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You will have guessed it from my recent activity here: I am busy writing lecture notes for an introduction to homological algebra.
This will proceed incrementally at:
There are 14 sections in 5 chapters. Currently sections 1) and 2) exist as coherent and complete text (not perfect, but coherent and complete). Sections 3) and 4) exist as incoherent and incomplete text, and 5) to 14) + Outlook as just keyword lists. Ah, and the References-section exists.
Of course everything is hyperlinked to the $n$Lab. I’ll be developing this further during the next weeks until the labels “coherent and complete” apply to all 14 sections. Then maybe this might be something to move to the $n$Lab and potentially develop there further? Not sure. But I am posting it here now in any case, also so that you can see what all that editing activity on the $n$Lab is related to.
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The only homological algebra book which (like your lectures) starts also with simplicial objects is, to my knowledge, Gelfand-Manin Methods of homological algebra (please distinguish from the other, encyclopaedia book on homological algebra by the same authors, which is quite different). The first English translation of the Russian original has introduced quite a few typoi, including in diagrams. Unfirtunately, the authors have given up on writing the second volume, which was planned at the time of Russian original). In any case I think the Gelfand-Manin Methods book is the closest to your choice and order of the topics.
Thanks, Zoran. I am kind of supposed to be using Weibel’s book. Of course, that book is somehow at the opposite end of where I am – but maybe that’s a good thing, we can meet in the middle :-) First I thought, ah, well, for the sake of it I’ll just linearly go through Weibel’s book, that’s why I made the detailed list at An Introduction to Homological Algebra. But it seems that the moment I try to explain anything I am departing from Weibel. This is how my present notes come about, the result of two opposing forces. So far it’s being received well, we’ll see how it goes. And apart from some ordering of topics there is not that much of a choice. Weibel discusses chain homology etc. just as well, only that it’s hidden somewhere in the middle of something else and not highlighted as the raison dêtre of the subject.
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Unsurprisingly for those who saw the original layout, I ended up splitting off the discussion of mapping cones as a separate session:
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I see you are doing classical derived functors (i.e. in terms of abelian categories, a la Cartan-Eilenberg and Tohoku), rather than the modern concept of derived functors (including the total derived functor) for triangulated categories. Is this for lack of time, or you have some deeper reason to be so classical in this part of the course ?
Yes, I am under severe constraints. I need to cater for the students. The running assumtion is that I am allowed to speak of categories and functors as such, but not allowed to do or even assume any actual category theory. Right from the start the premise was that this course does not do localization theory.
As I said, I am actually supposed to be following Weibel. But I did try to emphasize and explain heuristically in the course how taking homology groups on functors applied to resolutions is the poor-man’s version of descending a functor to the derived category.
This is what is currently (and hopefully still in the future) remark 34.
I would not like to be in that skin. I had so much freedom 4 years ago or so, when I had part of a semester devoted to the derived categories as triangulated categories at the level of Gel’fand-Manin’s, Methods of homological algebra, what was so nice teaching experience (I forgot myself much of the interesting details which I was at the top when lecturing that, as I did not use this for a while). My goal was to use that machinery to do the various important derived functors in abelian sheaf theory (the course was about sheaf theory with some cohomology), but did not really had time to do much of that at the end of the course.
You know, that’s why I like the idea of a set of lecture notes fully hyperlinked to the $n$Lab, as I am doing it: it’s actively an entry point to a larger universe, for those who want to walk through that door.
There is one very good student who asks the kind of questions that one would answer in a more general abstract setup of the course. I reply in as far as there is time, and then I say: follow the links in the notes and find plenty of further in-depth information and further pointers.
Not allowed to do category theory? That seems very strange, and even incomprehensible. Probably I misunderstand.
There is just no time to do any category theory. I can write down a direct sum for modules explicitly and point out that it satisfies some properties characteristic of what is called a biproduct, but if I’d embark on a general introduction to limits and colimits, that would spoil the course.
The students have very little background. Nobody had seen a tensor product before, not even of vector spaces. One had seen singular homology before. All these things want to be introduced and discussed, and then we haven’t even started speaking about homological algebra.
There is just no time to do any category theory. I can write down a direct sum for modules explicitly and point out that it satisfies some properties characteristic of what is called a biproduct, but if I’d embark on a general introduction to limits and colimits, that would spoil the course.
Okay, I see. It sounds like the type of thing I used to do when teaching say introductory real analysis or functional analysis – I would scarcely breathe the words “category” and “functor”, but still I would be secretly hammering home some points like various universal constructions (e.g., the universal property of Cauchy completion in terms of uniformly continuous maps, or the universal property of $l^1(X)$ in terms of short maps, etc.). I would also secretly work in adjoint functor manipulations, frequently at the level of posets (image and inverse image, etc.).
I hope you’re having some fun with this! :-)
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The second part, 10b), is only as of this morning and still needs a bit of attention and a bit of glue here and there.
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I have also added a Thanks-section for Todd! :-)
Grr, there is some flaw in the automatic numbering. Prop 86 is automatically referred to as prop. 85 further below.
I must have some typo somewhere in the Proposition-environments. But I haven't spotted it yet. This is a bit tedious...
Of course you’re welcome, Urs!
BTW: if ever there’s some presumably classical technical lemma you’d like but rather not spend time hunting down a proof of, just ask! If a lemma elf knows a proof and has some time, one might appear by magic. ;-)
BTW: if ever there’s some presumably classical technical lemma you’d like but rather not spend time hunting down a proof of, just ask! If a lemma elf knows a proof and has some time, one might appear by magic. ;-)
Yes, good. I will!
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(or at least a first version, am now proof-reading…)
This is excellent stuff, Urs; really impressive!
Ah, you think so? Thanks for the feedback, that’s useful to hear.
I am also grateful for criticism. Myself, I certainly see room for fine-tuning various things in the notes, concerning the exposition and concerning the content itself. Maybe I find time to do further edits.
One aspect that I noticed while giving the course is that it is a bad idea of the administration that allocates these courses to not formally harmonize the Homological Algebra course with the Algebraic Topology course. The optimal situation would be the course on algebraic topology and that on homological algebra running in parallel (different days of the week) and students forced to hear them in parallel, with close harmonization of the topics, AT providing the examples, HI the abstract theory and both playing the ball back and forth.
Maybe next time….
a bad idea of the administration that allocates these courses to not formally harmonize the Homological Algebra course with the Algebraic Topology course
Yes, that would make sense. Without that, motivation for homological algebra becomes a lot trickier (although not impossible).
Do you have assigned texts for this course? I get the impression that for the most part you are developing the text yourself (but maybe refer to others as well). Do you have homework sets?
Yes, the text is all mine, though of course I am using all the standard sources. (I just don’t see a single standard source that develops the material the way I feel it is needed, otherwise I would have just sticked to such a source.)
I am not preparing extra homework sets, but there is plenty more material in the notes than I actually write to the board. I keep assigning the proofs that I skip over as homework (“first try it yourself and if you get stuck look at the notes for the solution”).
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But mostly I have been further working on prettifying section 12).
It looks like I’ll stop here and leave section 14) and the “Outlook” undone (not entirely surprisingly). Maybe some other day I find time to fill in material there.
To expand briefly on my #23: I’ve looked at Weibel’s book, but it didn’t really mean much to me, because I don’t have much of an intuition for abelian stuff — I had wondered if there was a text that presented similarly basic material from a more abstract, or at least more homotopical, point of view. So I really like how you’ve linked the traditional homological algebra with more high-level homotopical ideas, in the earlier sections anyway.
So I really like how you’ve linked the traditional homological algebra with more high-level homotopical ideas, in the earlier sections anyway.
That was one of my motivations, anyway, to indicate for each topic what its whereabouts are, what the reasons for it are on top of the fact that it just so happens that somebody dreamed up some construction and it just so happens to turn out to be useful.
I remember years back when I was a theoretical physics student trying to understand the derived category of branes of the topological string; I picked up Weibel’s book and read it (in Vietri, southern Italy on the beach, during one of those small but nice annual mathematical physics meetings which they (used to?) have there) and my constant and increasing impression was an “Okay – but why on earth?”
Sorry, Jim, I don’t understand what you are meaning to say. Could you elaborate?
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