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I intend to try working through Bredon's seminal sheaf theory text prior to graduating (I am currently a second year undergraduate), but it is at a level which is far beyond my own (friends of mine who study algebraic topology have gone until their second and third years as graduate students before touching it). However, I am interested in algebraic geometry (though the material treated in Bredon's text is certainly of relative interest to me) and find Bredon's "Topology and Geometry" to be a superb treatment of the algebro-topological tools which may have some utility in my future studies (Bredon takes a more geometric approach). Is there another text which might be a better 'crash course' on algebraic topology for someone at my level (a bit of algebra, analysis, and point-set topology, with a good deal of category theory), or am I on the right track with Bredon's text? Thanks!

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To be frank, my reaction to this is "what's the rush?" I am perhaps blinded by my own personal limitations, but I believe that the undergraduate years are best spent doing as much mathematics as possible using only your "bare hands" and learning as many concrete examples as possible, whether in topology or algebraic geometry. All the fancy abstract machinery is much more meaningful, if you have first tried to do things without it. Also, I have seen too many precocious students try to answer relatively simple questions using too much machinery. –  Deane Yang Jun 6 '10 at 23:43
Deane, I was going to write exactly the same thing. Learning something as dry as sheaf theory before encountering a real need for it (in alg. geom., several complex variables, etc.) is a very unwise idea (like learning homological algebra in the absence of applications). Retention will be negligible. Sheaf theory is a powerful body of techniques for solving certain kinds of problems, but this stuff is best understood only in the service of an application (e.g., cup products, deRham comparison isom, etc.). Anyway, Godement's sheaf theory book (in French) is better than Bredon's. :) –  BCnrd Jun 7 '10 at 1:15
Alright, well I am not unsure as to my capacity to at least learn some geometry/topology; is the Bredon text mentioned above a good place to start learning these topics in a 'more advanced' light? I have the basic results of point-set topology and analytic geometry in my ken, so to speak. –  lambdafunctor Jun 7 '10 at 1:56
So learn more geometry and topology! There are plenty of books that aim to teach you complex differential or algebraic geometry, where just enough sheaf theory is introduced as needed. If you have not already learned everything in the context of Riemann surfaces, that's a really nice easy place to start. Also, if you have not read Bott-Tu (which really is a graduate text), I suggest studying that before Bredon. –  Deane Yang Jun 7 '10 at 3:40
@Deane For some stupid reason I can never comprehend,I never really warmed up to Bredon.And I have no idea why-he seems to be doing everything right! It just seems to me if you're working your way through Lee,Bott/Tu,May and Hatcher,Bredon becomes superfluous and strangely cold and forbidding in comparison. –  Andrew L Oct 8 '10 at 16:14

3 Answers 3

Firstly, as you say you are interested in algebraic geometry, Bredon's book may be a slightly unfortunate choice. It very much emphasizes the point of view of the "espace étalé"; it's not much harm to translate things back into the "site" perspective, which is the only one that generalizes to algebraic geometry.

No offence to this great book, but it is extreeeeeeeeeeeemely technical and certainly written for people who want to know all the possible subtleties of sheaves on topological spaces (e.g. it is full of beautiful examples that show how badly things can go wrong ;-); but that's certainly not the kind of stuff a beginner in algebraic topology should learn; and needless to say, it's about subtleties in topology that the Zariski topology certainly does not have.

My advice would be the excellent introduction by Demailly in his online book http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf - the chapter about sheaf cohomology is essentially self-contained and avoids the use of higher homological algebra (unlike the books of Shapira / Kashiwara etc., Schneiders).

Have fun! Sheaves rock!

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Thank you, Kai! It looks like a great text! –  lambdafunctor Jun 7 '10 at 1:55

I think you are on the right track in the sense that it's a good idea to read Bredon's Topology and geometry while you're an undergraduate. It starts from scratch more or less and contains the necessary prerequisites for many more advanced books in different areas: algebraic topology (Spanier etc), differential geometry (Besse), Lie groups, you name it.

On the other hand, I don't think that it is formally a prerequisite to reading Sheaf theory. To do this you only need some point set topology and some basic homological algebra (complexes and cohomology but no abelian categories).

On yet another hand (how many hands can one have?), chances are you'll never need to read Bredon's Sheaf theory -- Godement or Iversen are (in my opinion) far more suitable as a first introduction to sheaves than Bredon. There are also two books by Kashivara and Schapira. One is "Categories and sheaves" and the other is "Sheaves on manifolds". Both develop the theory from scratch (especially the first one) but they are a bit dry for a first reading. In fact, the only thing that you can find in Bredon's Sheaf theory but not in any other introductory text I know of is the Smith spectral sequence.

Since this is already way too long to for a comment, let me mention some other sources similar to Geometry and topology that you may find useful. Bott and Tu's Differential forms in algebraic topology can't be recommended high enough. This book starts at an elementary level but contains some of the key ideas in topology, which come up over and over again in many contexts. There are no sheaves there, but if you read it carefully (i.e. try to reconstruct all the proofs in detail), the definition of a sheaf will come as no surprise to you afterwards. Or you may even discover it yourself.

Milnor's Morse theory is arguably one the best introductory mathematical texts ever written and it also falls into the category "what every geometer or topologist should know".

Lee's Introduction to smooth manifolds is quite long but it presents the material in excruciating detail.

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I first learned sheaves in the context of Riemann surfaces, where all you needed to know was one-variable complex analysis. Is that just a too easy and elementary place to start for serious algebraic geometers? –  Deane Yang Jun 7 '10 at 2:46
Deane -- to study sheaves and their basic properties one doesn't need even one-variable complex analysis. I wouldn't say that the notion of a sheaf is very difficult (certainly not more so than the notion of a real number). But the thing is, it is different from what is usually included in the standard school and undergraduate curriculum. So the examples one starts with should play a psychological role as well - they should help convince the mind that sheaves are useful. Where these examples come from (analysis, algebraic geometry, topology) doesn't really matter as long as they ring the bell. –  algori Jun 7 '10 at 3:33
Hartshorne's description of sheaves is rather bad. It's never explained that sheaves are a formalization and abstraction of what you're doing when you glue continuous maps along an open set in topology. He makes it seem a lot harder than it actually is, and I suspect that many people first encounter sheaf theory through Hartshorne. –  Harry Gindi Jun 7 '10 at 3:41
Milnor, Bott/Tu, and Lee ARE all very good texts from what I've perused. That is, they are informative and non-intimidating to the relative beginner, and something can be gleaned from them (something profound, in fact) with relatively few prerequisites. That you suggest all three exhibits very good taste in mathematical literature on your part, methinks :) –  lambdafunctor Jun 7 '10 at 3:42
^And you make a great point, HG; from what I've read of sheaves in Hartshorne, he presents things in a rather convoluted manner. –  lambdafunctor Jun 7 '10 at 3:44

Although it's one of my favourite books, Kashiwara and Schapira 'Categories and Sheaves' is incredibly dry and abstract (and somewhat nonstandard in notation in many points).

When I first learned the material I found extremely valuable the seventh chapter of Taylor's Several Complex Variables with Connections with to Algebraic Geometry and Lie Groups. It's an all round great book and it made me understand things much more clearly.

You're supposed to read it after chapter six (which might be a little bit of an overkill) but if you're not too much into abstract nonsense you can substitute 'sheaves' or 'modules' whenever he speaks of objects in an abelian category.

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Kashiwara and Schapira is an awesome book,but man,it's brutal. Then again,it's easier then plowing through MacLane and Adult Bredon. –  Andrew L Oct 8 '10 at 5:23

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