I have a question which is, to a large extent, a more focused version of the existing question Homological Algebra texts

To be more precise: a student, at the precocious undergraduate/starting graduate level, has asked me to recommend some material for learning homological algebra. He seems at the moment to be interested in algebraic topology and diff. geometry, which is not how I came to learn hom alg, and I am therefore slightly uncertain which of the many options would be best. So what I would like to ask is something like the following:

I am looking for a recommendation of one or two books on homological algebra, which would be accessible to an advanced undergraduate/beginning graduate student (North American level) and which would complement the student's existing interest & self-taught progress in algebraic topology & diff geometry . Ideally, I would like people answering this question to have used their recommended text for teaching, or else used it themselves in graduate studies; and I am after a text which will connect to applications in topology/geometry. The two I am currently looking at (from which I gained most of my own small understanding of Hom Alg) are Hilton-Stammbach and Weibel: my worry is that the first, while accessible, is a bit old, while the second is more up-to-date but is quite diffuse, and assumes familiarity with some examples from algebra that the student might not have covered.

Do people think that this question will elicit any new information not hitherto stated in the answers to Homological Algebra texts?

Well, I don't know the student all that well, and (especially since he is still an undergraduate) what he thinks he ought to learn may turn out not to really be what he wants or needs. He's already worked through most of the stuff in Munkres' book on Alg Top (not something I ever did) and we will be meeting to discuss what he wants to get out of any such reading. But since he specifically asked about homological algebra (rather than, say, spectral sequences) I wanted to have some suggestions to hand.

@Shevek: In your infinite wisdom, perhaps you could explain why it's a terrible idea =p. Have you actually read Quillen's Homotopical Algebra?

I think you may be endowing yourself with an authority on algebraic topology (and homotopy theory) that you do not possess.

Likewise.

The primary thing that model categories have given algebraic topology proper is a clarification of the relationship (well, equivalence) between classical algebraic topology of topological spaces and the homotopy theory of simplicial sets.

Are you not familiar with the whole area of stable homotopy theory as well as the many model structures on the many different categories of spectra? Are you suggesting that so-called categorical homotopy theory is not at all a branch of topology? It's easy to dismiss me because I'm arguing here on meta with my real name. If you'd like to call my suggestion into question and assert your own expertise in the matter, please use your real name. If not, then be quiet. I'd rather not be engaged by nameless critics.

Also, I have no interest in "proving to you" that starting with model categories is not a good way for someone to start learning homological algebra, which is why my original remark was a single, unclarified sentence. I'm confident that my position is sufficiently obvious to others (if not to you personally).

That's awfully conceited of you.

Harry, please do not use this thread to argue with someone who disagrees with you about something off topic. Shevek, please don't poke with sticks.

(My post above was implicitly saying that I think the question is a nice one. But making explicit what the student wants to get out of the book would be good)

@Mariano, @ajtolland: I will talk to the student tomorrow - but based on previous interaction, it may be hard for me to coax him into articulating specific goals. Most probably I will lend him a copy of Weibel to start with and see if that meshes with other courses or reading that he is currently engaged in. Thanks for the suggestions.

@ajtolland:

I don't think I claimed that I was helping anyone. I was merely suggesting that Anonymous User Shevek "put up or shut up", because he has a history of anonymous nastiness on meta. I consider his dismissal of my suggestion with a snide remark to be a continuation of his behavior in other threads.

If you're implying that my suggestion is unhelpful, then I respectfully disagree. Even though model categories don't cover every case of derived/triangulated categories (minor adjustments to the axioms allow us to cover them, however), that doesn't mean that it's not a worthwhile way to think about things. Homotopical algebra is no more abstract than homological algebra, and it provides an immense amount of motivation for it. Most of the constructions in homological algebra have their analogues in pointed and stable model categories (and Quillen's Homotopical Algebra discusses the pointed case and its relationship with homological algebra in depth).

@Yemon: The only complaint I've heard about Weibel's book is that it doesn't discuss model categories (they are discussed in Gelfand and Manin, so what I'm saying isn't so outlandish). From the bits of it that I've read, it is pretty comprehensive, although the classical perspective on, for example, Ext, in Hilton-Stammbach is also extremely useful.

You then proceed to give condescending lectures to people who disagree with you (eg "Are you not familiar with the whole area of stable homotopy theory..."). These people almost certainly are far more experienced and knowledgeable than you (I know this is true for Yemon, and from his comments I surmise that it is also true for Skevek). I have no idea why you continue to do this. Is this a lame attempt to impress us with how "hardcore" you are or how much math you know? Rest assured that it has the opposite effect.

I did not lecture Yemon, because, as you say, he is far more experienced than I am in these matters and mathematics in general. My objections were made towards Shevek, who has made an argument that is as fatuous as it is condescending. I am pretty sure that I know who Shevek is, and if I'm right, then I refuse to defer to his "expertise".

Edit: Actually, my mistake, here is who Shevek is: link to meta discussion where he revealed himself to be [[name removed by request - Anton]]. No offense to the gentleman, but I trust my judgement on this particular subject more than I trust his.

@Harry : I'm reluctant to get involved here, but I can't help myself. Though you're acting innocent, I think that even you know that it would be foolish to try to understand homotopical algebra before acquiring a serious understanding of homological algebra, algebraic topology, etc. Suggesting it just adds noise to the conversation.

Then I guess I'm a fool, because my introduction to homological algebra and algebraic topology was from the point of view of homotopical algebra. I find that these discussions are often dominated by people with extremely conservative views of the order in which one should approach particular mathematical subjects. If I had suggested that the person in question try to learn from Lurie's DAG I and learn about stable ∞-categories, then I would absolutely have deserved to be reprimanded. My suggestion is not nearly so outlandish.

I would argue that trying to understand topics like the Dold-Kan correspondence, Eilenberg-Zilber, simplicial resolutions, André-Quillen cohomology, derived functors, derived categories, etc. are all more natural from a homotopical rather than a homological perspective. That is why I suggested what I did. I stand to gain nothing from "adding noise".