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Can someone give me a recommendation on homological algebra textbooks? I would like something that are accessible to a beginner (i.e., someone who have studied abstract algebra) and that have

1) an account on preadditive, additive, monoidal, abelian, triangulated categories, respectively,

2) motivations from algebraic topology exploited, (an elementary example that comes into my mind is the mapping cylinder/cone construction explained in contrast with Puppe sequence,)

3) an explanation on module theoretic topics like injective / projective resolutions,

4) a coverage of sheaf theory, cohomology of groups, and Galois cohomology.

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    $\begingroup$ Possible duplicate of Good introductory books on homological algebra $\endgroup$
    – Watson
    Mar 12, 2016 at 12:52
  • $\begingroup$ @eltonjohn You don't close a question because you got stisfatory answers. You upvote, the useful answers you got and accept the most useful one. If your question was answered in the comments, you kindly ask if the person who answered the question could turn his comment into an answer. If, in the mean time you found the answer yourself, then you can post an answer yourself (and accept it). $\endgroup$
    – gebruiker
    Mar 12, 2016 at 13:59
  • $\begingroup$ @Watson: The question looks duplicate, but (luckily for me) the answers do not. In fact the answers I got are more suited to my concern 1) through 4) above than the ones posted in math.stackexchange.com/questions/28646/…. $\endgroup$
    – eltonjohn
    Mar 13, 2016 at 12:33

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[Weibel, An introduction to homological algebra] covers almost everything you mention, except monoidal categories.

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  • $\begingroup$ Thanks! My impression is Weibel is a good book except the first chapter which is too sketchy. (Of course I would not care if the book were not titled "introduction". ) --- upvoted --- $\endgroup$
    – eltonjohn
    Mar 14, 2016 at 6:46
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As suggested by the OP, I move my comment into an answer with minor extensions.

Well, here we go:

For (basic) monoidal categories theory and applications I recommend Kassel's "Quantum Groups" book. If you want to dive into homological algebra starting from algebraic topology, then pick up Gelfand Manin's textbook (I refer to "Methods of Homological Algebra").

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  • $\begingroup$ Thanks! I browsed Gelfand and Manin. it looks very good (but a bit marred by the remaining typos that escaped revision.) --- accepted --- $\endgroup$
    – eltonjohn
    Mar 14, 2016 at 6:49
  • $\begingroup$ Thank you. Yep, the book contains some typos, but doing the proofs by yourself you will be able to correct them ;) $\endgroup$
    – Avitus
    Mar 14, 2016 at 18:40
  • $\begingroup$ Note that the book of Gelfand and Manin, although wonderful, contains many errors and misprints, so tread with care! $\endgroup$
    – Pedro
    Feb 23, 2019 at 9:37
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The best books have already been mentioned - these are the books by Weibel and by Gelfand-Manin.

A rough guideline to reading these book is as follows:

One picks up elementary facts (snake lemma, complexes, homology, abelian categories) from algebra/algebraic topology courses; then, on the intermediate level, one uses most of Weibel's book to learn yoga of derived functors and spectral sequences in a hands-on way (chapters 2-5, then 6-9 are various specific applications which you might read later on when you need it); finally, for the cleanest modern account of derived categories machinery one reads chapter 3 of Gelfand-Manin. (I personally found chapter 4 on triangulated categories very helpful in my work too).

As far as your specific needs go:

1) Some algebra textbooks have this material (my favorite being Aluffi "Algebra. Chapter 0"). Appendix and chapter 1 in Weibel and chapter 2 of Gelfand-Manin explain abelian categories in depth, albeit in a rather dry fashion. For triangulated categories read chapter 4 of Gelfand-Manin.

2) Both Weibel and Gelfand-Manin have some sketchy remarks on classic algebraic topology, but maybe it is better to directly use algebraic topology textbook for motivation, e.g. May's book "A concise course in algebraic topology". However, both textbooks contain nice accounts of simplicial approach to algebraic topology.

3) Chapter 2 of Gelfand-Manin and chapters 1,2 of Weibel.

4) Sheaf theory is scattered over Gelfand-Manin's book and despite being conceptually explained really well, it might be too hard for a beginner. It might be better to go directly to an introductory text in algebraic geometry such as Vakil's notes or Hartshorne. Group and Galois cohomology at some introductory level could be found in chapter 6 of Weibel.

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