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I have to begin a course in algebraic topology-homological algebra and I know the professor will be following an old approach defining simplicial-singular and cellular homology in this order following Mumkres and Hatcher.

Are there books presenting all these concepts and using a more modern-categorical approach without forgoing the geometry?

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  • $\begingroup$ Maybe you could try "A concise course in algebraic topology" ? $\endgroup$ Commented Sep 15, 2017 at 14:00
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    $\begingroup$ Yes I think you are right, then maybe something like the book by Spanier or the book by Dieck (in both case the name is "algebraic topology") ? $\endgroup$ Commented Sep 15, 2017 at 14:58
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    $\begingroup$ @Nicolas I just skimmed through both books and Spanier seems to deal with the problem more like I want it. If you want to make of it an answer I would accept it. If you expand with more reference I would gladly look at them. $\endgroup$
    – Lolman
    Commented Sep 17, 2017 at 12:11
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    $\begingroup$ I can't make an answer (except repeating what I said in the comments) because I don't know well these books enough, I just know that I didn't like Hatcher because it was not concise enough, and maybe not using enough algebra. Dieck's book was a bit more homotopy-oriented though, so indeed Spanier seems more adapted. Maybe another nice reference is also Davis&Kirk, I only did read few pages about local homology and really enjoyed it. Good luck for your studies :) $\endgroup$ Commented Sep 17, 2017 at 12:28
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    $\begingroup$ (I didn't see that Davis&Kirk was already suggested in the answer, sorry about that) $\endgroup$ Commented Sep 17, 2017 at 12:33

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I would recommend Lecture Notes in Algebraic Topology by Davis and Kirk. It begins with simplicial homology and gets quite categorical from there. When one wants to approach algebraic topology specifically homology from this direction one does not necessarily need simplicial structures which is one reason it is not defined in A Concise Course in Algebraic Topology and the equally interesting follow up More Concise Algebraic Topology both by May, the latter with Kate Ponto.

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  • $\begingroup$ I know that It is not a needed tap in the categorical journey of homology, but because of my needs I need to be thorough on this journey. Thank you for the reference but in the notes they do not delve too much in simplicial constructions and do not even show it forms a category. $\endgroup$
    – Lolman
    Commented Sep 17, 2017 at 11:53

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