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  • Sheaves, Cosheaves and Applications - University of Pennsylvania
    We develop applications to persistent homology, net-work coding, and sensor networks to illustrate the utility of the theory The driving computational force is cellular cosheaf homology and sheaf cohomology
  • Math 216A. Sheafification Introduction
    3 Separated presheaves By de nition, a sheaf is a presheaf satisfying two conditions: local uniqueness (i e , s; an open cover of U must satisfy s = t) F and gluing of compatible local data If satis es just t e rst of these two conditi
  • Lecture 3: Interlude on sheaves with values in D Z - math. berkeley. edu
    Let's nish up our discussion of abstract matters We want to work in the ∞-category of sheaves with values in D(Z) instead of always ha
  • Lecture 11 - Decomposition of Sheaves - University of Toronto . . .
    In this lecture we will develop a way to understand Etale sheaves on a space by cutting it up into `simpler' pieces Speci cally, we consider the scenario where X is a scheme, Z is a closed subscheme, and U = X Z is the complementary open subscheme We denote the maps by Given a sheaf F on X, we get sheaves F1 = i F and F2 = j F on Z; U resp
  • Sheaves - University of California, San Diego
    on 6 3 A sheaf F on a topological space is a presheaf which satis es the following two axioms: (1) Given an open cover Ui of U an open subset of X, and a col-lection of sections si on Ui, such that sijUij = sjjUij then there is a section s on U such that sj 0, then s
  • LECTURE NOTES ON SHEAVES AND PERVERSE SHEAVES
    Part 1 Sheaves 1 Sheaves: the lightning tour 1 1 Category theory Sheaf theory requires some category theory, as summarized in Appendix A Don’t try to read it all at once I have added references to this section as needed 1 2 Let Rbe a commutative ring (with 1) Let Xbe a topological space
  • Di erential cohomology theories as sheaves of spectra
    In this paper we use these adjoints in order to deconstruct a given sheaf into its under-lying homotopy invariant part, cycle data and a characteristic map which contains the information how the homotopy invariant part and the cycle data is glued together, see De nition 3 7
  • Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology . . .
    homology, and CW complexes, in particular With the above motivation in mind, this book consists of two parts The first part con-sisting of the first seven chapters gives a cras -course on the homological and cohomologi
  • Section 59. 31 (04FQ): Supports of abelian sheaves—The Stacks project
    Let F be an abelian sheaf on Se´tale The support of F is the set of points s ∈ S such that Fs¯¯¯ ≠ 0 for any (some) geometric point s¯¯¯ lying over s
  • Sheaf Theory (London Mathematical Society Lecture Note Series)
    In this course we build enough of the foundations of sheaf theory to give a broad definition of manifold, covering as special cases the algebraic geometer's schemes as well as the topological, differentiable and analytic kinds; and to define sheaf cohomology for application to such objects
  • Section 6. 7 (006S): Sheaves—The Stacks project
    We are going to construct a sheaf of sets Π from this data For U ⊂ X open set Π(U) =∏x∈UAx For V ⊂ U ⊂ X open define a restriction mapping by the following rule: An element s = (ax)x∈U ∈ Π(U) restricts to s|V = (ax)x∈V It is obvious that this defines a presheaf of sets We claim this is a sheaf
  • LECTURE NOTES ON SHEAVES AND PERVERSE SHEAVES
    The sheaf of chains, however, is a naturally occurring entity with complicated cohomology sheaves But it is not so obvious how to construct a sheaf on X that corresponds to (say) the singular chains
  • 1. Defining Sheaves - Denison University
    sets of a topology We can get the open sets to be arbitrarily small but we can also build up information from the local information in a coherent way (th nks to the gluing) There are also maps (or morphisms) from one sheaf to another; sheaves (of a speci c type, such as sheaves of Abelian groups) with their morphisms on a xed topological sp
  • Section 96. 6 (06TU): The structure sheaf—The Stacks project
    The structure sheaf of X is the sheaf of rings OX = p−1O For an object x of X lying over U we have OX(x) =O(U) = Γ(U,OU) Needless to say OX is also a Zariski, étale, smooth, and syntomic sheaf, and hence each of the sites XZar, Xe´tale, Xsmooth, Xsyntomic, and Xfppf is a ringed site This construction is functorial as well Lemma 96 6 2
  • Section 92. 19 (09DF): The Atiyah class of a sheaf of modules—The Stacks . . .
    Let A → B be a homomorphism of sheaves of rings Let F be a sheaf of B -modules Let P∙ → B be the standard resolution of B over A (Section 92 18) For every n ≥ 0 consider the extension of principal parts see Modules on Sites, Lemma 18 34 6





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