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I am studying Algebraic Topology and increasingly find that when I turn to the internet for help, the explanations and even definitions I need are given in terms of Category Theory (which I know nothing about).

This seems to be true especially for the algebraic, but also the topological, aspects of the topic. (The sites I consult most are Wikipedia and Wolfram.)

My question is this: Is an understanding of Category Theory becoming a prerequisite for a good understaning of Algebraic Topology? If so, I would appreciate any suggestions concerning sources on Category Theory relevant for studying Algebraic Topology (rather than for studying Category Theory, per se).


marked as duplicate by Dietrich Burde, drhab, user91500, user228113, gebruiker Mar 13 '16 at 16:28

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  • $\begingroup$ I sat a course in algebraic topology last semester, and while in principle you can formulate the fundamental group and its relation to top spaces say, with category theory, you really don't need to and we didn't. It was just mentioned to us that it's a way of formulating things $\endgroup$ – snulty Mar 13 '16 at 15:16

You can start studying Algebraic Topology without knowing anything about Category Theory. It is also true that Algebraic Topology consists in traveling (I mean transforming Topological problems in Algebraic ones, solving the second ones and then come back to the topology land and try to deduce some consequences of the solution for the algebraic problem). However, for example, you could start by working on chapters 0 and 1 of Hatcher without necessity of Category Theory.

Anyway some Category Theory is always useful. If you want a very brief overview designed for Algebraic Topology, then I recommend you to take a look at chapter 0 of Rotman's book: An Introduction to Algebraic Topology. He gives a very quick picture of the basis of Category Theory and then he develops it during the book when he needs it.

Once you are confident with some Category Theory Tammno Tom Dieck book is great. In fact, more Category Theory you know, more related the stuff is in your mind. But from my own experience, the Geometry is what matters and motivates what you do. The Category Theory and Algebra are the ways to make the statements trivial and provide you with a powerful machinery.

To sum up. Start by looking at the Geometry, once you have understood it, turn to Categories so all will follow easily and will be more related.

If you give more details about what you are studying, then I will be able to give you more accurate recommendation.

I hope this helps.

  • $\begingroup$ That helps a lot. I am working thru Hatcher on my own, so my only resource for basic definitions and explanations is the internet - which as I indicated seems to relie more and more on category-theoretic definitions and explanations - which does not help me at all. $\endgroup$ – MPitts Mar 13 '16 at 18:14
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    $\begingroup$ The notion of groupoid is a special case of that of category, and many aspects of the theory of the fundamental group are to my mind better seen in the context of groupoids, for example van Kampen Theorem (for dealing with unions of non connected spaces), covering spaces, orbit spaces. Many aspects of groupoids are also usefully studied using categories. See my book "Topology and Groupoids": www.groupoids.org.uk/topgpds,html . $\endgroup$ – Ronnie Brown Mar 13 '16 at 18:29
  • $\begingroup$ @RonnieBrown. Thanks, and I have ordered it. $\endgroup$ – MPitts Mar 13 '16 at 18:57

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