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Atrium > Mathematics, Physics & Philosophy: A learning roadmap for Homotopical Algebra (in a broad sense of the term)

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  1.  
    • CommentRowNumber1.
    • CommentAuthortret3jtt
    • CommentTimeNov 18th 2016
    • (edited Nov 18th 2016)
    Good morning, day or evening to you, people of nForum! I hope that this question is appropriate for the forum. Of course, we also have mathoverflow, but I think here I will get a more qualified answer.

    I'm a student of mathematics interesting in learning "higher" mathematics, in particular, homotopical algebra in its reincarnations as model category theory, simplicial category theory, infinity-category theory and then higher algebra (as in Lurie's book "Higher Algebra").
    The subject of homotopical algebra is vast nowadays, as I can see it, we have many long (800-1000 pages) books on a different topics of it. The ones I came across are:

    1)Abstract and simplicial homotopy theory (model categories and simplicial sets, that is, the prerequisites for "higher" mathematics):

    - E. Riehl "Categorical Homotopy Theory" (as I understand, it's not a very long book that gives a general picture for seemingly diverse areas of homotopy theory, but E. Riehl says that it's better to have an acquaintance with simplicial sets, model categories and quasicategories before reading that book, but it's not a strict prerequisite; I'm not sure if it means whether the exposition is concise or something else)

    - Z. L. Low "Notes on homotopical algebra (very long summary of definitions of result, but for proofs the author refers to other books or articles, 1000 pages long)

    - P.G.Goerss, J.F.Jardine "Simplicial Homotopy Theory (not very short book on simplicial sets and categories, I'm not sure if all the material there is required for going further (for example, for reading Lurie's books))

    - A.Joyal, M.Tierney "Notes on simplicial homotopy theory" (much shorter than the aformentioned book by Goerss and Jardine, only 100 pages, but, as I understand, it covers the same topic; is it a better choice to read before Lurie's or Simpson's book?)

    - M.Hovey "Model categories" (from what I gathered, it's kinda the standard reference for model categories; I don't know if you need the whole book for HA though)

    2)Higher category theory and higher algebra, that is, modern reincarnations of homotopical algebra:

    - famous J. Lurie's books: "Higher Topos Theory, "Higher Algebra" (and "Spectral Algebraic Geometry", but it is more advanced). As I understand, "HTT" contains a modern introduction to modern categories in appendix, so, probably, something like Hovey's book is not a prerequisite.

    - C. Simpson "Homotopy Theory of Higher Categories"

    As you can see, there are a lot of books. It's hard to choose among them, and even harder to determine what sections to read and in what order. From the vague understanding I have, it's this order: simplicial homotopy theory -> model categories -> higher category theory ("HTT" or "HToHC") -> higher algebra (and higher geometry from "SAG" or Toen's papers after that).
    Of course, I understand that one has to know category theory before starting on homotopical algebra. But after that I'm kinda lost in the sea of topics and those huge books. But, again, from the vague understanding I have, this route could be the fastest and deep enough: A.Joyal, M.Tierney "Notes on simplicial homotopy theory" -> appendix of Lurie's "Higher Topos Theory" on model categories -> E. Riehl "Categorical Homotopy Theory" -> the rest of "Higher Topos Theory" and Simpson's "Homotopy Theory of Higher Categories" -> Lurie's "Higher Algebra"

    I would be truly thankful for a helpful answer.
    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeNov 18th 2016

    The older books on model categories, such as Hovey and Hirschhorn do suffer from not covering combinatorial model categories, which is of paramount importance to any modern exposition of model categories.

    Riehl’s Categorical Homotopy Theory does cover model categories, so I’m not sure what would be the benefit of reading Lurie’s appendix first.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 18th 2016

    You are lucky, as now there is this text:

    • Introduction to Classical homotopy theory (web, pdf)

    which gives a pedagogical account, starting at the very beginning and using the modern theory, with full details and full proofs.

    You simply read that from first line to last line. When you are done, let me know. Then we check what happened to you in the process and depending on that I’ll give you a suggestion as to what to read next.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 18th 2016

    Urs, I’m looking through these notes for the first time. These look really great; thank you!

    Small comment: at the top of 7/100, would you rather say unions of finite intersections of subbasis elements?

    Another minor comment: while it might not be crucial for your development, in the general theory of topological categories, source and sink diagrams are even allowed to be large (not just indexed over sets), and this makes a difference in the general development, as The Joy of Cats makes clear. For example, it’s a theorem that the topological category axioms on U:C→SetU: C \to Set force UU to be faithful, but this isn’t the case on omitting the largeness condition.

    I may have more comments later, as I read…

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 18th 2016
    • (edited Nov 18th 2016)

    Hi Todd,

    thanks for these comments, and for all further ones you may have.

    Small comment: at the top of 7/100, would you rather say unions of finite intersections of subbasis elements?

    So you are saying the word “elements” needs to appear, right? Sure, thanks.

    Since my laptop died last week, I am on a small “web-book”, and this stalls when I ask it to edit that big page. Might you have a second and be so kind to make the edit?

    I mean in the nLab entry Introduction to Stable homotopy theory – P

    Thanks!

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 18th 2016
    • (edited Nov 18th 2016)

    Sorry I wasn’t more clear. No, what I meant was that you had “finite intersections of unions of subbase [elements]”, whereas it should be “unions of finite intersections of subbase elements”. I’ll make the edits in a bit.

    (Even under the distributive law, the former collection is usually strictly contained in the latter. For example, the product topology of ℝ×ℝ\mathbb{R} \times \mathbb{R} takes as subbase elements sets of the form π 1 −1(U)\pi_1^{-1}(U), π 2 −1(V)\pi_2^{-1}(V) where U,VU, V are open in ℝ\mathbb{R}; an arbitrary union of such can be written in the form π 1 −1(⋃ iU i)∪π 2 −1(⋃ jV j)=π 1 −1(U)∪π 2 −1(V)\pi_1^{-1}(\bigcup_i U_i) \cup \pi_2^{-1}(\bigcup_j V_j) = \pi_1^{-1}(U) \cup \pi_2^{-1}(V) for some U,VU, V, and then finite intersections of those will not look much different from rectangles or finite unions of rectangles. So the interior of a circle will not be a finite intersection of unions of subbase elements.)

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 18th 2016

    Ah, right, thanks for catching this!

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 18th 2016

    Edited. The compile time was about 2 minutes (not too bad).

  9. Different people learn in different ways, but I personally would suggest not to bother with a roadmap at all. Try to just get a ’big picture’ overview, reading whatever you need in whatever random order you find in order to do so. For instance read introductions, surveys, lecture slides, …, dipping into details a little if something intrigues or mystifies you. And then, as soon as possible, try to find some open problem, no matter how small, to think about. You can start with a big idea (construct K3 cohomology, for instance!), and narrow it down further and further to something you can actually realistically get into. You’ll learn immeasurably more that way, by trying to find techniques that might help you solve the problem, and to understand what goes into those techniques.

    • CommentRowNumber10.
    • CommentAuthortret3jtt
    • CommentTimeNov 18th 2016
    • (edited Nov 18th 2016)
    Dmitri Pavlov,

    > The older books on model categories, such as Hovey and Hirschhorn do suffer from not covering combinatorial model categories, which is of paramount importance to any modern exposition of model categories.

    > Riehl's Categorical Homotopy Theory does cover model categories, so I'm not sure what would be the benefit of reading Lurie's appendix first.

    Thank you for your response. I had a feeling that a more modern exposition is available, that's why I was inclined to read Lurie's appendix. Riehl says that the only strict prerequisite is the knowledge of ordinary category theory, but (here's the quote from the preface of the book):

    > An ideal student might have passing acquaintance with some of the literature on this subject: simplicial homotopy theory via [32, 55]; homotopy (co)limits via [10]; model categories via one of [24, 36, 38, 58, 65]; quasi-categories via [40, 49]. Rather than present material
    > that one could easily read elsewhere, we chart a less-familiar course that should complement the insights of the experienced and provide context for the naive student who might later read the classical accounts of this theory. The one prerequisite on which we insist is an
    > acquaintance with and affinity for the basic concepts of category theory: functors and natural transformations; representability and the Yoneda lemma; limits and colimits; adjunctions; and (co)monads. Indeed, we hope that a careful reader with sufficient categorical
    > background will emerge from this book confident that he or she fully understands each of the topics discussed here. While the categorical prerequisites are essential, acquaintance with specific topics in homotopy theory is merely desired and not strictly necessary.
    > Starting from Chapter 2, we occasionally use the language of model category theory to suggest the right context and intuition to those readers who have some familiarity with it, but these remarks are inessential. For particular examples appearing in the following, some
    > acquaintance with simplicial sets in homotopy theory would also be helpful. Because these combinatorial details are essential for quasi-category theory, we give a brief overview in Chapter 15, which could be positioned earlier, were it not for our preference to delay
    > boring those for whom this is second nature.

    Urs, thank for for the link. The text looks very interesting. In the context of Dmitri Pavlov's comment, can I ask if it covers combinatorial model categories? And am I right that it is a text mainly on "abstract" homotopy theory (that is, homotopy theory in model categories) that uses classical topology as an example? Would this text suffice for reading Lurie (along with something on simplicial homotopy theory, like Joyal's notes)? If yes, Joyal + your text looks like a good start before reading Lurie and Riehl's book.

    Richard Williamson, thank you for your response! The method of "learning by demand" looks reasonable enough. I don't have much expertise, but I think that you are also right in that "different people learn in different ways", so, for now, I think I'll be more confortable with reading literature. I'll see how I feel once I get some grounding in it, maybe, I will follow your advice.
    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 19th 2016

    can I ask if it covers combinatorial model categories?

    It discusses the concept of cofibrantly generated model categories. A combinatorial model category is a cofibrantly generated model category whose underlying category is locally presentable.

    And am I right that it is a text mainly on “abstract” homotopy theory (that is, homotopy theory in model categories) that uses classical topology as an example?

    Yes. And that should be exactly what you are after.

    Would this text suffice for reading Lurie (along with something on simplicial homotopy theory?

    Yes, you just need in addition some basic idea of simplicial sets and the classical model structure on simplicial sets.

    Originally I had planned to include discussion of that in the notes, but then it became too long. But the core of the material exists, it is at

    The proofs that this entry presently omits (but we could fill them in as we go along) are all in the excellent textbook

    which I recommend.

    • CommentRowNumber12.
    • CommentAuthoramg
    • CommentTimeNov 19th 2016

    @tret3jtt, I’d like to share my own experience, which echoes Richard’s comments. I’ve found that all of this stuff is extremely difficult to learn if you’re just doing it “for the sake of it”, trying to read things cover-to-cover in The Recommended Order. That’s not how math works, mostly because that’s not how our brains work. I’ve found that having a more specific and directed goal in reading makes it far easier to read difficult mathematics, and it makes it much easier to actually retain and mentally organize the things you’re reading. As I think is common, as an undergrad I sort of had this sense that e.g. I’d “know everything there is to know about abstract algebra” if only I read Dummit & Foote cover-to-cover. Not only is that false, but it presupposes that there is a binary meaning of the word “know”. All of this is just to say that I would strongly encourage you to remain flexible in your expectations and dynamic in your approach.

    • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeNov 20th 2016

    Re #10: I’m not sure if Lurie’s technical appendix is the best place to start. Perhaps Dwyer and Spaliński’s survey or Hovey’s book is easier to read.

    And I can only concur with Aaron’s opinion in #12: things are much easier to learn if you have a certain goal in mind, e.g., a research project. Reading things in linear order cover to cover is rarely the best idea.

    • CommentRowNumber14.
    • CommentAuthortret3jtt
    • CommentTimeNov 20th 2016
    • (edited Nov 20th 2016)
    amg, Dmitri Pavlov,

    That's a very reasonable perspective. But you see, for me it's really "the basic" material, I'm not learning "abstract" homotopical algebra only so I can solve some specific problems in more "concrete" areas, like classical algebraic topology or, say, scheme-theoretic algebraic geometry. I'm interested in it per se (and I also have been led to believe that it gives the most natural and the least ad-hoc perspective on concepts in related areas, but you all probably agree with this, since we're on nLab which advocates the so-called "nPOV"). Of course, that's not to say I'm not interested in related areas like topology and geometry, it's rather that I'm also genuinely interested in HA on par (or more) with those other more conrete subjects.
    So, my point is how can I even start research before I know some of the theory I'm interested researching in? But you probably do not advocate against reading books at all (if you don't, don't be offended, there are some people who would actually advise that, and some of them are smart and good mathematicians, for example, I read the story about how Joe Harris advised his students against reading books on scheme theory per se in favor of learning specific results by demand by doing research in classical algebraic geometry) , maybe, you just mean that I shouldn't read every page of every 1000-page book, as Lurie's books are 900/1100/2000 pages (for "HTT", "HA", "SAG", respectively). In that case, I can't question your advice, I absolutely understand it.

    amg, but I don't agree with you about books like Dummit and Foote. I think it's not a very long book (in terms of coverage of material it's much less than Lang), and, for example, Harvard math graduate department thinks that every graduate student in math must know all the material in it (see their qualyfing exam syllabus). After all, everything in D-F is "basic", like everything in Rudin's "Mathematical Analysis". It appears that every mathematician (even a classical analyst-to-be) should go through the material in D-F (maybe, from other books, since D-F is not the best one IMO, considering more modern ones, Aluffi comes to mind). That's not to say you can't be a mathematician if you forget a word from D-F, but at least it's possible and sometimes even must to read the book from cover-to-cover, for example, for a graduate school's exam.
    And if you want some personal experience, I liked my algebra book to be as complete as possible, but, probably, not in the most conventional way. I didn't think it should cover all the topics from algebra, like Lang, but rather that what it covers, it should cover very thoroughly. I felt that I really understand the material this way (that's why I found Bourbaki's book on algebra invaluable, even if a little outdated) rather than learn a bunch of unconnected specific results.

    But that's only my experience with self-studying algebra. I'm in no way claim that the same thinking applies to the material in Lurie's books. I only suspect that different people may learn in different ways, especially, since you have mentioned algebra and D-F.

    Dmitri Pavlov, I noted your opinion about Lurie's appendix. I think I'll go with the notes linked by Urs (and Joyal's notes for simplicial stuff) before tackling more advanced material, like Riehl's book. As I understand it, they cover model categories in a modern way, but also they are not very technical, which should do for a first introduction. Also, they connect the material with classical topology, which can't be a bad thing. Thank you.
    • CommentRowNumber15.
    • CommentAuthorRichard Williamson
    • CommentTimeNov 20th 2016
    • (edited Nov 20th 2016)

    So, my point is how can I even start research before I know some of the theory I’m interested researching in?

    Firstly, one can start by trying to get an overview, as I suggested. Mathematical breakthroughs are achieved by creativity, not through knowing a mass of material. What is important is that one gets a ’feeling’ for a topic; that one develops ’intuition’. You could read through all of the formal proofs of Lurie’s work, and still not truly understand much about (∞,1)(\infty,1)-category theory. Much more important than knowing formal proofs is to understand the motivation, the context, the pioneering ideas that shape the subject, the big problems that people are working towards.

    Secondly, you do not have to take a research problem. You could just take an important result, and work towards understanding that. For instance, you could study the statement that there is a model structure on groupoids that is Quillen equivalent to a certain model structure on simplicial sets, of 1-types. And then you could study the stable version of this. You would learn lots of techniques for this, much more quickly than if you attempt to read something from cover to cover; and you will learn something of the motivation for the notion of (∞,1)(\infty,1)-categories. Or you could study the model structures on two notions of (∞,1)(\infty,1)-category, say quasi-categories and complete Segal spaces, and the Quillen equivalence between them. You could try to vary things; can you in a different way give a proof from scratch in the case of (2,1)(2,1)-categories, for instance?

    Or, for something more classical, you could take as motivation something like one of the goals of Urs’s notes, namely to compute the first few stable homotopy groups of spheres. Just working through one of the approaches here, using the May and Adams spectral sequences, will teach you a lot, and give you a springboard to chromatic homotopy theory, which is where much research involving (∞,1)(\infty,1)-categories is focused.

    Or, if you’re more ambitious, you could try to understand elliptic cohomology, beginning with say the computational stuff in the Hopkins-Mahowald paper (if you understand that, you’ll already have learned an incredible amount), moving to the Goerss-Hopkins obstruction theory, …, and then trying to understand Lurie’s paper.

    There are hundreds of other similiar concrete projects.

    It appears that every mathematician (even a classical analyst-to-be) should go through the material in D-F

    This is a complete misconception. Did Euler need to go through the material in this book? Did Grothendieck? Of course not. Mathematics is not defined by any one piece of knowledge. One forgets things after a while anyway! What is important is that one has depth of understanding, then one will be able to quickly refresh oneself with regard to a certain topic, should one need to.

    • CommentRowNumber16.
    • CommentAuthortret3jtt
    • CommentTimeNov 20th 2016
    Richard Williamson,

    First of all, thank you very much for the advice, I can see that it's very deep and profound.

    I, of course, understand that reading Lurie's books is not enough to start research in homotopical algebra. I realize that one needs to study a lot of articles. To see what are the topics people are working on. But I wasn't even thinking about starting research very soon, just getting a solid grounding in homotopical algebra. I was convinced that one needs to get a solid mathematical education (an a broad one at that, like Harvard's quals syllabus) before even thinking about research by people (this advice wasn't about category theory per se, rather about math in general, and an advisor's (as in "the one who gives the advice" not as in "the one who supervises your thesis in a grad school) area is far from category theory, so, maybe, things are different here, I don't know).
    I was just thinking about reading some books about the stuff I'm interested in, getting a solid foundation in the subject before moving to concrete topics, that is, recent articles by people working there. I am not in a rush, you see, I was just interested in a subject, I didn't have a specific goal in mind.

    What you are saying about learning concrete results sounds fun. Still, don't Lurie's books contain such results? Surely, not all, but quite a bit? There has to be something between those 1000 pages. :) Or do you advise not to focus on proofs? I also heard an advise not to learn anything as "black boxes". Also, another person I've heard said that Lurie is a great expositor, so I assumed his books are as pedagogical as something of this scope gets. Of course, I could be wrong. I haven't read them after all. But I didn't intend to stop on Lurie's books. There is also Simpson's book I meantioned. I've heard it brings a reader up to speed in to a modern research state of higher category theory.

    It appears from your words that Lurie's books lacks motivations. Or did I get the wrong impression? In that case, I've heard that there are some surveys on the topic of higher category theory. There was one by Tom Leinster, if I remember right, and someone said you should read those before attempting Lurie. Your advice confirms it. There were also notes about different definitions for a higher category. I suppose it's a better start, is it not?

    >What is important is that one has depth of understanding, then one will be able to quickly refresh oneself with regard to a certain topic, should one need to.

    But one needs to get that depth first, doesn't he? Again, I've never said that one needs to remeber everything from D-F their entire life, only that he better read it at least once during undergraduate studies or a graduate school to get a grounding in a basic algebra. If you disagree than we should at least agree on the fact that one needs it in order to get through the formalities of a graduate school at least. :) After all, if you hope to refresh it one day, you need to have something to refresh. But who am I to tell you, since you are probably a research mathematician. Still, there are successful research mathematicians that share those views about broad education ("you better learn a lot before you start doing research even if not everything you would have learned would be immediately applied in your research", but again, it was said in the context of "basic" math: abstract algebra, algebraic geometry, complex analytic geometry, representation theory, differential geometry, number theory; not abstract homotopy theory).

    Still, I though it was a consensus that everything needs to go thourgh courses on abstract algebra, real and complex analysis, smooth manifolds and some algebraic geometry at least.

    One more thing: I hope I doesn't looks rude for me to "talk back" like that, since I'm obviously a student and I don't know things compared to you. In fact, I'm not arguing, I don't have any expertise or knowledge to do so. I'm just trying to get to the bottom of things. You know, different people say different things. It could be hard to understand what they really mean if you just nod to all different opinions (different from each other, not from yours) and pretend that you understood it right away (when you didn't).
    • CommentRowNumber17.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 20th 2016

    I’ve never even seen a copy of D-F… Not every mathematician goes through the US system, you know.

    • CommentRowNumber18.
    • CommentAuthorRichard Williamson
    • CommentTimeNov 20th 2016
    • (edited Nov 20th 2016)

    Hi tret3jtt! Let me just say first that it’s great that you ask here. If you are a student and wish to get into homotopical algebra, that is fantastic, and nobody should put you off doing that. You should choose what you feel is best for you, don’t listen to other people too much :-). You have not been rude at all.

    My suggestions are only intended to give you an alternative point of view to think about, or have at the back of your mind. You by no means have to follow the advice; many would no doubt disagree with it :-).

    My opinion on mathematical education is that what one actually learns is very much secondary to the process of learning it. One’s development ’as a mathematician’ through one’s university courses is, I would say, more important that what one actually learns. Part of developing as a mathematician is certainly to be exposed to a range of different ways of thinking, different styles of mathematics.

    Regarding Lurie’s works, I was not suggesting at all that they are poorly written or lack motivation. You can certainly learn a lot from his expository material: introductions, little comments, etc. But I personally would not regard reading through the works in depth as a good idea; I would rather suggest to dip into them as one needs. Just like most algebraic geometers do not read EGA/SGA from beginning to end (though there are no doubt exceptions, such as Mochizuki, apparently); they dip into them as needed.

    I would indeed argue that Lurie’s three books do not contain very much material of the kind I was getting at, suitable for an ’end goal’. The really original ’end goal’ stuff is contained in other papers of his, that make use of the material in the three books. This is not to put those books down at all, I am just trying to explain that the innovations are, for a beginner, technical. It would be better (only in my opinion, again) to have something with more of a story to tell to get into.

    Regarding proofs: I completely agree that one should, as far as possible, not rely on black boxes (I myself like to understand the tools that I use thoroughly), though again, one should not make blanket rules. And proofs are very important. But digesting them second hand is not so important, I would say. What is important is that you make a proof ’live’ for yourself: you have a need for it, you have some feeling as to why it is hard or easy, unexpected or expected, and the details can then tell you a lot.

    But I do not wish to suggest that you would not learn anything by following a ’roadmap’: you certainly would! I am just trying to suggest something that might possibly be even more fulfilling. But ultimately, you know yourself best, and I’m sure you’ll be very successful whatever route you take!

    • CommentRowNumber19.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 21st 2016

    I largely agree with Richard Williamson’s comments, but it might help to know if you are in graduate school and have an adviser you feel good about; talking with a good adviser can help immensely in sketching possible plans of study or plans of attack.

    Please excuse the following words if they seem preachy or overly generalized.

    On a general emotional level: the idea that there are thousands of pages that one must know to get into a subject can be extremely daunting and dispiriting. My own personal experience is that math also involves a great deal of sheer fooling or fiddling around on one’s own; the more dignified word is ’experimentation’. It’s much easier to work hard if one is having fun discovering and exploring for one’s self or with trusted mathematical friends. It need not be about anything especially important. (John Horton Conway discovered his theory of games and numbers largely by playing… games.)

    Part of the point is that acquiring mathematical knowledge often happens very accidentally in the course of happy explorations, and in the course of grappling with specific and sometimes very concrete problems.

    Also a spirit of independence helps. In my own case, it often takes the form of a general dissatisfaction with how mathematics is presented, and wanting to re-do it on my own terms. It can be in the feeling that one can do things more generally and more cleanly, or conversely in the feeling that sometimes there is needless or over-elaborated generality that loses sight of the really critical driving examples. It can reside in the feeling that one can reorganize a proof and make it much prettier. It can even reside in choice of notation. The general idea is to make mathematics one’s own.

    Also: feel free to go “slumming around” in a relaxed way, attending talks or seminars on areas outside of immediate interest. The really top-notch category theorists have an impressive breadth of knowledge and are open to thinking about just about any kind of mathematics, equipped with the kind of “inside edge” that the powerful general methods that category theory affords. Lawvere and Joyal are impressive examples of that spirit, and I notice that John Baez is moving into areas which seem at first glance very applied but which are amenable to graphical calculi rooted in category theory.

    I agree too with Richard that knowing the contexts and narratives that abstract developments are a response to is extremely valuable, and there a good adviser can be invaluable.

    I guess the last bit of preaching is that much of knowledge acquisition occurs on a “need-to-know” basis. Don’t worry too much about “ought to know” – there is no end to it. Instead, in the course of grappling with a problem, there will arise stuff that you realize you absolutely must know – that’s the kind of thing you throw yourself energetically into, and in that spirit there is nothing that gets in the way of mastering whatever it is you need to know.

    • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeNov 21st 2016

    My own personal experience is that math also involves a great deal of sheer fooling or fiddling around on one’s own…In my own case, it often takes the form of a general dissatisfaction with how mathematics is presented, and wanting to re-do it on my own terms.

    Hear hear. I think I still have pages and pages of notes that I wrote to myself in grad school explaining what various aspects of category theory and homotopy theory were “really about”. Nowadays I do the same sort of thing by writing nLab pages.

  21. Also a spirit of independence helps.

    This expresses elegantly the heart of what I am getting at,

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeNov 21st 2016

    If “Be more independent.” is going to be the main reply you suggest to give to young students who have the guts to come to the nForum and ask for advice, I’d ask you to reconsider.

    It’s evident that the status of the textbook literature in homotopy theory is lacking behind the enormous development of the field. Any student who asks for advice on literature here deserves to get some helpful pointers (check out over at MO for inspiration).

    This should be especially so here on the nForum, whose purpose is not to be a general chat box, but to be the talk pages that accompany the edit of useful information on the nnLab.

    The optimal attitude of expert regulars here, when faced with a puzzlement about literature, would be: What are you looking for? Is it not on the nnLab yet? Then let’s get going and start creating pages for it!

    • CommentRowNumber23.
    • CommentAuthorRichard Williamson
    • CommentTimeNov 21st 2016
    • (edited Nov 21st 2016)

    Encouraging a «spirit of independence» is different from a command «be more independent». I have worked rather a lot with students of different levels, and my suggestions (which, to emphasise once more, are certainly not a claim to be a universal truth, only something for tret3jtt to consider) come from that experience to a large degree, not only my own experience of learning.

    I made several concrete suggestions of ’end goals’ as opposed to textbooks. I am happy to elaborate. But my point is precisely that one should keep in mind whatever literature one stumbles upon, all kinds of different things, rather than sticking to a prescribed ’roadmap’. Clearly tret3jtt has already come across many standard texts, so he/she is more than capable of finding relevant literature. Picking out a roadmap amongst this literature is the point I’m contending: just keep it all in mind, and be driven by more particular goals, whether they be working towards an existing result, or exploring something new.

    One might agree or disagree with that, but let’s not misinterpret it.

    This should be especially so here on the nForum, whose purpose is not to be a general chat box, but to be the talk pages that accompany the edit of useful information on the nLab.

    1) I think that using the nForum for discussion of category theory/higher category theory in general, sometimes independently of the nLab, is perfectly fine.

    2) In my opinion, most discussion on the nForum is in fact related usually related to nLab pages, at least to a certain degree. Even I, who rarely edits the nLab, am occasionally drawn to do so, after an nForum discussion.

    The optimal attitude

    Let’s not be prescriptive. If tret3jtt is given some advice that he/she one day may find somewhat useful, then something has been achieved as well. There is more to the world than the nLab :-). (Even if it is of course important that matters of the nLab are not drowned in other discussion; but I don’t think there’s any evidence of that.)

    • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeNov 21st 2016

    I also understood “a spirit of independence helps” as simply advice for the appropriate mindset / study method while reading the literature pointers that had already been given.

    • CommentRowNumber25.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 22nd 2016
    • (edited Nov 22nd 2016)

    whose purpose is not to be a general chat box

    I don’t think that the “regulars” in general, and I in particular, make a habit of misusing the nForum in this regard. But now and then it may be okay to speak from the heart.

    I am sincerely sorry if I gave offense to anyone.

    • CommentRowNumber26.
    • CommentAuthorzskoda
    • CommentTimeNov 22nd 2016
    • (edited Nov 22nd 2016)

    17 I never even heard of D-F and consider myself a relatively broadly educated in mathematics. Google finds mixed, even strange, opinions about it, e.g. http://www.adequacy.org/stories/2001.10.14.163749.94.html

    Happily, there is an (extensive) errata of the 3rd edition available: pdf.

    As far as the statement that Lang covers “cover all the topics from algebra”, I find rather untrue. It goes for long on some topics like Galois theory, while almost none on many others, which I find basic in my own experience in algebra. If I am gonna teach a graduate course on algebra, I will simply have to supply many supplements (if Lang were a textbook). Look for example a comparable size first volume of Faith’s algebra to see how different a same-volume algebra book from about the same historical period. Not to mention that, more recently, some new topics came at forefront.

    • CommentRowNumber27.
    • CommentAuthorMike Shulman
    • CommentTimeNov 22nd 2016

    I’m not sure exactly what a “chat box” is, but I don’t think it’s wrong to have discussions on the forum that don’t relate specifically to the nlab. Isn’t that what the “Atrium” category and subcategories are for?

    • CommentRowNumber28.
    • CommentAuthorzskoda
    • CommentTimeNov 23rd 2016
    • (edited Nov 23rd 2016)

    27: Exactly. The classification of this thread is now “Mathematics, physics and philosophy”, which is a part of the Atrium, not part of the nLab classification (see https://nforum.ncatlab.org/categories.php).

    • CommentRowNumber29.
    • CommentAuthortret3jtt
    • CommentTimeDec 7th 2016
    Sorry, I've been away for a while and was unable to answer your comments.

    Richard Williamson,

    >My suggestions are only intended to give you an alternative point of view to think about, or have at the back of your mind. You by no means have to follow the advice; many would no doubt disagree with it :-)

    I absolutely understand. And I sincerely say that your advice is important, it actually provoked some thoughts for me. And I partially agree. As of now, I think that it's a good idea to find the right balance between "diving right into unknown territory" and "reading foundational managraphs". I can actually feel the appeal of your approach.

    >Just like most algebraic geometers do not read EGA/SGA from beginning to end (though there are no doubt exceptions, such as Mochizuki, apparently); they dip into them as needed.

    It's funny, at some point of time I actually wanted to read EGA "from beginning to end". And what stopped me wasn't it's lenght but rather the fact that there are more modern treatments available for many things there.

    >The really original ’end goal’ stuff is contained in other papers of his, that make use of the material in the three books.

    This brings as to a rather philosophical question: what should we regard "end goal"? I understand that "homotopical algebra" is regarded is a very abstract technical machinery used mainly for applications. But wasn't the same true for algebraic topology at once? For many other areas of mathematics?
    For some reason, I was interested in homotopical algebra itself. I know it might sound strange, but I find those constructions "beautiful" and "interesting". Many people I know of disregarded HA, saying it's a "useless subjects" or that it's "1000 pages and 0 results", but then I saw there are quite a lot of successful mathematicians doing research there (don't get me wrong, I'm not interested in HA because "successful mathematicians do research there", it's rather I'm interested in HA, and the fact that people work in this area gives me hope that the area is viable and living). For example, as I understand, not everything that Lurie created is directly applicable to some classical problems, some of his work are namely for "higher algebra" (or, as I call it, homotopical algebra). Same can be said about Clark Barwick, Moritz Groth, Gijsbert Heuts, Saul Glasman.

    But I don't know. Maybe, those are not "research subjects" for most mathematicians. But I got an impression that "homotopical algebra" (subsuming higher category theory, higher algebra, model categories, infinity-operads, higher topoi, derived, homotopical and spectral algebraic geometry) is a living and breathing field of mathematics. Maybe, I was wrong.

    Again, my logic was something like this:

    Originally, abstract algebra was a technical tool to deal with number theory or Euclidean geometry (or later Klein geometry). But now we study it for its own sake, we actually don't care about many questions it was originally intended to study (such as groups of transformation on plane, though we still care about number theory and, probably, will always care ). Sorry, I've chosen my words wrong. Actually, not everyone study algebra for its own sake. My point was that some do, and, what is more important, it that many of those who study abstact algebra do not care about concrete problems (such as groups of transformations of a Euclidean plane) it was originally created to study. Of course, they care about modern number theory, modern algebraic geometry, but those subjects actually more modern than abstract algebra, they are not what "stable homotopy theory" to "higher algebra", they are more like what "homological algebra" to "noncommutative algebraic geometry", if you can understand my analogy. But the point it that many of those who study abstract algebra will never study number theory or Klein's geometry, Or, even if they will, abstract algebra is not dependant anymore on those topics.

    So, I thought it's only fair to assume that "homotopical algebra" is so grand with many people working in this area, it's actually obtained some sort of "independence" from "concrete" (again, a pholosophical question: what does "concrete" mean? Mathematicians of 1960 probably thought that projective varieties are concrete and schemes are not, but now for most of us schemes are almost as concrete as varieties) fields of mathematics it's originally intended to generalize. That's not to mean that I presume that applications of HA to other areas are not important. They are now even more than ever (at the very leasy, more applications HA gets, the easier it would be to shut haters of homotopy-categorical mathematics who claim it to be pointless). That's only to presume there is something more to it other than just applications.

    But, of course, I may be wrong. And HA can still be tool and only a tool for classical topology and geometry. Or it can be a "dead area" (doesn't seem like it to me, though). If it's so, please, correct me.

    I'm not writing my thoughts to convince you of something, don't get me wrong! I write it so you can understand my logic and, maybe, correct me.

    David Roberts,

    >I’ve never even seen a copy of D-F… Not every mathematician goes through the US system, you know.

    Neither have I! And I'm not from US either. I wasn't talking about D-F as a "book", I was rather talking about the material in it, that is is regarded as something "every mathematician should know".
    (I actually don't like D-F. I'm more of a fan of Aluffi and Hungerford)
    • CommentRowNumber30.
    • CommentAuthortret3jtt
    • CommentTimeDec 7th 2016
    Todd Trimble,

    > but it might help to know if you are in graduate school and have an adviser you feel good about; talking with a good adviser can help immensely in sketching possible plans of study or plans of attack.

    No, I'm actually far away from a graduate school yet. Don't be surprised by seeing "a schoolboy interested in homotopy and category theory", I've also have finished the school already, unfortunately. I actually don't have any "plans of attack" yet. In fact, I also don't have any plans of "plans of attacks". I've just been self-studying mathematics, I got convinced by my expertise that I would be extermely interested in "homotopical algebra". So, why not study what you like? Of course, maybe, it's pointless (see my aformationed writings about this), but I was convinced it isn't and that it is a very active area of research (of course, "it is a very active area of research" isn't the reason for my interest in it, it only fuels it a bit giving me a bit more motivation).

    > On a general emotional level: the idea that there are thousands of pages that one must know to get into a subject can be extremely daunting and dispiriting

    Funny, but it's not to me. It actually encourages me. What kind of creature am I !?
    Probably, it's due to the fact that I'm at the early stage of my "mathematical career". I can understand a 2-3rd year graduate student or even a postdoc not having much time an/or energy to study thousands of pages just to get some result.

    >Also a spirit of independence helps. In my own case, it often takes the form of a general dissatisfaction with how mathematics is presented, and wanting to re-do it on my own terms. It can be in the feeling that one can do things more generally and more cleanly, or >conversely in the feeling that sometimes there is needless or over-elaborated generality that loses sight of the really critical driving examples. It can reside in the feeling that one can reorganize a proof and make it much prettier. It can even reside in choice of notation. The >general idea is to make mathematics one’s own.

    I can definitely relate to that! I actually had the same experience with studying. I didn't like even the best of books. I wanted to re-do everything in maximum proximity to a categorical language (for abstract algebra, and, of course, only where it's useful) and maximum generality (only where it can enlighten the understanding). But I never wanted to "reduce" generality yet. Of course, maybe this time is still to come. Perhaps, I'm not mature enough yet.

    >Also: feel free to go “slumming around” in a relaxed way, attending talks or seminars on areas outside of immediate interest. The really top-notch category theorists have an impressive breadth of knowledge and are open to thinking about just about any kind of mathematics, >equipped with the kind of “inside edge” that the powerful general methods that category theory affords. Lawvere and Joyal are impressive examples of that spirit, and I notice that John Baez is moving into areas which seem at first glance very applied but which are >amenable to graphical calculi rooted in category theory.

    Seems like a good advice. Though it's probably not easy in terms of energy and motivation, but it's definitely useful. But I never intended to be a "narrow" specialist. At the very list, I have always intended and still intend to learn basics of such fundamental areas as algebraic geometry, differential geometry, Galois theory, representation theory, homological algebra, K-theory. But one probably needs more than "basics" for it to actually be useful.

    zskoda,

    >I never even heard of D-F and consider myself a relatively broadly educated in mathematics

    Exactly! If you claim that you are "relatively broad educated in mathematics", than you probably know more algebra than there is in D-F. Or, at the very leasy, not much less.
    My point wasn't about D-F itself, but rather about the material in it. If you like, you can substitute D-F for Lang, Rotman, Grillet, Hungerford, Aluffi or Knapp.

    >As far as the statement that Lang covers “cover all the topics from algebra”, I find rather untrue. It goes for long on some topics like Galois theory, while almost none on many others, which I find basic in my own experience in algebra. If I am gonna teach a graduate course >on algebra, I will simply have to supply many supplements (if Lang were a textbook). Look for example a comparable size first volume of Faith’s algebra to see how different a same-volume algebra book from about the same historical period. Not to mention that, more >recently, some new topics came at forefront.

    Well, do you mean Carl Faith's book? I think it also doesn't contains many things Lang does. I have been led to believe that the mateial in Lang (or it's reduced version, that is, D-F) is "the" basic. Of course, everyone is entitled to their opinion. My opinion is not even my own! You can't do much worse than me ;)
    (It seems you have a refreshing and interesting prespective on how to teach abstract algebra. I would love to see the book on the matter author by you one day! ;)

    P.S. I hope this topic can still be "revived'.
    • CommentRowNumber31.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 7th 2016

    (I hope there is no concerted objection if the discussion here is not necessarily with a view toward the creation or editing of nLab pages, since the response from tret3jtt indicates a more personal discussion might not be remiss.)

    May I ask, tret3jtt, where you are in your schooling? Your use of the word “schoolboy” makes me think you might be in high school, or recently so. (If that is the case, I’m very impressed, and you seem to have large appetites for mathematics, which is of course fantastic.)

    Reading EGA from beginning to end is something that some people would strongly recommend for learning modern algebraic geometry, but see also this MathOverflow thread for various other suggestions. (Do you tune in to MO?)

    Regarding Lang’s Algebra: it’s rather slanted towards those who plan to work in algebraic number theory and algebraic geometry (what Rota calls “Algebra I”). It’s a fine book, and I consult it often, but it’s far from comprehensive. I like Jacobson’s Algebra books for a bit of counterbalance.

    • CommentRowNumber32.
    • CommentAuthortret3jtt
    • CommentTimeDec 7th 2016
    >May I ask, tret3jtt, where you are in your schooling? Your use of the word “schoolboy” makes me think you might be in high school, or recently so. (If that is the case, I’m very impressed, and you seem to have large appetites >for mathematics, which is of course fantastic.)

    At the moment, nowhere. You can say I'm "self-schooling". I was saying that while I'm far from graduate school yet, I'm also not in a high school, which I've already finished some time ago. I got into a fine university right after school (but not into mathematics, in our country we have to choose a specialization right before applying), then I realized I wasn't interested in studying what I had been indending to study in the first place, and quit. Then I went on thinking what I truly want to do. As it happended, I fell in love with mathematics. Currently I'm self-studying and thinking about "formal" higher education: where to go earn my undergraduate degree. As for future, I, of course, wish to go to a graduate school.

    >Reading EGA from beginning to end is something that some people would strongly recommend for learning modern algebraic geometry, but see also this MathOverflow thread for various other suggestions. (Do you tune in to >MO?)

    Thank you, yes, I check on MO from time to time, and this thread is quite helpful. I understand that "to sit and read entire EGA" is not the best idea for everyone. But I was dissatisfied with the level of generality and "rigor" in regular textbooks. In particular, I would always choose clear and abstract exposition over informal motivational one. That's just my cup of tea. (But of course, clean abstract motivational one is even better)

    >Regarding Lang’s Algebra: it’s rather slanted towards those who plan to work in algebraic number theory and algebraic geometry (what Rota calls “Algebra I”). It’s a fine book, and I consult it often, but it’s far from >comprehensive. I like Jacobson’s Algebra books for a bit of counterbalance.

    I see. I've seen Jacobson book, and it looks quite neat. Still, the topic on "what every mathematician should know" is not an easy one. Some won't need even 50 % of Lang, and some will need all of Bourbaki and all of Jacobson. Maybe, something small but conceptual like Aluffi would do for "everyone", and those who are interested in other topics, such as those in Lang or Jacobson, would learn them when they want.

  33. Hi tret3jtt,

    it actually provoked some thoughts for me

    That’s great, then I am happy :-).

    Many people I know of disregarded HA, saying it’s a “useless subjects” or that it’s “1000 pages and 0 results”

    I certainly would not agree with those people. It is far from useless; indeed, it is clear that it is going to be vitally important in many areas of pure mathematics for the foreseeable future.

    For example, as I understand, not everything that Lurie created is directly applicable to some classical problems, some of his work are namely for “higher algebra” (or, as I call it, homotopical algebra). Same can be said about Clark Barwick, Moritz Groth, Gijsbert Heuts, Saul Glasman.

    Absolutely, there are many innovations within higher algebra in the work of Lurie and all of these people, and these innovations are very important. It is just important to appreciate the broader context into which these innovations fit.

    Maybe, those are not “research subjects” for most mathematicians. But I got an impression that “homotopical algebra” (subsuming higher category theory, higher algebra, model categories, infinity-operads, higher topoi, derived, homotopical and spectral algebraic geometry) is a living and breathing field of mathematics. Maybe, I was wrong.

    You are not wrong, and your impression is very much correct. Indeed, I would say that is a very fashionable field just now (only ten years ago, there were only a handful of people thinking about these things): we see (∞,1)(\infty,1)-categories cropping up in papers all the time. And this research certainly does fully include research in higher algebra for its own sake.

    But, of course, I may be wrong. And higher algebra can still be tool and only a tool for classical topology and geometry. Or it can be a “dead area” (doesn’t seem like it to me, though). If it’s so, please, correct me.

    Again, in my opinion you are correct: it is not only a tool for other areas, and it is very far from a dead area.

    My only purpose is to encourage you to consider the context from which higher algebra has arisen, and to consider the context in which it is being used as a tool in research today; because both of these things will lead you to a deeper appreciation and understanding of higher algebra itself.

    But my impression from your thoughts in this thread are that you are very capable and very motivated. If you would like to follow a roadmap to learn higher algebra for its own sake, from beginning to end, by all means go for it! It could be the best thing for you. You will undoubtedly learn an enormous amount, which will be very useful to you whatever you come to do in mathematics later. You could try just jumping into the beginning of Lurie’s books, and then looking up relevant literature whenever you do not understand something. But you will know better than any of us what would suit you best.

    Good luck! Feel free to ask questions here on the nForum, I am sure that many of us would be happy to answer.

  34. Regarding EGA and SGA:

    I understand that “to sit and read entire EGA” is not the best idea for everyone. But I was dissatisfied with the level of generality and “rigor” in regular textbooks. In particular, I would always choose clear and abstract exposition over informal motivational one

    I have a lot of sympathy for this point of view: when one wants the details, I agree entirely should be presented clearly, rigorously, and thoroughly, and have followed this philosophy in the lectures notes that I have written for my own teaching. EGA is indeed excellent in this respect (SGA a little less so, though still very good). The point is again just that there is more to the subject that a series of results; the broader context in which they fall is equally important, and for this one would need to look beyond EGA. Studying something like the classification of surfaces, such as in the book by Beauville, would be a much better project, in my opinion, using this as a springboard to jump into EGA, SGA, and whatever else one might like.

    Reading EGA from beginning to end is something that some people would strongly recommend for learning modern algebraic geometry,

    Very few algebraic geometers that I have encountered would recommend this. A very useful reference: yes. Reading from beginning to end: no. Of course there are rare exceptions, but I have very met very few (who actually have some experience to draw on, as opposed to just espousing a point of view).

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