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. Nonetheless, its a very methodical development of the theory and its applications to group actions. Bredons writing is terse, but the newer edition has more exposition than the first. In the event that the starting precosheaf was equivalent to a cosheaf bredon calls. Applications of local smith theory and generalized manifolds jim davis some possible applications are group actions on tori or the recent paper of bridsonvogtmann. In this chapter we shall define the sheaftheoretic cohomology theory and shall develop many of.
Singular support in the derived category of sheaves. Master was created as a result of apple computers abandonment of the dos 3. The golden age of mathematicsthat was not the age of euclid, it is ours. A topological space is locally euclidean if every p2mhas a neighborhood uand a homeomorphism u. The theory is developed quite concisely in the book. What books should i read before beginning masaki kashiwaras. Sheaf theory is also important in other fields of mathematics, notably algebraic geometry, but that is outside the scope of the present book. Firstly, as you say you are interested in algebraic geometry, bredons book. The theory is presented in a way that is meant to be read and appreciated by a broad audience.
Hirzebruch, friedrich 1995, topological methods in algebraic geometry, classics in mathematics, berlin, new york. A presheaf is a sheaf if it is separated and it has the gluing. The origins of sheaf theory, sheaf cohomology, and spectral sequences haynes miller february 23, 2000 jean leray november 7, 1906november 10, 1998 was con ned to an o cers prison. Bredon back in 1968 was rather pessimistic on the issue. Is bredons topology a sufficient prelude to bredons sheaf. More on sheafs and singular support 121 chapter 10. The sheaf theoretic morse lemma and applications 114 3. Elizabeth gasparim, a first lecture on sheaf cohomology ravi vakil, introduction to algebraic geometry justin curry, 3. Bredon and a great selection of related books, art and collectibles available now at. Primarily concerned with the study of cohomology theories of general topological spaces with general coefficient systems, the parts of sheaf theory covered here are those areas important to algebraic topology. An updated treatment along the same lines is that of bredon. Overall bredon gets very complicated, but it has a pretty modern style and chapter i should be pretty readable. The fact that sheaftheoretic cohomology satisfies 1 the homotopy property is proved for general topological spaces. A fairly complete account of sheaf theory using resolutions was later given by h.
For orbifolds there is a generalization of k ktheory which is closely related to the bredon cohomology rather than usual equivariant cohomology. It finds its main applications in topology and modern algebraic geometry where it has been used as a tool for solving, with great success, several longstanding problems. Glen e bredon this book is primarily concerned with the study of cohomology theories of general topological spaces with general coefficient systems. Springerverlag, isbn 9783540586630, mr 35917 updated edition of a classic using enough sheaf theory to show its power. Honkasalo, a sheaftheoretic approach to the equivariant serre spectral sequence, j. There are plenty of books that aim to teach you complex differential or algebraic geometry, where just enough sheaf theory is introduced as needed. Another possibility is just having a look at the first chapter or so of bredons sheaf theory. Derived morita theory and hochschild homology and cohomology of dg categories german stefanich in this talk we will explore the idea that an algebra aover a eld ring, spectrum kcan be thought of as a way of encoding a category, namely amod its category of modules. In the sheaf of twisted integers example 2 each sx is isomorphic to the ring z, but this sheaf is not a sheaf of rings. Another possibility is just having a look at the first chapter or so of bredon s sheaf theory. This style fosters the highly involved approach to learning through discussion and student presentation which the author favors, but also allows instructors who prefer a lecture approach to include some of. Curry march, 20 abstract this note advertises the theory of cellular sheaves and cosheaves, which are devices for conducting linear algebra parametrized by a cell complex.
The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. They then use ktheory techniques to get an orbifold invariant over the rationals q. Sheaf theory omt lovering september 24, 2010 abstract in this essay we develop the basic idea of a sheaf, look at some simple examples and explore areas of mathematics which become more transparent and easier to think about in light of this new concept. Concerning relative cohomology, it should be noted that sheaftheoretic cohomology is usually considered as a single space theory. Anyway, godements sheaf theory book in french is better than bredons. This book is primarily concerned with the study of cohomology theories of general. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext.
Chapter 1 sheaf theory the theory of sheaves has come to play a central role in the theories of several. Bredonstyle homology, cohomology and riemannroch for. Sheaves are defined next, along with several examples of sheaves that illustrate the sometimes nonhausdorff topology of sheaves. The theory of sheaves provides a framework for solving. Sheaf theory is also important in other fields of mathematics, notably. Here, by liouvilles theorem in complex function theory, the only global sections are. References 1 adem, davis, topics in transformation groups 2 borel, seminar on transformation groups 3 bredon, introduction to compact transformation groups 4 bredon. Sheaf theory is also important in other fields of mathematics, notably algebraic. The most basic concept in sheaf theory is that of a sheaf generated by a given presheaf.
It would be very helpful to the newcomer to sheaf theory if the author could explain why the functor from open sets to abelian groups is a contravariant functor, instead of a covariant one. In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. Is bredons topology a sufficient prelude to bredons sheaf theory. Anyway, godements sheaf theory book in french is better than bredon s. The book by kashiwara and schapira discusses sheaves with motivation from homological algebra, abelian sheaf cohomology and homotopy theory, leading over in the last chapter to the notion of stack. In fact, the only thing that you can find in bredons sheaf theory but not. Applications of sheaf theory to symplectic topology 105 chapter 9. Sheaf theory provides a means of discussing many different kinds of geometric objects in respect of the connection between their local and global properties.
Scribd is the worlds largest social reading and publishing site. One of the main reasons that it fails to give the results that we would want is that if xis a complex variety and fis a constant sheaf then hi zarx. This course is an introduction to sheaf theory, as a general localtoglobal tool relevant in pure and applied settings. Topology, cohomology and sheaf theory tu june 16, 2010 1 lecture 1 1. Keyser this time of writing is the hundredth anniversary of the publication 1892 of poincares first note on topology, which arguably marks the beginning of the subject of algebraic, or combinatorial, topology.
Also, relative cohomology is introduced into sheaf theory. Translation groupoids and orbifold bredon cohomology arxiv. However, formatting rules can vary widely between applications and fields of interest or study. If tennison is too hard, im not sure this will be a better suggestion, but a good gentle introduction is swans the theory of sheaves.
There are several ways to build up sheaf theory with di erent axioms. However i should say that the epigraph to this very good book is do not shoot the pianist, and maybe not without a reason. Thus a more descriptive title for this book might have been algebraic topology from the point of view of sheaf theory. Bredon wrote it as a commercial program during the late 1980s where it experienced widespread success. What would be a roadmap to learning sheaf theory and topos. Here are a few things you could use as guiding lights. The origins of sheaf theory, sheaf cohomology, and spectral sequences haynes miller february 23, 2000 jean leray november 7, 1906november 10, 1998. A categorical introduction to sheaves daping weng abstract. In categorical terminology this is the concept of a re. I personally wont recommend bredons book, rather iversens cohomology of sheaves especially if you are interested in the topological aspectsapplications of sheaf theory. Though we attempt to avoid being too dependent on category theory and homological. I have the basic results of pointset topology and analytic geometry in my ken, so to speak. Dec 06, 2012 sheaf theory is also important in other fields of mathematics, notably algebraic geometry, but that is outside the scope of the present book. Details about the papers the goal of the midterm and nal papers is to produce nice expository articles about topics in algebraic.
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