Although it is rarely presented this way in introductory texts e. Now we just need to generalize the conditions for a presheaf to become a sheaf. Sheaf theory wikibooks, open books for an open world. Sheaf theory is also important in other fields of mathematics, notably algebraic geometry, but that is outside the scope of the present book. Thus a more descriptive title for this book might have been algebraic topology from the point of view of sheaf theory. It is interesting to me how g g is right adjoint to f f, but also g. Most problems could be phrased and perhaps solved without sheaf theory, but the. Presheavesovera basis of opens throughout this section, k is a category which admits projective inverse limits.
Dec 07, 2016 now we just need to generalize the conditions for a presheaf to become a sheaf. 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. We go back to the conditions that make a classical presheaf a sheaf. Newest sheaf theory questions feed subscribe to rss newest sheaf theory questions feed to subscribe to this rss feed, copy and paste this url into your rss reader. Topics in abstract algebrasheaf theory wikibooks, open. The first step prior to applying the four observations is to consider the singleton map. Use this tag with the broader algebraicgeometry tag. The notions of a local section and a global section of a presheaf of rings, and of the restriction of a presheaf of rings is exactly as in the case of a presheaf of sets. 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. Sheaf theory provides a means of discussing many different kinds of geometric objects in respect of the connection between their local and global properties. Though we attempt to avoid being too dependent on category theory and homological.
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. There are always two ways to think of topos theory. Another possibility is just having a look at the first chapter or so of bredons sheaf theory. 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 camp \o ag in austria for the whole of world war ii.
Sheaves do not belong to algebraic geometry let f f be the functor from presheaves to spaces over x x, and let g g be the functor from spaces over x x to presheaves. They can be summarized in the language of category theory by saying that. Contents 1 lecture 1 1 2 lecture 2 5 3 lecture 3 9 4 lecture 4 15 5 lecture 5 21 6 lecture 6 27 7 lecture 7 31 8 lecture 8 35 9 lecture 9 41 10 lecture 10 47 11 lecture 11 55. Sheaves and homotopy theory daniel dugger the purpose of this note is to describe the homotopytheoretic version of sheaf theory developed in the work of thomason 14 and jardine 7, 8, 9. The collection of measures is a sheaf, the collection of measurable real valued functions. A slightly different definition is given in voisin, in hodge theory and complex algebraic geometry i definition 4. Overall bredon gets very complicated, but it has a pretty modern style and chapter i should be pretty readable. Now an intuition about cohomology is that it measures how many more sections you gain when you go more local. Spv, wa, sheaf theory is a natural language in which to make such a notion precise. So, we will not try to give a full view of these subjects. A categorical introduction to sheaves daping weng abstract.
Sheaves are defined next, along with several examples of sheaves that illustrate the sometimes nonhausdorff topology of sheaves. There he took up algebraic topology, and the result was a spectacular. Let xbe a topological space, ba basis for the opens of xand k a category. In this way, we have no doubt that the reader will. The sheaf theoretic morse lemma and applications 114 3. Godement, topologie algebrique et theorie des faisceaux, hermann 1958 2 r. The theory of characteristic classes is a very well developed branch of mathematics and the literature concerning riemannrochtheorem is huge. So a way to see sheaf cohomology geometrically is to make the sheafs into geometric objects this way then we have both schemes and sheaves living in the same environment and see that sheaf theory is about maps from schemes to sheaves. A fairly complete account of sheaf theory using resolutions was later given by h. Discrete morse theory for computing cellular sheaf cohomology. Ramabhadran tata institute of fundamental research bombay 1956. 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. Elizabeth gasparim, a first lecture on sheaf cohomology ravi vakil, introduction to algebraic geometry justin curry, 3.
Serre in the early 1950s on algebraic varieties greatly influenced the development of sheaf theory. Sheaf models of type theory goal of the talk sheaf models of higher order logic have been fundamental for establishing consistency of logical principles e. Newest sheaftheory questions feed subscribe to rss newest sheaftheory questions feed to subscribe to this rss feed, copy and paste this url into your rss reader. Sheaf semantics in constructive algebra and type theory. 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. Compactlysupported sheaf cohomology is expressed as the coend with the image of the constant sheaf through this equivalence. Consider the full subcategory of opx whose objects are the elements of b.
We prove that, for the noncommutative infinitesimal topology of an associative algebra over a field of characteristic zero, the cohomology of the structure sheaf modulo. The major virtue of sheaf theory is informationtheoretic in nature. We recall that this equivalence is given by associating to an algebraic vector bundle f x the sheaf f of sections of f. We will often use the equivalence between the category of algebraic vector bundles on x and the category of locally free sheaves.
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. F g \circ f makes a functor from presheaves to sheaves sheafification which is itself left adjoint. Inspired to provide fast algorithms for persistence, we prove that. Manifolds and varieties via sheaves purdue university. 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. The driving computational force is cellular cosheaf homology and sheaf cohomology. A sheaf theoretic approach to measure theory matthew jackson, phd university of pittsburgh, 2006 the topos shf of sheaves on a. Usually you think of a sheaf on a space as the data of functions defined on that space, although there is a more general interpretation in terms of category theory. Tennison, sheaf theory article pdf available in bulletin of the american mathematical society 831977 january 1977 with 275 reads.
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. There he took up algebraic topology, and the result was a. Singular support in the derived category of sheaves. For questions about sheaves on a topological space. Newest sheaftheory questions mathematics stack exchange. There are several ways to build up sheaf theory with di erent axioms. Pages in category sheaf theory the following 58 pages are in this category, out of 58 total. Piecing together solutions of such local problems in a coherent manner to describe, e. We develop applications to persistent homology, network coding, and sensor networks to illustrate the utility of the theory. Booleanvalued models here, the connection is more direct. We have listed them here because, just by having a quick glance at them, one can realize that sheaves play a relevant sometimes crucial role.
What would be a roadmap to learning sheaf theory and topos. The theory is presented in a way that is meant to be read and appreciated by a broad audience. Manifolds and varieties via sheaves in rough terms, a manifold is a topological space along with a distinguished collection of functions, which looks locally like euclidean space. More on sheafs and singular support 121 chapter 10.
An introduction to symplectic topology through sheaf theory. Here are a few things you could use as guiding lights. Sheaves are defined next, along with several examples of sheaves that illustrate the. In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. Applications of sheaf theory to symplectic topology 105 chapter 9. The equivalence is further used to establish relations between sheaf cohomology and a herein newly introduced theory of cellular sheaf homology. The sheaves for the canonical topology on b form a boolean topos shb equivalent to the category of sets and functions with jf is a functioni1 of the boolean sheaf models for set theory 99 valued universe v see again jech, p.