that will chase you through a long list of lemmas spread out all over the book, eventually culminating in something in the appendix. Much of the actual work is happening here.
Thanks for all the suggestions! I was just trying to sink my teeth linearly into chapter 2 and was getting very confused by some of the more exotic model structures on simplicial sets (though Joyal’s paper on quasicategories helped clarify the earlier proofs). So I guess I will skip around a bit. Also the Crossed Menagerie and Charles Rezk’s article look helpful.
I’d certainly be interested in helping out with any such nLab page once I know enough to. Our reading group will only start in the fall, though. I’m not ready to do much at this point (other than give examples of quasicategories and left fibrations), but when I am could definitely help with an “Understanding higher topos theory” project.
Someday when I have a lot of spare time (i.e. never) I would like to experiment with rewriting HTT in terms of derivators. If it works, it might be a much faster route to the analogous content of chapter 6.
I spoke to the other person that plans on doing this (apparently he already had communication with Akhil without me knowing!) and we both would love to use the nlab to do this.
Remember, the words in the box on the home page of the lab:
The purpose of the nLab is to provide a public place where people can make notes about stuff. The purpose is not to make polished expositions of material; that is a happy by-product.
We all make notes as we read papers, read books and doodle on pads of paper. The nLab is somewhere to put all those notes, and, incidentally, to make them available to others. Others might read them and add or polish them. But even if they don’t, it is still easier to link from them to other notes that you’ve made.
That seems great and even if only the first few chapters are handled that would be very useful for everyone.
I am on writing a (one more) small reading guide to HTT.
At the beginning of his appendix on simplicially enriched model categories (A.3) Lurie says:
”Among the many different models for higher category theory, the theory of simplicial categories is perhaps the most rigid. This can be either a curse or a blessing, depending on the situation. For the most part, we have chosen to use the less rigid theory of ∞-categories (see §1.1.2)”
I see that in his setup many important theorems are formulated in the simplicially enriched way, but maybe one can say more precisely where the advantages and disadvantages of the two viewpoints are and why he considers simplicial sets less ”rigid” that simplicially enriched categories. Does he comment on these differences somewhere else in the text?
What is rigid about simplicial categories is that in them the horizontal composition of n-morphisms is strictly associative.
It may be helpful to consider this for the case that the $\infty$-category has a single object. Then in general the hom-space of the single point is an A-infinity space: composition is associative only up to higher coherent homotopy. But if once you pick a model of this by a simplicial category, it suddenly becomes an ordinary monoid, with strictly associative product.
This is maybe most familiar in the case that the $\infty$-category is in fact an $\infty$-groupoid with a single object: then in general the hom-space of the single object is an infinity-group, but once you pick a presentation by a simplicial category, it becomes an ordinary simplicial group.
On the other hand, if you model the $\infty$-category by a simplicial set, a quasi-category, then there is no such strictification going on. In that case there is not even a notion of “horizontal composition” in general, at least not without further work and further choices.
So this is what makes simplicial categories a strict and easy model for $\infty$-category theory. What makes them, on the other hand, a difficult model is that they are in general far from being cofibrant.
This means. If you have two simplicial categories presenting two $\infty$-categories, then only very rarely are the ordinary simplicial functors between them models for all $\infty$-functors between the corresponding $\infty$-categories. Instead, one has to replace the domain simplicial category by a suitable cofibrant resolution and then consider simplicial functors out of that. (Assuming here that the simplicial categories are in fact Kan-enriched and hece fibrant).
This is actually an example of a very general phenomenon or rule of thumb, at least: the more strict algebraic structure you put on an object, the easier it becomes to handle, “the more it becomes fibrant” but “the less it becomes cofibrant”.
It is a general trade-off between fibrancy and cofibrancy.
I am on writing a (one more) small reading guide to HTT.
Very good, that’s the way to go. I think that’s a pretty good roadmap.
Thanks, Urs.
strictly associative.
I see. so I will rather use the self-explaining word ”strict” in place of ”rigid”.
This is actually an example of a very general phenomenon or rule of thumb, at least: the more strict algebraic structure > you put on an object, the easier it becomes to handle, “the more it becomes fibrant” but “the less it becomes cofibrant”.
Do you think one can explain this phenomenon in terms of monadicity: If $K$ is a model category and $(F\dashv V):K\stackrel{V}{\to} Set$ is a morphism of model categories (a Quillen adjunction), perhaps on can somehow argue that the assumption on $(F\dashv V)$ to be monadic (in that it is of the form of a free forgetful adjunction based on the category of algebras over a monad) is less well behaved with cofibrant objects than with fibrant ones?
I am not sure if I can explain it the way you suggest. Mike might have more to say here.
But I think it can be understood quite straightforwardly from the very nature of model categories: they are built on the notion of factorization systems, and these are very much by definition so that if the right class is small, then the left class is large, and vice versa.
Or put another way: the essence of how model structures capture homotopy-phenomena is in the interplay of the left and right classes of morphisms. You can move the complexity around, to be “more in the righ class” or “more in the left class”, but in the end the total information must be retained, and so you can’t make the complexity just go away entirely.
Sorry, that’s a bit vague. But in a way I think this is all there is to it.
if the right class is small, then the left class is large
I am not sure how this kind of ”calculus” works out since a morphism can be in both classes of the wfs or in none. But ok, you said it is just a rule of thumb.
total information
If it is just the information in general (and not precisely the ”algebraic information”) then my idea above distinguishing between monadic adjunctions (=algebraic theories) and non-monadic ones is of course of no use.
I am not sure how this kind of ”calculus” works out since a morphism can be in both classes of the wfs or in none. But ok, you said it is just a rule of thumb.
This I didn’t mean with the “rule of thumb”; this is instead a basic fact: if you start with a factorization system $(L,R)$ and then make $R$ larger, $L$ will in general become smaller: some elements of $L$ that have left lifting again all elements in $R$ may no longer have left lifting against the new elements that you added.
So whenever you increase the class of fibrations and hence of acyclic fibrations, the class of cofibrations will accordingly shrink in general, and vice versa.
What I meant instead by the “rule of thumb” is that “making objects more algebraic also makes them more fibrant”. This is certainly true in most cases that one encounters in practice. But I am not sure how one might formalize it such as to be promoted from a “rule of thumb” to something more precise.
a basic fact
I thought of e.g. the trivial model structure where both classes (fibrations and cofibrations) are large (and do not differ in size) but probably your point is that in interesting cases this does not happen.
Yes, if both are large, then the homotopy theory encoded by them is not so interesting. The trivial model structure encodes an $\infty$-catgegory that is in fact just a 1-category.
Urs said:
if you start with a factorization system $(L,R)$ and then make $R$ larger, $L$ will in general become smaller
Stephan said:
I thought of e.g. the trivial model structure where both classes (fibrations and cofibrations) are large
It sounds to me like the two of you are talking about slightly different things. There are two weak factorization systems in a model structure: (acyclic cofibrations, fibrations) and (cofibrations, acyclic fibrations). Those are the things to which Urs’ observation (which is always true) applies: if there are more cofibrations, there must be fewer acyclic fibrations, and similarly etc. But that implies that in a model structure, if you have more cofibrations and you keep the class of weak equivalences the same, then there must be fewer fibrations in order to make there be fewer acyclic fibrations.
I think it’s also important to note that this is all relative: comparing one model structure or wfs on the same category to the other. I don’t think it says anything about absolute notions of “largeness”.
Finally:
perhaps on can somehow argue that the assumption on (F⊣V) to be monadic (in that it is of the form of a free forgetful adjunction based on the category of algebras over a monad) is less well behaved with cofibrant objects than with fibrant ones?
Perhaps something along these lines is that if the model structure on the category of algebras is created downstairs, then the fibrant objects (and indeed the fibrations) upstairs are no harder to understand than those below, but the cofibrations in general become much more complicated: they get the monad “built into them” in some sense.
Your explanation (saying, for a Quillen adjunction $(l\dashv r):A\stackrel{r}{\to} B$ we have that $A$ is ”fibrantly downstairs” (=easier) since the fibration-preserving functor is going out of it, and adjointly $B$ is ”cofibrantly downstairs” (=easier) since the cofibration-preserving functor is going to of it) as I understand it does not need to assume monadicity but only the definition of Quillen adjunction. But it answers the question why the cofibrations in $sSet Cat$ are more difficult to deal with than the ones in $sSet_Joyal$ (I don’t know how the model structure on $sSet Cat$ is called by Lurie) by referring to the adjunction
$(||\dashv N):sSet Cat\stackrel{N}{\to} sSet_Joyal$I was referring specifically to the model structure on algebras over a monad in which the fibrations are created, not merely preserved, by the forgetful functor.
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