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Tl;Dr: I know higher category theory and algebra is used ubiquitously in advanced algebraic topology. However, every time I ask someone, or try to find out, how one actually learns to apply the higher theory to genuine topological problems (to have enhanced perspective, new techniques, new results,… anything) I get no meaningful response. Books on higher category theory tend to focus solely on the algebra, and I am very puzzled as to how one is supposed to learn to turn this algebra into algebraic topology. Many people do seem to know about this, so I’m asking: how did you learn it and where did you learn it from?

This question on MO is similar in spirit. I want to preface this post with a few things: first of all, I really do know nothing. Any opinions and perspectives I have about the subject are probably quite naive, and this is because I do not (yet) have the luxury of a teacher or anyone to talk to. I am very happy to be put right on any points. I'm also aware that many responders might be tempted to say: "don't worry, this is all postgraduate stuff: you don't need to know this any time soon." And sure, I bow to that wisdom, but I am very deeply invested in the question I'm about to ask. Even if it is inappropriate ('jumping the gun') to pursue it now, I will pursue it eventually and I hope to get answers that are helpful for that journey.

For the last few months, I have been very keen to study algebraic topology. I quickly got the impression that there are essentially two levels to the subject. The first is content such as may be found in Hatcher, Von Dieck, May, Rotman, ... I haven't come close to finishing (or starting) any of these books, but skimming through the contents it's clear these books really care about the topology. The abstractions which are made still feel fairly concrete to those with mathematical maturity, and one is always learning about spaces, maps between spaces and what you can do to them. One learns how to prove the classical theorems and one learns the classical techniques and machinery.

I think of this kind of study as pertaining to 'real' topology. I do not at all mean to dismiss the other areas of algebraic topology, but I use the adjective 'real' to describe the feeling of "actually doing topology". Even if the methods are very algebraic, one is always doing topology or algebra necessary for doing topology.

There is then what I consider the 'second' level. I very quickly got the impression that whatever one learns in the 'first' level, one can learn through a highly categorified perspective, with vastly more powerful tools, perspectives and language at one's disposal; the ‘real thing’, as it were, that we should strive to understand and use which has been the subject of much research and is still being developed in current research. Indeed, it seems to be that a sizeable chunk of modern activity in, and motivation for, category theory is for doing algebraic topology and geometry.

I'd already learnt some category theory and took quite a liking to it, so I decided I'd rather learn algebraic topology, from the get-go, as categorically as possible. I don't want to learn something and then re-learn it in a 'better' way later on: I think putting the work in to understand the abstractions and then coming to do the 'real' topology later, with more power and perspective, would be really nice... it fits the way I like to think about things. To that end, I studied:

  • Convenient categories of spaces, detailed proofs of cartesian closure etc.
  • All of Riehl's "Category theory in context", the vast majority of Mac Lane's "Categories for the working mathematician", various odds and ends (e.g. Freyd's monograph on Abelian categories)
  • The fundamentals of working with simplicial sets (from the Kerodon, Lurie)

The first item on the list has proven exceptionally useful. Sadly, the other two haven't appeared yet in my studies of more classical topics. For instance, I really enjoyed playing with monads, from Riehl's book, but I fear I will never see them again.

The 'second' level is extremely opaque. I've asked several people online questions similar to: "So, you tell me simplicial sets are really important for modern algebraic topology. Could you please say a bit more about how and why they are used, and some references for applying them to topology?" Or: "So, you tell me higher category theory, such as may be found in HTT, HA (Lurie) or "Categorical Homotopy Theory", "Some aspects of $\infty$-category theory" (Riehl), is very closely connected to, and useful for, algebraic topology. Could you please say a bit more about that?" The response always consists of very few meagre examples and has never yielded a reference: it no longer seems all that useful, after all.

If anyone reads this and recognises such a conversation, I mean no offence. I am just asking here for further elaboration, and I thank you for your time anyway.

Yet, when I browse the nLab, MO, or an online algebraic topology community, it seems as if they are doing higher category theory all the time: the algebraic topology of 'the second level' is everywhere: in the face of this evidence, I don't doubt that it's useful and interesting. But where do you learn it?

I mentioned four books on higher stuff already. I haven't started any of them, but I've taken the time to browse the contents and examine a chapter of two. Overwhelmingly, these seem to be algebraic and it is very hard for someone with my background to see where the topology is, or how one could use anything in these tomes to actually do something in topology (ideally, to do things that one couldn't do before). They seem mostly about algebra for algebra's sake - which is fine, but this doesn't interest me with respect to algebraic topology.

My question is:

  • How does one learn the higher algebra and machinery with a view to solving (previously unsolvable) problems in topology?

Everyone seems to have learnt this by magic. I just cannot find anything, anywhere. For example, Rotman and May both reference and use categorical language to some extent, but that extent is not one I consider to be of the 'higher' kind. Riehl investigates very abstract and powerful categorical ideas, but I cannot find a single meaningful application to topology (very happy to be corrected!). It seems as if one is expected to read the higher category theory and automatically become well versed in the higher algebraic topology. Of course, that isn't true, so I reiterate my question: "how does one (how did you) learn to apply one to the other (bridge the gap)?"

I've made partial progress towards this. For the last month I've enjoyed reading the first chapter of: "Simplicial homotopy theory" by Goerss and Jardine. I'm about to learn about the equivalence of simplicial homotopy theory with the ordinary homotopy theory of spaces. I intend to read this book up to chapter 3, at least, since I can clearly see a fair few uses and items of interest. This book has given me hope that the goal of studying the 'second' level is attainable and worthwhile. However, I am sorely disappointed to leaf through many of the standard algebraic topology textbooks (of the 'first' kind) to see no mention of simplicial sets. Either "delta-complexes" or "simplicial complexes" are used, presumably for pedagogical reasons: they are less abstract. But I already know about simplicial sets, and I know they are more general than both of the above, so e.g. I would love to learn about simplicial homology with the full power of simplicial set theory and categorical tools.

This leads me to some adjacent questions:

  • Is there a textbook that teaches the 'basics' but using the theory of simplicial sets?
  • What's a natural next step after having read (select pieces of) Goerss-Jardine? I intend to read (pieces of) "Model Categories" by Hovey, but it's hard to know what else to read

I would be really grateful for any references. I don't have a university to fall back on, so my main source for finding out what to learn and how to learn it is MSE. In the linked post, it is quoted:

Algebraic topology is a subject poorly served by its textbooks

I am hoping that there is a way around this: it would be very useful to get constructive feedback from the many experts in algebraic topology and higher category theory who use this site.

N.B. I fully intend to study from the texts I mentioned in the 'level 1' category. I am asking about how I might progress from there, or supplement my reading with more advanced perspectives.

FShrike
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    @MarianoSuárez-Álvarez That makes sense. And I am keen to learn things with a view to applications. I can't stomach reading about infinity categories (say) without being concretely, believably introduced to why they apply (even, why they are 'essential') to algebraic topology. Do you have suggestions for sources that motivate the weird stuff? – FShrike Jan 05 '23 at 23:04
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    At least for some authors, the goal of infinity categories is to replace or improve on ordinary category theory: to recognize some of the shortcomings of category theory as a framework for doing mathematics, and to suggest that infinity categories present a better option. There are applications to algebraic topology, but maybe that's too narrow a view. Take a look at the introduction to Charles Rezk's manuscript https://faculty.math.illinois.edu/~rezk/quasicats.pdf, for example. – John Palmieri Jan 05 '23 at 23:18
  • @JohnPalmieri Algebraic topology isn’t that narrow of a view, since it is what ‘everyone’ says originally and continually motivates developments in category theory. Algebraic topology is what I’d rather study, anyway. In an ideal world with lots of free time, I’d be happy to study those notes for their own sake, but it is rather a large investment of time. I am told quasicategories are of importance to algebraic topologists, so, to supplement a course like Rezk’s, I’d like to have sources that link it all back to the original, “extremely concrete” problems Mariano hints at. – FShrike Jan 05 '23 at 23:32
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    I once asked myself the same question as in the first bullet point and ended up writing about 140 pages of notes on this topic, which you may (or may not) like: http://dmitripavlov.org/notes/topology.pdf. The notes use simplicial sets from the very beginning, but treat fairly elementary topics: (co)homology, fundamental group(oid) and covering spaces, etc. The level of exposition is more elementary than Goerss–Jardine since no prior knowledge of algebraic topology is assumed. – Dmitri P. Jan 06 '23 at 01:05
  • @DmitriP. I am very grateful for those notes, they seem to address my second subquestion well. Thank you – FShrike Jan 06 '23 at 07:19
  • @MarianoSuárez-Álvarez I think they present a better option to do algebraic topology, or perhaps an option to do newer, more advanced algebraic topology. The sole reason I think that is because that is what all internet research (and conversations with people) has told me in no uncertain terms. The essence of my question is: how can I understand why that is the case / where to learn about why this is the case? – FShrike Jan 06 '23 at 22:50
  • That's not quite right. I don't think they supplant the algebraic topology that already existed. But it seems as if, to get anywhere beyond graduate level in algebraic topology, you need a lot of categorical firepower. I am not planning to be beyond graduate level any time soon of course but it bugs me that it seems impossible to learn about this stuff. All textbooks discuss graduate topology, higher category, but never both (so far, the only exceptions I've found have been Goerss-Jardine and the odd text on model categories: but even so, Goerss-Jardine is limited in topological scope) – FShrike Jan 06 '23 at 22:55
  • @MarianoSuárez-Álvarez I completely agree! That type of useful information is exactly the type of information I’m looking for (sources to learn it from). – FShrike Jan 06 '23 at 23:28
  • Let us continue this discussion in chat. – FShrike Jan 06 '23 at 23:29
  • @MarianoSuárez-Álvarez Right. I don’t even want to learn the analogue of the ‘measure theory’ unless I appreciate the reasons for doing so. – FShrike Jan 06 '23 at 23:30
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    I just want to acknowledge that this is such a well-written and well-elaborated question. And I am already curious about the answers. – Martin Brandenburg Jan 07 '23 at 00:05
  • @MarianoSuárez-Álvarez: I pointed to Rezk's manuscript: in the introduction he identifies several ways in which ordinary categories fall short. – John Palmieri Jan 07 '23 at 00:46
  • @MartinBrandenburg That means a lot to me, thank you. – FShrike Jan 07 '23 at 08:10
  • At the risk of sounding contrarian, I'm not sure you're approaching category theory correctly. While it does provide some tools and insights to study classical topology, it's well and truly its own discipline; as such, many of its results may be important to category theorists, but less so to outsiders of the field (this is true of any two distinct fields of maths). If you really want to learn applications to topology, I'd recommend by starting with topology. Eventually you'll need cohomology which is a nice introduction to homotopy theory (eg derived categories) and then to categorical ideas. – Brevan Ellefsen Jan 07 '23 at 12:33
  • @BrevanEllefsen I know it is well and truly its own discipline, and that’s why all the higher category textbooks are about categories purely for their own sake. My question is: people who know a lot of algebraic topology always indicate a great importance of category theory, but I’m finding it very difficult to find learning resources that allow someone (who knows both the relevant topology and the relevant category theory) to apply the category theory to gain new techniques, results, etc. I am pleading for someone who knows about this stuff to say how they came to know about it! – FShrike Jan 07 '23 at 12:53
  • In other words, you recommend starting with topology. That’s also what I want to do. It’s just very hard to find a topology text that actually brings in the more advanced categorical tools: I’m worried that the only way to learn this stuff is by struggling through research papers, which is very unsystematic and honestly not always educational. But surely such texts should exist, because all the experts learnt their stuff somehow, even as far back as 50 years ago. This is the essence of the question – FShrike Jan 07 '23 at 13:02
  • It depends somewhat on what you mean by algebraic topology, and I feel that's important to include in your question. If your goal is to classify $4$-manifolds then I wouldn't expect any higher categories at all to be strictly required, at least given current techniques. On the other hand, if you consider the study of properly structured Grothendiek topoi to be algebraic topology, then yeah: there's plenty of categorical tools to be had. What exactly are your goals for topology? Given you mentioned not having read through topology books, what applications do you even have in mind? – Brevan Ellefsen Jan 07 '23 at 13:08
  • @BrevanEllefsen I ask this question more as a future reference for myself. As it stands, I know that the subject fascinates me in a rather general way, and I intend to become well versed in the material of the more standard textbooks. I know that when I eventually (or presently) want to study at a more advanced level, higher category theory should feature everywhere (given that it’s all the homotopy theorists, say, ever seem to talk about online). I’ve tried quite hard to get a picture of how one learns about these links, but I’ve come up with little success (Goerss-Jardine is my only ‘hit’) – FShrike Jan 07 '23 at 13:12
  • I also am not qualified to talk about genuine applications of higher category theory to algebraic topology. That’s another aspect of my question! Every time I’ve asked someone who knows: “why should I even study (-)?” I don’t get a response beyond: “topology was the original motivation for (-) and (-) is very important in modern topology”. It would be nice to see texts that actually demonstrate why (-) might be motivated by, and useful for, topology. For instance Prof. Álvarez hinted at many concrete applications earlier, so I’d like to know where one learn about them (what even are they?) – FShrike Jan 07 '23 at 13:15
  • This discussion continues to feel odd to me. Speaking for myself as a mathematician, I will learn techniques or theory because of their applications to particular problems, or I might be intrigued by a theory and want to learn more about it on its own. But for me, and I'm guessing for many people, it is unfamiliar territory to say, "I want to learn about a theory because it's supposed to be good for some problems, but I don't have any problems in mind, so tell me some problems and tell me where to learn about this theory as it applies to those problems." – John Palmieri Jan 07 '23 at 18:03
  • In particular, if my guess is right, then there just may not be very many relevant texts: people may recognize that the theory is useful and decide to write about the theory in some generality, not with a focus on a particular area. It sounds like you're looking for a book which would be called "Algebraic topology from the infinity-category point of view," and I don't think such a thing exists, in part because infinity-category theory is a pretty recent development, maybe in part because people view Hatcher's book and May's books, among others, as filling the relevant gaps in the literature. – John Palmieri Jan 07 '23 at 18:07
  • @JohnPalmieri I would certainly love to see such a text, “algebraic topology from the $\infty$-category point of view”. I don’t feel like Hatcher or May fill the relevant gaps though, as I feel certain the contents of the conversations about higher category in topology that I observe are quite beyond the machinery used by May in their “concise” courses. Anyway, your first comment does raise a good point. Judging by responses from others, I seem to have come across as fumbling pretty chaotically in the dark. – FShrike Jan 07 '23 at 18:25
  • I confess to being a little frustrated, I suppose. One the one hand I see people talking about this stuff everyday online, and it seems interesting and powerful. On the other hand, finding concrete motivations seems impossible. I enjoy category theory so I “want an excuse” to study the higher category theory in more depth, but I do have a time budget as well as being restricted by what’s more important for me to learn in the short term, considering I will soon have to be sitting undergraduate exams. But if someone revealed a few common tools and landmark results of the higher theory (cont.) – FShrike Jan 07 '23 at 18:28
  • As applied to topology, I’d have reasonable justification (as well as, importantly, a list of references to study from!) to pursue the higher category theory without feeling like I’m misusing my time. This isn’t in the short term (I have A-levels coming up ;)) but it is a concern for the foreseeable next few years. Perhaps I will have to reboot this question in two years’ time with a more focused motivation – FShrike Jan 07 '23 at 18:30
  • @DmitriP. I notice in your text that many images are entirely missing. For example, your definition of a covering map in section $32$ goes: "...for any commutative square of the form: " but there is no square. Do you have a version of the pdf without this bug? – FShrike Jan 08 '23 at 15:54
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    @FShrike: Thanks for pointing this out, for whatever reason some diagrams failed to compile. I recompiled the file and all diagrams seem to be present again. – Dmitri P. Jan 08 '23 at 18:24
  • @DmitriP. Thanks. I notice you also don't require the diagonal filler in the covering diagram to be unique, as, say, Lurie and Gabriel/Zisman do in their respective texts. If one requires uniqueness, then there is a very close analogy with topological covering maps and their lifting properties. Was that an accidental omission or a deliberate choice of definition? Uniqueness is necessary and equivalent to being able to treat the pullback as fibrewise isomorphic to $\Delta^m\times\operatorname{dis}(S)$, I think, so perhaps it is a typo? – FShrike Jan 08 '23 at 18:30
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    @FShrike: Yes, I forgot to include “unique” in the first definition. Fixed now. – Dmitri P. Jan 08 '23 at 18:33
  • @DmitriP. Ok. It is a very pleasant surprise to see a Galois correspondence of fundamental groups and covering maps for simplicial sets, too. – FShrike Jan 08 '23 at 18:36
  • @DmitriP. Sorry to bother you again. In definition 32.6, it's not clear to me that colim(p):Recon(M)-->Y is actually a covering map. Would you mind giving an indication? – FShrike Jan 16 '23 at 22:26
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    @FShrike: Thanks for pointing this out, I uploaded a new version with an expanded proof of Lemma 32.7, which addresses precisely the question. – Dmitri P. Jan 17 '23 at 02:37
  • @DmitriP.: Your notes seem very nice, but unfortunately all the boxed hyperlinks make the text virtually unreadable. Wouldn't it be possible to mark the hyperlinks just by making the text blue, for example? – Hans Lundmark Mar 30 '23 at 14:04
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    @HansLundmark: I suppressed the boxes and uploaded the new PDF file. (This phonomenon of displaying boxes around hyperlinks appears to be specific to the older Linux apps like xpdf, evince, okular, and some older libpoppler-based apps for Android. The newer mupdf-based apps, as well as Firefox, Chrome, and macOS and Windows apps all seem not to display boxes.) – Dmitri P. Mar 30 '23 at 14:49
  • @DmitriP.: Great! Thank you. :-) – Hans Lundmark Mar 30 '23 at 15:08

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