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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 27th 2012
• (edited Aug 27th 2012)

At homological algebra I have worked on the Idea-section, expanding and also re-writing what was there, in an attempt to make the overview be more comprehensive and at the same time more systematic, hence more useful.

I think what I have improves over what was there, but I am not claiming that what is there now is perfect in any way. Feel free to fiddle with it.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeAug 27th 2012

I think this is good, thanks. I fiddled with it some, attempting to make it even better.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeAug 27th 2012

Thanks, that’s good.

I have just made one further change: I replaced the direct pointer to section 6.5 in HTT by a link to the entry Whitehead theorem.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeAug 27th 2012
• (edited Aug 27th 2012)

I noticed that the Idea-section was missing a pointer to stable model categories, so I added one.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeAug 30th 2012

I have added further references to homological algebra, and a bit on the history.

• CommentRowNumber6.
• CommentAuthorTim_Porter
• CommentTimeAug 30th 2012
• (edited Aug 30th 2012)

In the Ideas section, there seems to be some attempt to make homological algebra more part of algebraic topology than of algebra! Homological algebra, to me, is the study of algebra via homological methods. At present this aspect is under-represented in the ideas section of the entry. What do others feel about this? Should the ideas section be enlarged to talk a bit about homological algebra as used in algebraic geometry and commutative algebra, or about homological group theory, etc.? It could be left as it is, but the idea that stable homotopy theory is the raison d’etre of homological algebra does seem to me a bit strange.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeAug 30th 2012
• (edited Aug 30th 2012)

Here is the explanation (now added as item 5 in the entry):

1. Algebra in stable homotopy theory is higher algebra over E-∞ rings, and homological algebra provides approximations to that: by the stable Dold-Kan correspondence chain complexes of $R$-modules are a presentation for HR-module spectra. Moreover, A-infinity algebras in $HR$-module spectra are presented by dg-algebras, hence by ordinary associative algebras in chain complexes. Similarly E-infinity algebras are presented by commutative dg-algebras, hence by commutative algebras internal to chain complexes. By variation of this theme a multitude of notions in higher algebra finds their representation in homological algebra, for instance L-∞ algebras in terms of dg-Lie algebras: Lie algebras internal to chain complexes.
• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeAug 30th 2012
• (edited Aug 30th 2012)

Hey Tim,

I saw your further edits at homological algebra. All right, if you insist.

I have just slightly edited a bit further:

Homological algebra in its traditional form is the study of algebra (in particular of modules over rings …

I have changed to

Homological algebra in its traditional form is the study of properties of modules over rings

I have slightly edited a bit more here and there, check out the diff.

One general comment: I find it remarkable how much resistence the nPOV receives on the $n$Lab. Why is that? Why the insistence on starting entries with historical points of view that have been superceded?

I find this is analogous, in the context of physics, to insisting, that, for instance, the entry on electromagnetism starts with the words

Electromagnetism in its classical form is the study of the luminiferous aether.

or that the entry on celestial mechanics starts with

Celestial mechanics in its classical form is the study of epicycles.

You see: yes, our forefathers could compute most phenomena in electromagnetism using some complicated constructions formulated in terms of some aether, and they could indeed compute most phenomena in celestial mechanics using a huge machinery of epicycles. Yes, they founded the subject and for decades and centuries this is how it was done.

But nevertheless, in physics, when people realize that a traditional subject has been conceived from an unnatural point of view all along, people change their point of view. They do adopt the new paradigm as the natural new point of view. They don’t disregard any of the old insights, but they rewrite their textbooks and their wiki entries such as to teach their readers the new and better point of view. Discussion of aether and epicycles is still found in the textbooks, but in later chapters, not at the beginning, not in the Idea-section. Because it is not the right idea. It’s just an idea that happened to be historically relevant.

Much of the traditonal homological algebra that you amplify is the study of epicycles. It’s still useful for concrete computations (and the various nLab entries discuss that at length), but it is not useful as an explanation of the subject.

• CommentRowNumber9.
• CommentAuthorTim_Porter
• CommentTimeAug 30th 2012

I understand your point, don’t worry. What I feel is that a ’debutant’ may look at o general entry like this one and be put off as their only view so far has been the ’traditional one’ and I would like them to appreciate the nPOV and not immediately shy away from the entry. There are also very many homological algebraists who would not recognise the subjects as it was described and would have a very negative attitude to the nPOV as a result. I have worked with people who do the type of (commutative) algebra which uses homological methods extensively, and yet they would not think of themselves as doing stable homotopy theory.

I must admit that as I was a low dimensional / fundamental group person in my initial research and had all of Adams’ students pushing stable homotopy theory as the only thing worth knowing, I am not that happy at seeing stable homotopy theory pushed as ’what homological algebra is about’. This is a prejudice of mine! :-)

• CommentRowNumber10.
• CommentAuthorTodd_Trimble
• CommentTimeAug 30th 2012
• (edited Aug 30th 2012)

For what my opinion may be worth here, I think the current entry reads fine. The “history” is very short and unobtrusive, and I think it leads well into the very next nPOV paragraph. Pedagogically, I think it’s sound.

• CommentRowNumber11.
• CommentAuthorTim_Porter
• CommentTimeAug 30th 2012
• (edited Aug 30th 2012)

I tend to think we have to pay some attention to the pedagogical aspect, without letting it dominate. Make it easy for people to see the nPOV, and to understand it and they are more likely to contribute to the overall development both of the subject and the nLab itself. I also wonder about the impact on others of the use of ’really’ as in ’it is really the study of…’.

I am wary about this sort of thing as I experienced the downright hostility to category theory in the 1970s due to some unwary comments about what was the ’right’ way to do things!

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeAug 30th 2012

all of Adams’ students pushing stable homotopy theory as the only thing worth knowing, I am not that happy at seeing stable homotopy theory pushed as ’what homological algebra is about’.

How do these two sentences hang together? Of course stable homotopy theory is not the only thing worth knowing. But homological algebra is a topic in stable homotopy theory, as demonstrated at some length in the entry.

And this is nice, for the following reason: stable homotopy theory as such is conceptually very simple. It is simply the study of $\infty$-categories with pullbacks and 0 in which looping is an equivalence. This makes it nice and easy conceptually. As we discussed elsewhere, this is so very easy that it seems possible to formally axiomatize this in type theory.

Of course working within these axioms may require lots of computational tools. It turns out, as discussed in the entry, that homological algebra provides plenty of such tools. And so thereby all these tools find their conceptual meaning and conversely the general abstract axiomatics is equipped with a powerful computational toolbox.

And so everything is maximally nice.

• CommentRowNumber13.
• CommentAuthorTim_Porter
• CommentTimeAug 30th 2012
• (edited Aug 30th 2012)

homological algebra is a topic in stable homotopy theory

I suspect that is not the case for the majority of people who consider themselves homological algebraists. Homological algebra is very closely related to stable homotopy theory, YES, and there is a large overlap in methods and insights, but homological algebra is distinct from stable homotopy theory because of having differing objectives. (I have worked in homological algebra and would not consider what I did was stable homotopy theory as that is far from the considerations that were involved in what I had done.)

As to the two sentences, I was merely pointing out that I am prejudiced against stable homotopy theory as I had to put up, as a postgrad student, with being told by others postgrads that it was the ’best thing since sliced bread’ (as we say in Britain). (They also said that my work on the beginnings of shape theory and then strong shape theory and hence the nPOV approach to things, was uninteresting and unimportant. They were wrong.:-) but as I said ’I am prejudiced’.)

To see stable homotopy theory ‘taking over’ the older subject of homological algebra, as the wording seems to imply, rather than growing alongside it in parallel to the mutual benefit of both areas, (which is how I see the nPOV influencing both) seems worrying to me.

My main point however is the pedagogic one. If we give too heavy a dose of nPOV early on in ideas sections, it may put off the reader, which is not a good idea. To my mind the idea section should explain the idea of the topic in fairly non-technical terms, so as to persuade the intending reader to continue. This needs to be a balanced approach as we don’t want the reader to go away with just the old view in mind.

• CommentRowNumber14.
• CommentAuthorTim_Porter
• CommentTimeAug 30th 2012
• (edited Aug 30th 2012)

On a separate matter but still on homological algebra, I would suggest starting a separate entry on the History so as to get it away from the bottom of the page. (If any of you have note read Chuck Weibel’s article on this it is well worth looking at.)

Another hopefully non-contentious point: the subsection on Entries on Homological Algebra has homological algebra which leads straight back to the page on which it occurs. (This reminds me of Peter Freyd’s reference to something in his index (to the Abelian Categories book) which was just reffing the reader to that entry on that page of the index!) Can we handle this in better way as it gives a strange impression to me?

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeAug 30th 2012

I suspect that is not the case for the majority of people who consider themselves homological algebraists.

Also the majority of mathematicians will not agree that much of their work is absorbed by category theory or by higher category theory. To me the point of the $n$Lab is that we do not feel that majority decides about what is good in mathematics and how we want our wiki to be. If we did that, we would have to rewrite the entire $n$Lab and call it the 0Lab instead. But instead here we do have the nPOV even if other people may not enjoy it.

I do understand these discussions about not saying that something is part of higher homotopy theory. I can have these discussions with lots of people around me. What I don’t understand is why we have them on the $n$Forum and concerning the $n$Lab. I am thinking of the $n$Lab as the natural and canonical place where the truth may be promoted and explained: Homological algebra is a subject in stable homotopy theory / stable $\infty$-category theory.

I know that it’s easy to find lots of people who won’t like this statement. I can easily find lots of people who don’t like homotopy theory at all, don’t like category theory, don’t like higher category theory. But that doesn’t affect how I think certain topics should be presented on the $n$Lab.

• CommentRowNumber16.
• CommentAuthorTim_Porter
• CommentTimeAug 30th 2012

Urs, Please do not think that you know ’the truth’, that just gets peoples backs up. I suggest we drop this discussion as it is getting nowhere.

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeAug 30th 2012
• (edited Aug 30th 2012)

I suggest we drop this discussion as it is getting nowhere.

The thing is that we keep rehashing this discussion in the context of various topis.. It comes up again and again. Maybe it would be good if we sorted it out once. I keep writing entries, you keep complaining that I am amplifying a homotopy theoretic point of view and not presenting it from the point of view of long ago.

Please do not think that you know ’the truth’,

It just happens to be a premise of the $n$Lab, the nPOV. It’s just different here than it is on Wikipedia or on the 0Lab. Contributors not comfortable with the $n$POV, who feel it should be de-emphasized and the 0POV should dominate over it might have to consider the fact that they are not quite in accord with the idea of the $n$Lab.

On the $n$Lab it is not supposed to be considered “strange” that homological algebra is presented as a part of stable homotopy theory. On Wikipedia and on the 0Lab that would be strange. Here it would be strange if we did not amplify this relations. That’s what the $n$Lab is for. Otherwise we could just edit Wikipedia.

• CommentRowNumber18.
• CommentAuthorTim_Porter
• CommentTimeAug 30th 2012
I think 90% of our disagreement is a question of style.. I like a lead in whilst you like to plunge straight in.

I twice wrote a long reply to other points, but the system hung and lost them both.

I understand your point but I do not oppose an nPOV at all, I do think however that homological algebra is more than just a subtopic of stable infinity homotopy theory, mainly because it treats the relationships between algebraic properties of objects and the properties of the corresponding stable infinity homotopy theory. It thus uses stable infinity homotopy theory (in the form of chain compleses etc. in the traditional form of the theory) as a tool, so the relationship is more complex than you seem to think.

I must stop now. We won't resolve the 'lead in' versus 'plunge' as it it a question of taste really. The other point needs thinking about a bit more.

All for now.
• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeAug 31st 2012
• (edited Aug 31st 2012)

I think 90% of our disagreement is a question of style.. I like a lead in whilst you like to plunge straight in.

No, this is not the disagreement. I will try to clarify once more.

I am all in favor of pedagogy and expositions. I think I write considerably more pedagogical expositions in $n$Lab entries than most other contributors do. It is true that I also write many entries without pedagogical introduction, for lack of time and energy and according to the explicit Purpose of the nLab. I think I simply write quite a few more entries in general. The fact that some of them are lacking something or other is not a sign of an agenda.

In particular here at homological algebra I had spent quite some time with writing the entry, and writing it as an exposition. I made it start by saying right away that the archetypical example of homological algebra is singular chains on a topological space, which seems to be what you want to hear.

Then I had created the explicit subsection titled “As a toolbox in stable homotopy theory”, where I listed step-by-step which traditional construction in homological algebra is re-interpreted as presenting which construction in stable homotopy theory.

I did all this as a means of an exposition. I don’t need this information myself, because I already know it. But I thought readers might benefit from seeing this explained. Did you read the explanations? The one announced in #10, for instance?

As far as I can see you just added a few words to this (and without announcing this here, by the way) and I re-edited those in turn. So I am happy with how the entry looks.

What I am objecting to is your complaint in #6, that all this exposition which I wrote you find to be advocating a “strange” point of view. I think given that the entry goes to some length at trying to justify this point of view with plenty of concrete data, it would be nice if you could provide a more substantial contribution to the discussion than just dismissing everything outright on the basis of the prejudice that you admit in #9:

I must admit that as I was a low dimensional / fundamental group person in my initial research and had all of Adams’ students pushing stable homotopy theory as the only thing worth knowing, I am not that happy at seeing stable homotopy theory pushed as ’what homological algebra is about’. This is a prejudice of mine! :-)

Since you seem to be in disagreement, hence, not just with me but with various experts on this topic, maybe the burden of further justification is not so much on me. I don’t claim that the entry is perfect yet, but I think I did invest quite a bit of time writing an exposition. I’d be happy to see others join in making it an even better exposition. But I am wary of claims that what is being exposed there is “strange” on the basis of a “prejudice”.

And now I would suggest we move on and try to keep the ratio of opinion pieces over discussion of concrete contributions small. Let’s invent this rule: for each opinion piece here on the $n$Forum one needs to “pay” with one that announces a genuine contribution. Just so as to keep us not too distracted form getting work done.

• CommentRowNumber20.
• CommentAuthorTim_Porter
• CommentTimeAug 31st 2012
• (edited Aug 31st 2012)
(i) If I nit pick, then 'the archetypical example of homological algebra is singular chains on a topological space' is not part of Homological Algebra. It is part of Algebraic Topology. The object of study in the example is the space not an algebraic object.

(ii) 'As far as I can see you just added a few words to this (and without announcing this here, by the way)' .If you must know I had to ruch off to do something non-mathematical as I finished that edit. When I came back you had changed things, and I think improved my wording. :-)

I agree with the idea of a rule of thumb about the ration of comments to entries. I also would like someone else to look over the entry to see if there are any other improvements that could be made. I do think the wording does not quite succeed in getting your point across. I will not try to do this myself as I would be likely to create another stream of comments! :-)

(Minor matter: by the way I did change 'allows' to 'allows us' in Waldhausen category. The usual convention is that 'allows' is a transitive verb so 'allows us to..' or 'allows for the ...' are more correct. This is not directed at you but is to flag up the grammatical problem as it is a frequent glitch in articles and is in some nLab enties as well I think.)
• CommentRowNumber21.
• CommentAuthorDavid_Corfield
• CommentTimeAug 31st 2012

Worth linking to the cosmic cube, where it says that the variety of homotopy theory for

strictly abelian strict $\infty$-groupoids correspond to homological algebra.?

• CommentRowNumber22.
• CommentAuthorTim_Porter
• CommentTimeAug 31st 2012
• CommentRowNumber23.
• CommentAuthorMike Shulman
• CommentTimeSep 1st 2012

’best thing since sliced bread’ (as we say in Britain)

You consider that a British saying? We say it plenty here in America too.

• CommentRowNumber24.
• CommentAuthorMike Shulman
• CommentTimeSep 1st 2012

Tim, I am confused. You seem to be saying (and the first paragraph of the entry now says) that homological algebra is defined by interest in properties of algebraic objects, studied in a particular way. But that’s not how I learned it, even back before I understood anything about higher categories or stable homotopy theory. In my intro algebraic topology course, we used chain complexes to compute homology of spaces, and the forrmal stuff we did with chain complexes was called “homological algebra”.

So I always thought that just as “category theory” refers to the abstract study of categories and can be applied to study many different kinds of categories depending on what one is interested in, “homological algebra” referred to the abstract study of chain complexes, and can be applied to many different kinds of chain complexes depending on what one is interested in, e.g. singular or cellular chains on a spaces (for algebraic topologists), bar resolutions of modules (for group theorists), injective resolutions of sheaves (for algebraic geometers), etc. I think this is similar to the point Urs is making, but what I’m saying is that I don’t think it has anything to do with nPOV-ness or stable infinity-categoies. Saying “homological algebra is the study of algebra by homological methods” sounds to me as wrong as saying “category theory is the study of topology by categorical methods”.

• CommentRowNumber25.
• CommentAuthorTim_Porter
• CommentTimeSep 2nd 2012
• (edited Sep 2nd 2012)

I go back further than you to when I learnt HA :-(. That is not to say there is a right or wrong in this. I learnt HA more as the study of algebra via the use of homological tools and the study of those tools, thus group homology would be seen as being part of homological algebra (as well as being part of group theory). This formed part of a Master’s course in Algebra so perhaps was slightly slanted to homological ALGEBRA rather than HOMOLOGICAL algebra! By that I am suggesting that the first is studying algebra via homology, the second the algebra of Homology. From that perspective the study of singular chains on a space is thought of not so much as being part of homological algebra, but of homology theory which would be more part of algebraic topology, (but see also further down this comment.) This is just a difference in emphasis. In MacLane’s Homology, there is very little mention of topological side, but to see that book as the study of chain complexes is too restrictive, I feel.

The assignment of a particular part of a subject to a given labelled part of mathematics is largely a waste of time as it is the interconnections that are important and if by so doing we block understanding of those relationships, I feel that that is not good as it puts up barriers to the transfer of ideas. I felt that whilst there was a huge and important overlap between homological algebra and stable infinity homotopy theory, to say one was a topic in the other risked forgetting or underestimating that part of homological algebra that is the ’study of algebra by homological methods’, i.e. if you like the applications to group theory, algebraic geometry etc. From my point of view, the applications of homological algebra form part of the subject, so I prefer to see homological algebra as a wide area including its applications (and that includes its applications to algebraic topology and in particular to stable homotopy theory).

There were two posts that I lost due to problems with my login, in which I tried to say that a mathematical topic area such as homological algebra is not just determined by the objects it studies but by the questions asked, its ’style’ almost and therefore its paradigms. (What I wrote was not that well formed so it is probably a good thing that it was lost. :-) That poses, however, an interesting question at a somewhat philosophical level as to how we delineate topics and the use we mathematicians make of such delineation. I am now perhaps more inclined to think that the boundaries between topics are too fuzzy to bother with, and that the labels used are for convenience only.)

I agree with you that the difference between Urs point of view and mine is not primarily on the nPOV. I thought his statement was a bit too one sided and ignored large parts of the subject of homological algebra, especially those parts that deal with the applications. (I am working with people at Lyon (including Computer Scientists) on homological aspects of rewriting theory. I am sure that this forms part of homological algebra, (and of other areas as well), yet not sure that it is fair to burden stable infinity homotopy theory with it!)

You object to my "homological algebra is the study of algebra by homological methods". I agree with your criticism, I should perhaps have also said … ’and of those methods’, otherwise my view errs too much on the other side, only mentioning one part of the area and also ignoring the applications to homotopy theory.

• CommentRowNumber26.
• CommentAuthorMike Shulman
• CommentTimeSep 4th 2012

I agree with basically all of that. The problem is that we have to say something when introducing a topic! I rewrote the introduction to mention three different possible points of view; what do you think?

• CommentRowNumber27.
• CommentAuthorTim_Porter
• CommentTimeSep 4th 2012
• (edited Sep 4th 2012)

Perhaps the first and the third are too alike and need merging. Why not simply add ’The study of algebraic objects (groups, Lie algebras, etc.) using homological methods. This gives different subtopics such as the study of group (co)homology.’ or something along those lines.)

Not ignoring the red herring principle, non-abelian homological algebra forms a part of homological algebra and (in the shape of non-abelian cohomology) was one of the aims of Grothendieck’s pursuit of stacks. It is thus central to the nPOV. Perhaps it could be mentioned more centrally here.

I leave you to adjust things as you see fit.

• CommentRowNumber28.
• CommentAuthorjim_stasheff
• CommentTimeSep 4th 2012
I have long been bothered by the phrase non-abelian homological algebra'. If it's a part of homological algebra
(which I beleive), then how does one identify the splitting - the complement being abelian homological algebra'??
• CommentRowNumber29.
• CommentAuthorTim_Porter
• CommentTimeSep 4th 2012

I suppose, Jim, that whilst abelian group theory means the theory of abelian groups, and group theory that of all groups that might just happen to be abelian, so homological algebra should refer to all aspects of the subject and not just to the abelian part….. That is essentially where I came in on this debate as, to me, the reference to modules, chain complexes etc, put too much emphasis on the abelian side and the simplicial side (not only stablised) should also be there somewhere, as should the applications. It is but a name … and a rose by …., (Shakespeare), but it worried me a bit.

• CommentRowNumber30.
• CommentAuthorMike Shulman
• CommentTimeSep 4th 2012

I leave you to adjust things as you see fit.

I decline, since I already wrote what I saw fit. But you should feel free to adjust things as you see fit. (-: I wouldn’t disagree with merging 1 and 3 or with mentioning the nonabelian case.

• CommentRowNumber31.
• CommentAuthorTim_Porter
• CommentTimeSep 4th 2012
• (edited Sep 4th 2012)

On a serious point, what would be the consensus about where simplicial techniques within homological algebra should be put. Is it a historical accident that they tend to be mentioned as being homological algebra, whilst ’homotopical algebra’ would be more appropriate if that term was not almost exclusively used for model category theory etc. (I am sort of pondering this regardless of the structure of the nLab etc.)

• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeOct 30th 2012
• (edited Oct 30th 2012)

I took the liberty of adding a pointer to my lecture notes Homological Algebra - An Introduction to the list of References at homological algebra.

• CommentRowNumber33.
• CommentAuthorzskoda
• CommentTimeOct 30th 2012

Homological algebra from modern point of view (which includes nonabelian version) is precisely equal to homotopical algebra. This is the point of view which has been already 25 years ago taken in Gelfand/Manin book on homological algebra where the first chapter is about simplicial techniques and the last solely about model categories. By homotopical algebra, of course, one does not mean only the “algebra of” model categores, but also derivators, A-infinity categories, and other versions.

Of course, traditionally people who do modern homogebra and say homotopical vs. homological divide across the social line – being either more on the topological side, or more algebraic side. So homological part of community works mainly with enhanced derived categories (like A-infinity, dg-categories) while homotopical more on quasicategories, model categories and alike.

It is not entirely true that what the algebraic community does absorbs only into stable infinity-categories. For example, dg- and A-infty categories which belong to this realm are those which are pretriangulated (plus being in characteristic zero). There are dg-categories which are interesting and not in that class. And there are other examples…

• CommentRowNumber34.
• CommentAuthorMike Shulman
• CommentTimeNov 3rd 2012

Are you saying that all stable $(\infty,1)$-categories are characteristic zero? I don’t think that’s true.

• CommentRowNumber35.
• CommentAuthorDavid_Corfield
• CommentTimeAug 15th 2018

Can anyone figure out why this link in references isn’t working?

Urs Schreiber, Homological Algebra - An Introduction?

The intended page has the redirect ’HAI’.

• CommentRowNumber36.
• CommentAuthorDavid_Corfield
• CommentTimeAug 15th 2018

Added bibliographic details to Gelfand and Manin’s ’Methods of homological algebra’. I was alerted to this by an answer to an MO question on the possibility of teaching Lurie’s Higher Algebra to graduate students, where someone writes

One could argue this is the next logical step of a progression. Older books in homological algebra refused to use spectral sequences. Then Weibel’s highly praised book does the opposite and introduces them early on, but relegates derived categories to a final chapter. Then Gelfand-Manin take it one step further and start with derived categories. They discuss dg-algebras and model categories at the very end and stop short of discussing non-abelian derived functors. Lurie’s higher algebra is the next step but it’s also quite big and not meant to be used for lectures… (https://mathoverflow.net/questions/225712/teaching-higher-algebra)