Beam theory

In mathematics, a bundle F on a given topological space, X, yields, for every open set U of X, a set F(U), with a richer structure. In turn, these structures, F(U), are compatible with the operation of restriction from an open set to smaller subsets and with the operation of pasting of open sets to obtain a larger open. A prebundle is similar to a bundle, but pasting may not be possible with it. Bundles allow you to analyze and understand what it means to be a local property, as it is spoken of when applied to a function.

Introduction

Bundles are used in topology, algebraic geometry, and differential geometry whenever we want to keep track of the algebraic data that varies with each open set of the given geometric object. They are a global tool for studying objects that vary locally (i.e., depending on the open set). They function as natural instruments for the study of the global behavior of entities that are local in nature, such as open sets, or functions: continuous, analytical, differentiable...

To consider a typical example, be a topological space X for every open set U in X, the whole F(U), consisting of all continuous functions U ${displaystyle rightarrow }$ R. Yeah. V is an open subset U, then functions on U may be restricted to Vand we have an app F(U) ${displaystyle rightarrow }$ F(V). The "small" is about the following process: U_i are open sets whose union is Uand for each i We take an element f_i ${displaystyle in }$ F(U_i), i.e. a continuous function f_i: U_i ${displaystyle rightarrow }$ R. If these functions match where they overlap, then we can stick them together so that they give us a unique way to get a continuous function f: U ${displaystyle rightarrow }$ R with all f_i. The collection of sets F(U) together with the restricted applications F(U) ${displaystyle rightarrow }$F(V) form a beam of sets on X. Really, the F(U) are commutative rings and restrictive applications are ring homomorphisms, and F is also a beam of rings over X.

A very similar example is obtained considering a differentiable variety Xand for each open set U of X, taking the set F(U) like that of differentiable functions U ${displaystyle rightarrow }$ R. In this example the glued will also work and we will have a beam of rings over X. Another beam over X assigns to each open set U of X the vector space of all different vector fields defined on U. The restriction and the glue will work as in the case of the functions, and we will get a beam of vector spaces on the variety X.

History

The earlier origins of bundle theory are hard to discern - they are surely coextensive with the idea of analytic continuation. It took about 15 years to extract a self-contained bundle theory from the foundational work in cohomology.

• 1936 Eduard Čech introduces the construction of Nerve of an open coatingwhich associates a simplicial complex to an open coating.
• 1938 Hassler Whitney provides a 'modern' definition of cohomology, summarizing all the work done since Alexander and Kolmogórov defined the definition of cohomology. Cocadens.
• 1943 Steenrod publishes on homology with local coefficients.
• 1945 Jean Leray publishes work carried out in a camp of prisoners of war, motivated by the demonstrations on theorems of the fixed point in its application to the theory of EDP (deriving partial consequences). This is the beginning of beam theory and spectral sequences.
• 1947 Henri Cartan again demonstrates the Theorem of Rham by means of methods of theory of beams, in his correspondence with André Weil. Leray gives a definition of beam through the enclosed sets (the former Carapaces).
• 1948 The seminar of Cartan puts for the first time the theory of writing.
• 1950 The 'second edition' of Cartan's seminar on beam theory: where the definition of beam space is used (éspace étalé), with stem structure (stalkwise).

Supports are introduced, and cohomology with supports. Continuous applications give rise to spectral sequences. At the same time Kiyoshi Oka introduces the idea (similar to that) of a bundle of ideals, in various complex variables.

• 1951 The seminar of Cartan demonstrates the theorems A and B based on Oka's work.
• 1953 The theorem of finitude for making consistent in analytical theory is demonstrated by Cartan and Serre, as well as the duality of Serre.
• 1954 The article of Serre Faisceaux algébriques cohérents (published in 1955) introduces the beams into algebraic geometry. These ideas are exploited immediately by Hirzebruch, who writes a fundamental book on topological methods.
• 1955 Alexander Grothendieck in given readings in Kansas defines the Abelian category and the pre-release, and by using the injective resolution it allows you to use directly the cohomology of beams over all the topological spaces, as derivative funtors.
• 1957 Grothendieck article called Tohoku rewrite the homological algebra; test the duality of Grothendieck (i.e., duality of Serre for singular varieties).
• 1958 Godement's book on beam theory is published. At about the same time Mikio Satō proposes hyperfunctions, which end up being "made-theoretically."
• 1957 progressively: Grothendieck spreads the theory of beams by adjusting it to the needs of algebraic geometry, introducing the: schematics and making general about them, local cohomology, the derivative category (this with Verdier), and the Topology of Grothendieck. There also arise their influential and synthetic idea of 'six operations' in homological algebra.

At this point bundles have already become a fundamental part of the development of mathematics, and their use is by no means restricted to algebraic topology. Logic in bundle categories was later found to be intuitionistic (this observation is often called Kripke-Joyal semantics, but should probably be attributed to more authors). This shows how some of the facets of bundle theory can be traced back as far as Leibniz.

Formal definition

We will define the beams in two steps. The first is to introduce the concept of prebeam, which captures the idea of associating local information to a topological space. The second step is to introduce an additional axiom, called the pasting axiom or the bundle axiom, which captures the idea of pasting local information to obtain global information.

Prebeam definition

Let X be a topological space, and C a category (often the category of sets, of abelian groups, of commutative rings, or that of modules on a ring permanent). A prebundle F of objects in C over space X (a C- prebeam on X) is given by the following data:

• for each open set U in Xan object F(U) in C
• for each inclusion of open assemblies V ${displaystyle subset }$ U, a morphism F(U) ${displaystyle rightarrow }$ F(V) in category Cwhich is called the "restriction"

from U to V". We will write it as res_U,V. Two properties are required:

• for each open set U in X, we have res_U,U =_F(U), i.e., restriction U a U It's identity.
• given any three open sets W ${displaystyle subset }$ V ${displaystyle subset }$ U, we have res_V,W o res_U,V =_U,Wthe restriction F(U) a F(Vand then to F(W) is the same as the restriction F(U) directly to F(W).

This definition can easily be given in terms of the theory of categories. First we define the category of open sets on X as the Top category_X whose objects are the open sets of X and whose morphisms are inclusions. Top_X is then the category corresponding to the partial order ${displaystyle subset }$ about the open sets of X. A C-Hold on. X It's then a countervailing funtor from Top_X a C.

If F is a C-valued prebundle over X, and U is an open set of X, then F(U) is said to be the sections of F over U. (This is by analogy with the "fiber bundles" sections; see below) If C is a concrete category, then each element of F(U) is called a section. F(U) is often also denoted Γ(U,F).

The Axiom of Pasting

Bundles are prebundles onto which sections on open sets can be pasted to give larger sections on open sets. We will first state the axiom in a way that requires C to be a concrete category.

Let U be the union of the collection of open sets {U_i}. For each U_i, choose a section f_i about U_i. We say that the f_i are compatible if for all i j,

res_{Ui,Ui${displaystyle cap }$Uj}(f_i=res_{Uj,Ui${displaystyle cap }$Uj}(f_j).

Intuitively speaking, if the f_i represent functions, we are saying that any one of them will match another wherever they overlap. The beam axiom says that we can obtain with the f_i a unique section f over U whose restriction to each U_i is f_i, i.e., res_U,Ui(f)=f_i. Sometimes this is said with two axioms, one guaranteeing existence and the other guaranteeing uniqueness.

Paraphrasing this definition so that it works for any category, we note that we can write the objects and morphisms involved in it in a diagram that looks like this:

${displaystyle {mathcal {F}}(U)rightarrow prod _{i}{mathcal {F}(U_{i}){rightarrow atop rightarrow }prod _{i,j}{mathcal {F}}}(U_{i}cap U_{j}}}}}}$

The first application here is the product of the restricted applications res_U,Ui,:F(U)${displaystyle rightarrow }$F(U_i) and each pair of arrows represents the two res restrictions_{Ui,Ui${displaystyle cap }$Uj}:U_i${displaystyle rightarrow }$U_i${displaystyle cap }$U_jand res_{Uj,Ui${displaystyle cap }$Uj}:U_j${displaystyle rightarrow }$U_i${displaystyle cap }$U_j. It is worth noting that these applications exhaust all possibilities in terms of restrictive applications U, the U_iand the U_i${displaystyle cap }$U_j.

The condition that F is a bundle is exactly the condition that F(U) is the limit of the rest of the diagram. This suggests that we need to paraphrase the notion of cover in a categorical context. When we do this, we get a diagram that looks like the one above:

${displaystyle prod _{i,j}U_{i}cap U_{j}{rightarrow atop rightarrow }prod _{i}U_{i}rightarrow U}$

(It is important to note here that in order to form the products in the diagram, we must embed the Top_X category in an entire category) The condition that U is the union of the U_i is that U is a colimit of the rest of the diagram.

The gluing axiom is now that F turns all colimits into limits.

Examples

Apart from those we've already put, sections are important examples. Suppose E and X are topological spaces and π: E ${displaystyle rightarrow }$ X a continuous application. For each open set U in XI mean, F(U) the set of all applications f: U ${displaystyle rightarrow }$ E such that π(f(x) = x for everything x in U. Such a function fIt's called. section of π. It's not hard to prove that F That's it. a beam of sets on X. In fact, every beam of sets over X is essentially of this type, for very special applications π; see below.

Given a bundle F on X, the elements of F(X) are also called the < b>global sections, terminology motivated by the previous example.

Other examples:

• The constant beam.
• Any vector fiber provides a bundle beam, taking the sections.
• Look how you do them are used in the Riemann Surface article.
• Nestled spaces are made of commutative rings; locally ringed spaces are especially important, where all stems (see below) are local rings.
• Schemes are locally special ringed spaces, important in algebraic geometry; modules are important in the associated theory.
• You make straight. in the article: Simulation.

Further reading

• Bredon, Glen E. (1997). Sheaf theory. Graduate Texts in Mathematics 170 (2nd edition). Springer-Verlag. ISBN 978-0-387-94905-5. MR 1481706. (oriented towards conventional topological applications)
• de Cataldo, Andrea Mark; Migliorini, Luca (2010). "What is to perverse sheaf?" Notices of the American Mathematical Society 57 (5): 632-4. Bibcode:2010arXiv1004.2983D. MR 2664042. arXiv:1004.2983.
• Godement, Roger (2006). Topologie algébrique et théorie des faisceaux (in French). Paris: Hermann. ISBN [[Special:BookSources/2705612521}IND2705612521}] |isbn= incorrect (help). MR 0345092.
• Grothendieck, Alexander (1957). "Sur quelques points d'algèbre homologique". The Tohoku Mathematical Journal. Second Series (in French) 9 (2). pp. 119-221. ISSN 0040-8735. MR 0102537. doi:10.2748/tmj/1178244839.
• Hirzebruch, Friedrich (1995). Topological methods in algebraic geometry. Classics in Mathematics. Springer-Verlag. ISBN 978-3-540-58663-0. MR 1335917. (updated edition of a classic using enough sheaf theory to show its power)
• Iversen, Birger (1986). Cohomology of sheaves. Universitext. Springer. ISBN 3-540-16389-1. MR 842190. doi:10.1007/978-3-642-82783-9.
• Kashiwara, Masaki; Schapira, Pierre (1994). Sheaves on manifolds. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 292. Springer-Verlag. ISBN 978-3-540-51861-7. MR 1299726. (advanced techniques such as the derived category and vanishing cycles on the most reasonable spaces)
• Mac Lane, Saunders; Moerdijk, Ieke (1994). Sheaves in Geometry and Logic: A name Introduction to Topos Theory. Universitext. Springer-Verlag. ISBN 978-0-387-97710-2. MR 1300636. (category theory and toposes emphasised)
• Martin, William T.; Chern, Shiing-Shen; Zariski, Oscar (1956). «Scientific report on the Second Summer Institute, several complex variables». Bulletin of the American Mathematical Society (in English) 62 (2). pp. 79-141. ISSN 0002-9904. MR 0077995. doi:10.1090/S0002-9904-1956-10013-X.
• Ramanan, S. (2005). Global calculus. Graduate Studies in Mathematics 65. American Mathematical Society. ISBN 0-8218-3702-8. MR 2104612. doi:10.1090/gsm/065.
• Seebach, J. Arthur; Seebach, Linda A.; Steen, Lynn A. (1970). "What is a Sheaf." American Mathematical Monthly 77 (7). 681-703. MR 0263073. doi:10.1080/00029890.1970.11992563.
• Serre, Jean-Pierre (1955). «Faisceaux algébriques cohérents». Annals of Mathematics. Second Series 61 (2). pp. 197-278. ISSN 0003-486X. JSTOR 1969915. MR 0068874. doi:10.2307/1969915.
• Swan, Richard G. (1964). The Theory of Sheaves. Chicago lectures in mathematics (in English) (3 edition). University of Chicago Press. ISBN 9780226783291. (notes conference concise)
• Tennison, Barry R. (1975). Sheaf theory. London Mathematical Society Lecture Note Series (in English) 20. Cambridge University Press. ISBN 978-0-521-20784-3. MR 0404390. (pedagogical)