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.