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I know undergraduate algebra (groups, rings, fields, Galois etc), undergraduate differential geometry, undergraduate real/complex analysis and now I feel as though to get to the next level, i should be learning algebraic topology (so i can finally understand what people are talking about when they say homology). I like learning about the structure of things, and i was told that algebraic topology was one of the major sparks of category theory. Im looking for a reference on algebraic topology (a first course), which is as categorical of an approach as possible- given my background of basically 1 year grad.

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    $\begingroup$ Bredon.${}{}{}$ $\endgroup$
    – Aloizio Macedo
    Commented Jun 3, 2016 at 16:15
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    $\begingroup$ Most algebraic topology texts use groupoids, a special case of categories, only partially or not at all. Compare the discussion at mathoverflow.net/questions/40945/… $\endgroup$ Commented Jun 4, 2016 at 11:10

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A lot of category-theory (model cats) stuff can be found in May's "Concise" and "More Concise". I don't know if this is what you want.

If by "categorical approach" you mean an approach full of commutative diagrams (which is cat. theory, but not made explicit and well, scary) and a few figures and geometrical reasonings (like Hatcher's book or Milnor's characteristic classes), then tom Dieck's Algebraic Topology is definitely the way to go.

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I got a lot of help from Rotman's An Introduction to Algebraic Topology to get used to categories and commutative diagrams. It is written in quite an algebraic viewpoint, and it's easy to read.

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