# Algebraic topology, etc. for Mac Lane's “Categories for the Working Mathematician”

[NOTE: For reasons that I hope the question below will make clear, I am interested only in answers from those who have read Mac Lane's Categories for the working mathematician [CWM], or at least have a solid grasp of the minimal mathematical background that a prospective reader would have to have in order to profit from its exposition.]

I would like to read Mac Lane's CWM but I'm stymied by the fact that a great many (if not most) of Mac Lane's examples come from areas of mathematics I know little or nothing about. (I can't make any sense of category-theory writing without the aid of copious examples, so skipping Mac Lane's would be pointless for me.)

For example, algebraic topology and homological algebra seem to be greatly favored by Mac Lane as sources for illustrative examples. I could adopt the strategy of simply reading standard, full-length textbooks on these subjects (of which there's certainly no shortage to choose from), but this would take me many months, which is more than I want to devote to such preparatory reading, plus I suspect it may be overkill anyway.

(I should clarify that, when it comes to mathematical subjects that I'm completely unfamiliar with, I just have to read books from the beginning. IOW, I can't simply take the tack of consulting one of such books (or Wikipedia, etc.) as a reference whenever I ran into some unfamiliar example in CWM, and selectively looking up whatever I did not understand. At best, such a narrowly targeted excursion would fill me in some definitions, but it would almost certainly fail to make the example any more useful as an illustration of an abstract concept than it was before. I find examples useful only when I have some familiarity with the example's "case study".)

My only remaining hope is to find introductions to these subjects that are not only brief, but also (and this is crucial) that focus on those areas of their subjects from which Mac Lane draws his examples. (The reason the last requirement is my having found that the little algebraic topology that I know, which I learned several years ago from an introductory treatment by Henle, is of little help to me when I confront Mac Lane's algebraic-topology-based examples in CWM, which suggests to me that the focus of Henle's intro is not particularly well aligned with Mac Lane's point of view.)

EDIT: I'm comfortable with the basics of set theory, general group theory (shakier grasp of rings, monoids, abelian groups), general/point-set topology (first 2/3 of Munkres' book), real analysis (shakier with complex analysis and measure theory), linear algebra and linear/vector spaces, posets.

Thanks!

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Can you detail any area of mathematics that you do feel comfortable with. Especially anything involving algebra. –  Marc van Leeuwen Dec 2 '11 at 13:22
@MarcvanLeeuwen: thanks for your comment; that was indeed a necessary clarification. –  kjo Dec 2 '11 at 13:55
Are you in a great rush to learn category theory? Why not spend the time to get familiar with rings and modules, then read a book like Peter May's Concise Course before picking up CWM? –  Ryan Budney Dec 2 '11 at 16:02
My own path was that I just learned "concrete" math for a long time, and then all of a sudden category theory started to drop into place. It's good to be able to say "a tensor product is just a colimit" -- but the very notion of colimit doesn't make sense (and isn't that aesthetically appealing, to me at least) until you already know a few examples. Category theory -- or at least basic category theory -- is a breeze when you know examples, and nearly impossible when you don't. I can't tell you about any other path, but I know this one worked well for me. –  Aaron Mazel-Gee Dec 15 '11 at 21:36
Put another way: category theory will be all but useless to you until it's all but trivial. –  Aaron Mazel-Gee Dec 15 '11 at 21:41

All condescension aside, my first thought was that, in fact, category theory is an incredibly useful tool and language. As such, many of us want to read CWM so that we can understand various constructions in other fields (for instance the connection between monadicity and descent, or the phrasing of various homotopy theory ideas as coends, not to mention just basic pullbacks, pushforwards, colimits, and so on). So it is in fact relevant WHY you want to read it.

As an undergraduate, I started reading CWM, with minimal success. The idea being primarily that, as you say, I had very few examples. I thought the notion of a group as a category with one element was rather neat, but I couldn't really understand adjunctions, over(under)-categories, colimits or some of the other real meat of category theory in any deep, meaningful fashion until I began to have some examples to apply.

In my opinion, it is not fruitful to read CWM straight up. It's like drinking straight liquor. You might get really plastered (or in this analogy, excited about all the esoteric looking notation and words like monad, dinatural transformation, 2-category) but the next day you'll realize you didn't really accomplish anything.

What is the rush? Don't read CWM. Read Hatcher's Algebraic Topology, read Dummit and Foote, read whatever the standard texts are in differential geometry, or lie groups, or something like that. Then, you will see that category theory is a lovely generalization of all the nice examples you've come to know and love, and you can build on that.

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I started to read CWM with only few knowledge about algebraic topology and homological algebra. You don't really have to know much about these fields to benefit from CWM. Just read the basic definitions (fundamental group of a space, homology of a chain complex) and it will be enough. Of course you benefit even more when you have already some background in the areas where category theory has become one of the central corner point, but you can also learn both simultanously.

After all you also have to ask yourself: Why do I want to learn category theory? Any answer will make it clear which book you have to read first.

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These days, "why do I want to learn category theory?" is akin to what "why do I want to learn set theory?" would have been, say, 50 years ago. Incidentally, when you write "You don't really have to know much about these fields to benefit from CWM", thus flat-out contradicting the very reason I had for asking the question, you are basically implying that I don't understand myself well enough to know what I (not you) really need to profit from reading CWM. It is somewhat condescending, if not downright insulting, wouldn't you say? –  kjo Dec 2 '11 at 14:03
@kjo: You still haven't said why you're interested in category theory. It would certainly help people answer your question. –  Ryan Budney Dec 2 '11 at 16:14
@RyanBudney: I think kjo is saying that he's interested in category theory as a subject in its own right, rather than as a means to understand other areas. Perhaps kjo is drawn to the abstraction, or perhaps to the sheer generality of category theory, but I am just speculating. –  Jesse Madnick Dec 2 '11 at 20:47
@kjo: You should obviously correct me if I'm wrong, as I don't want to put words in your mouth. –  Jesse Madnick Dec 2 '11 at 20:47
@kjo: I didn't want to insult you. Also I'm aware of the importance of category theory and in fact use it every day. But nevertheless you should be able to answer the question "why do I want to learn category theory" for yourself. I think that it is quite important to have a specific motivation to learn a theory whatever. –  Martin Brandenburg Dec 4 '11 at 21:49

Except for some knowledge about ordinary topology you do not need to know much algebraic topology or homological algebra in order to follow (the examples in) the CWM book. In fact, I do not think that he is biased towards these fields at all in his treatment. Most examples are from algebra, so one should have some experience with groups, rings and modules.

Moreover, whenever he refers to such examples he actually explains enough of the respective area so that the reader can follow the example; for instance on p. 20 he introduces homotopy and the fundamental groupoid, homology of complexes is introduced in chapter VIII (abelian categories), chapter VII (monoids) includes sections on the simplicial category, compactly generated spaces, and loops and suspensions. He also gives references to the literature, so these references might be a good starting point if you want to go deeper into the subjects touched; but my point is, that it is not necessary to read them before starting on CWM.

I also found it very helpful to have a good knowledge about the basic concepts of order and lattices, because these provide important special cases for many categorical concepts.

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The examples in CWM are mostly from Algebra or Algebraic topology. I will concentrate on algebra in my answer. I would advise to read "A Survey of modern algebra" by Birkhoff and Mac Lane himself. In this book there is also a chapter on category theory "for the student mathematician", so the level is easier and draws on examples taken from the book. After that you should be able to follow perhaps 80% of the examples in CWM, the remaining 20% being examples in Algebraic topology.

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I started reading CWM just as I was taking my first introduction to general topology class. It's enough to just know a little bit about modules and general algebra --- free constructions, (co-)limit of a system of rings or modules, the tensor-Hom adjunction, etc. All you need to know about homology/homotopy is that it's a functor $\mathrm{Top}\to\mathrm{Grp}$.

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