I've been doing a lot of reading on homotopy theory. I'm very drawn to this subject as it seems to unify a lot of topology under simple principles. The problem seems to be that the deeper I go the more confused I get since the number of possible formalisms grows exponentially. The more i read the harder it gets to seperate the fundamental principles from the formal manipulations.

For example: I know the basics of cofiber and fiber sequences and why they work concretely but I still have no clear idea of how to organize them into a clear picture that would work the same in the homological setting.

Here are things i've been trying to understand axiomatically in terms of first principles and so far have been unsuccesful:

  1. General notion of a derived functor between categories with weak equivalences.
  2. Homotopy (co-)limits - cofibrant and fibrant replacements (Which as i understand are a special case of 1).
  3. Stable homotopy category.

From the reading i've done on nlab it seems a lot of homotopy theory can be expressed neatly in terms of $(\infty ,1)$-categories. For me it's a pretty good argument to learn that formalism.

1. Should I learn $(\infty ,1)$-category theory?

2. If not, is there a way to gain a formal unified undertanding of homotopy theory which feels less like walking around in a dark room and more like climbing a mountain?

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    $\begingroup$ There are two questions of this form you could be asking, one of which is, should I learn about some particular model of $(\infty, 1)$-categories, and the other of which is, should I learn some model-independent things that ought to be true in all models of $(\infty, 1)$-categories (e.g. formal statements about the behavior of homotopy limits and colimits). I think the latter is worth doing, although I don't know a good resource for doing it; I haven't personally needed to do the former yet so I'm not the person to ask about that. $\endgroup$ – Qiaochu Yuan Nov 27 '15 at 5:18
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    $\begingroup$ @QiaochuYuan I see your point. I don't mind commiting to a model as long as It's clear what are the model-independent critical stuff and what's the technical overhead. Trying to read about the bits of the general theory without any model is confusing as hell since i can't really prove anything. Would you say Lurie's Higher Topos Theory is an optional resource for this material? Or is it too heavy on irrelevant algebra? I'd love to learn the general theory without commiting to a model but so far it's not really working for me :( $\endgroup$ – Saal Hardali Nov 27 '15 at 5:32
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    $\begingroup$ Do you know anything about model categories yet? I find it's easier to learn first, mainly because it's a well-established field of study, so there is a large amount of places (books, lecture notes...) you can learn it from; and contrary to $\infty$-category, everyone agrees on what a model category is. It's getting there for $\infty$-categories, but it takes time... Besides model categories were introduced first historically, and a lot of the motivation/inspiration for $\infty$-categories comes from there, I think. See this MO thread. $\endgroup$ – Najib Idrissi Nov 27 '15 at 8:30
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    $\begingroup$ I don't know. I think the place I learn was Dwyer and Spaliński's Homotopy theories and model categories, and then Hovey's book Model categories, and then here and there as needed. But it's really the subject of a new question (don't be afraid of asking a new one, I searched and couldn't find one for "Where is a good place to learn model categories"). $\endgroup$ – Najib Idrissi Nov 27 '15 at 8:35
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    $\begingroup$ I eventually asked the question myself. I think it would be nice to know what people consider to be good books about it. $\endgroup$ – Najib Idrissi Nov 27 '15 at 12:27

I've recently gone through a bit of a journey in learning about $(\infty, 1)$ categories, and whilst I don't know whether OP will any longer be interested in my answer, perhaps I can write a few useful words for future readers. What I'll try to provide here is answers which I think that I would have found useful if I'd seen them written down elsewhere.

  1. A derived functor between categories is (as I've seen written in several places) a "homotopy theoretic version of the functor that you started with". The issue with this rather concise statement is getting to grips with what it actually means. I think the best way to address this difficulty is with a concrete example:

    Consider the category which consists of diagrams of (CW) topological spaces of the shape $$* \leftarrow * \rightarrow *.$$ I'll denote this category $\textbf{Top}^{\mathcal{D}}.$ In particular, the objects in this category are diagrams of the above shape, and morphisms are object-wise morphisms between the spaces in the diagram. For example consider the following two diagrams (which are objects in this category): $$D^2 \hookleftarrow S^1 \hookrightarrow D^2$$ and $$$$ $$* \leftarrow S^1 \rightarrow *.$$

    There is a morphism from the first to the second of these diagrams in this category which consists of the identity map on the $S^1$, and the constant map on the $D^2$s.

    Now, one can note that given diagrams of this shape, there is an obvious way to define homotopy equivalent diagrams. They're the ones which have homotopy equivalent objects. Alternatively put, they're the ones which have object-wise homotopy equivalences as morphisms between them. In particular, the two examples which I just gave could be said to be homotopy equivalent diagrams.

    The point about all this is that now I do not only have a category, but I've defined a particular class of morphisms to be "weak equivalences". This is a useful thing to do, as we know from doing it in the category Top.

    Now let us consider the functor $\text{colim}: \textbf{Top}^{\mathcal{D}} \rightarrow \textbf{Top}$. This is the functor which sends a diagram to the space which is its colimit. This is fine, no problem. Diagrams are sent to topological spaces, this is a functor, all well defined.

    However, what happens if I want to insist that objects which are weakly equivalent should be considered isomorphic? This is what we (at least, conceptually) do in Top, after all. We consider homotopy equivalent spaces as "basically the same". Of course, homotopy equivalent spaces are not necessarily isomorphic (homeomorphic), but really we like to let things vary up to homotopy. It makes life simpler. Anyway, this idea of "considering weakly equivalent thigs to be isomorphic" can be formalised, and is referred to as "localising at the weak equivalences" - we, in a category theoretic sense, add in inverse morphisms for all the weak equivalences in our category so that they actually become isomorphisms.

    Anyway, this is all rather formal and difficult in full generality. The actual construction of the category which you get after localising at the weak equivalences is tricky - it's not easy to immediately see what the morphisms are in your new category (strange stuff happens). Mostly (in particulary in Top), it suffices to just say "things that are homotopy equivalent should be considered isomorphic" and move on. This avoids a lot of tricky business. This is fine for most purposes.

    This leads to a quandry, though. In the case above, where we have the colimit functor, what happens after we localise at our weak equivalences (in this case, the object-wise homotopy equivalences)? Can we just expect the colimit functor to work as it did before? Objects which were simply homotopy equivalent are now actually isomorphic in our new category. Any well-defined functor must send isomorphic objects to isomorphic objects. This means that (once we've localised at the weak equivalences) weakly equivalent objects should be sent to weakly equivalent/isomorphic objects.

    However, and this is this big thing there is absolutely no reason to expect an ordinary functor to do that. That is, there's no reason to expect a functor to send homotopy equivalent/weakly equivalent things to homotopy equivalent/weakly equivalent things. Why should functors care about your class of morphisms which you've defined to be weak equivalences and then send objects which are weakly equivalent to objects which are either isomorphic or weakly equivalent in another category? The answer is that they don't.

    Concretely (again, a tip with this stuff is that it is always useful to return to concrete examples when things start getting a bit unclear), in the example above, whilst we have two "homotopy equivalent diagrams", their colimits are $S^2$ and $*$, respectively. Once we've localised at the weak equivalences, our colimit functor is sending two isomorphic objects to non-isomorphic objects. It is therefore not a well defined functor on this new category that we get after localising.

    So what do we do? We have our nice homotopy equivalences/weak equivalences, and our functors ignore them. More precisely, the functor which we started with is not a well defined functor on our new category which we get after localising at the weak equivalences.

    This is where derived functors come in. They are essentially designed (via a clever universal property) to be the "best version" of the functor that we started with which is well defined on the homotopy category (the homotopy category being the name for the thing which we get after localising at the weak equivalences). In our case, the derived functor for colim is the hocolim functor. These derived functors are constructed such that they do respect weak equivalences - weakly equivalent objects get sent to weakly equivalent/isomorphic objects.

    Now, motivation? Well, now that we know that the colimit functor ignores the fact of whether diagrams are/are not homotopy equivalent, we realise that it is basically useless for us (as homotopy theorists) to recognise an object as the colimit of a diagram - the homotopy type of the colimit changes even if the homotopy type of the objects in the diagram don't (again, this is exactly what happens in the example above). It can however be useful to find that something is the homotopy colimit of a particular diagram - we know that the homotopy type won't change so long as the homotopy type of the objects in the diagram doesn't change. There are an enourmous plethora of examples where this is extremely useful.

  2. Fibrant/cofibrant replacements of objects? In full generality, very difficult. This entirely depends on the so-called "model structure" that you're working with. What the term "model structure" means is simply that you're in a category where you're doing homotopy theory (i.e. you have defined weak equivalences, fibrations and cofibrations, and these classes of morphisms satisfy some nice properties with respect to one another).

    There's no one-size-fits-all construction. The most useful way I can think of to explain what I mean by this is give a specific example of what I was playing with when I learnt about this stuff. I happen to have diagrams of spaces which are very very special (Reedy diagrams; where all the morphisms are cofibrant inclusions). The examples which I give above are indeed examples of these very very special diagrams. Now, when we did stuff above, the weak equivalences were simple enough to define. However, I didn't actually give the full model structure (the cofibrations and fibrations as well as the weak equivalences) - this is because the cofibrations and fibrations are not easy to define in this category. Not only are the definitions tricky, but they also depend on which model structure you pick - there is more than one available choice! The two model structures which are usually available for diagrams of spaces are the projective and injective model structures. The weak equivalences are defined the same way for both, but the cofibrations and fibrations aren't. Anyway, it's hard to identify which objects are fibrant or cofibrant. In the case of Reedy diagrams (my special case), there exists a special model structure called the Reedy model structure - this is neither the injective nor projective model structure. It's different, and often a lot easier to handle. In a very special case of the Reedy model structure, this model structure coincides with the projective model structure (that is, the fibrations and cofibrations are the same for both). Then if an object is cofibrant in the projective model structure, it's cofibrant in the Reedy model structure and visa versa. It is also a theorem that if an diagram is cofibrant in the projective model structure, then its colimit is homotopy equivalent to its homotopy colimit. It is still a little complicated to determine whether an object is cofibrant in the Reedy model structure, or what a cofibrant replacement should be, but its okay. The point is, after realising about ten million special conditions, things become tractable for me. Still not easy, but tractable.

    TL;DR: Computing the homotopy colimit/homotopy limit is hard unless you have particularly nice stuff.

Finally, in answer to your second two questions: Yes, you should learn $(\infty, 1)$ category theory - I cannot put into words the breadth of understanding which I now have which I did not have before. Seeing the bigger picture is always beneficial in mathematics, even if its not directly related to your work. In answer to the second, I did not find a quick fix. I spent a lot of time "wondering around in the dark", and talking to a lot of people, before finally seeing the bigger picture a bit more clearly. And I still don't know much.

I will recommend some sources (all freely available online):

  • Daniel Dugger's Primer on Homotopy Colimits
  • A whole bunch of Emily Riehl's work. In my case, this was very helpful, but all of her work is really excellent
  • nLab. It's worth persisting. There are some well written gems in there amongst some rather more general difficult stuff.
  • I'll obnoxiously point you to the questions which I've asked on MathSE about this stuff. As with almost any topic, there are a vast number of experts here who can answer any questions which you have.
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