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!
[Math] Is Bredon’s Topology a sufficient prelude to Bredon’s Sheaf Theory
sheaf-theorysoft-question
Related Solutions
Maybe I am reading too much into your pseudonym and your partly apologetic and partly condescending comments about the course you are going to take, but please,
Don't disparage the "rules" and computational aspects of differential equations.
Firstly, it is a beautiful subject with direct scientific origin and arguably most applications (save only calculus, perhaps) of all the courses you'd ever take. Secondly, these scientific connections continue to motivate and shape the development of the subject. Thirdly, rigor and abstraction are not substitutes for the actual mathematical content. Bourbaki never wrote a volume on differential equations, and the reason, I think, is that the subject is too content-rich to be amenable to axiomatic treatment. Finally, I've taught students who were gung-ho about rigorous real analysis, Rudin style, but couldn't compute the Taylor expansion of $\sqrt{1+x^3}.$ Knowing that the Riemann-Hilbert correspondence is an equivalence of triangulated categories may feel empowering, but as a matter of technique, it is mere stardust compared with the power of being able to compute the monodromy of a Fuchsian differential equation by hand.
Having forewarned you, here are my favorite introductory books on differential equations, all eminently suitable for self-study:
- Piskunov, Differential and integral calculus
- Filippov, Problems in differential equations
- Arnold, Ordinary differential equations
- Poincaré, On curves defined by differential equations
- Arnold, Geometric theory of differential equations
- Arnold, Mathematical methods of classical mechanics
You will find a lot of geometry, including an excellent exposition of calculus on manifolds, in the right context, in Arnold's Mathematical methods.
Notation: $f:X \to Y$ is the map we're pushing forward along, and $F$ is our sheaf on $X$. In general the stalks of $f_*F$ at different points will not be isomorphic. For instance if $f$ misses the point $y \in Y$ and your space is sufficiently separated then the stalk of $f_*F$ at $y$ will be 0 while it will be nonzero for points in the image.
An extreme case is when the map has image a point. Then you get a skyscraper sheaf, which is very far from constant on most spaces and most points (Note: if you're hitting the generic point of $Y$ then the direct image will in fact be constant).
Edit: Another extreme case is when $X$ is a large discrete space. Then one can get direct image sheaves where no stalk is isomorphic to any other stalk. For instance this happens if all the fibers of $f$ have different cardinalities. I think you would usually need the axiom of choice to even define such a map.
Best Answer
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!