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The derived category of sheaves and the Poincare-Verdier duality
The derived category of sheaves and the Poincare-Verdier duality

... The resolution of a complex should be regarded as an abstract incarnation of the geometrical operation of triangulation of spaces. Alternatively, an injective resolution of a complex can be thought of as a sort of an approximation of that complex by a simpler object. The above result should be compa ...
S1-Equivariant K-Theory of CP1
S1-Equivariant K-Theory of CP1

... dimensional vector spaces, we call E = tx∈X Ex a G -vector bundle if it is equipped with certain topological structure and a continuous action of G such that g .Ex = Eg .x . Example. (The Hopf Bundle) A canonical example of a S 1 -bundle over CP1 is H ∗ := {([d], z) ∈ CP1 × C2 : z is a point in the ...
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Topology Ph.D. Qualifying Exam Jan 20,2007 Gerard Thompson
Topology Ph.D. Qualifying Exam Jan 20,2007 Gerard Thompson

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Test Assignment for Metric Space Topology 304a

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EXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY 1

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On the category of topological topologies

... Theorem 2.3 and the 4-morphism {f, g}: {B, C}--+ {B’, C’ j} is hom ( f , g ) , for any f : B ’ --+ B and g : c » C’ . If B ( Y , Y*) 6 q , it is known by [8] , As for the internal hom ...
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§5: (NEIGHBORHOOD) SUB/BASES We have found our way to an

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... which is a weak equivalence induces a weak equivalence between topologically enriched categories and therefore a weak equivalence of classifying spaces. For example, the usual inclusion O(n) ! GL(n, R) is a weak equivalence, hence we get an equivalence BO(n) ' BGL(n, R). Let G act on a space X. Taki ...
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... X, then, is the set containing all real numbers, and the elements of the collection τ can be defined in the following way. The subset A of X is open if for any point x ∈ A, there exists a ball of radius  (i.e. the set {y s.t. |x − y| < } ) that is a subset of A. Theorem 1.3. The above notion of an ...
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Click here

and x ∈ U y ∈ V ˆ N = N∪{∞} (d) Let a, b:ˆN
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seminar notes - Andrew.cmu.edu
seminar notes - Andrew.cmu.edu

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arXiv:0903.2024v3 [math.AG] 9 Jul 2009

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Quiz-2 Solutions

... show that B is dense, we want to show, if V is any non-empty open set in X then V ∩ B 6= φ. As U1 is dense, U1 ∩ V 6= φ. Further U1 ∩ V is open and thus U2 ∩ (U1 ∩ V ) 6= φ (since U2 is dense). Thus (U1 ∩ U2 ) ∩ V 6= φ. Let X be a discrete metric space and Y be any other metric space. Let f : X → Y ...
today`s lecture notes
today`s lecture notes

PGPRD-SETS AND ASSOCIATED SEPARATION AXIOMS
PGPRD-SETS AND ASSOCIATED SEPARATION AXIOMS

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Semi-closed Sets in Fine-Topological Spaces

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Global Calculus:Basic Motivations

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CLASSIFYING SPACES OF MONOIDS – APPLICATIONS IN

... simplex, R is the weakest equivalence relation identifying points (s, x) ∈ ∆n × Xn and (t, y) ∈ ∆m × Xm such that y = X(f )x, s = ∆f (t) for some monotone map f : [m] → [n] and induced lineal map ∆f : ∆m → ∆n which takes values ∆f (ei ) = ef (i) on vertices of ∆m . The geometric realization | · | is ...
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Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. 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. For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They exist in several varieties such as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets.There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category. On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of sheaf theory.Due to their general nature and versatility, sheaves have several applications in topology and especially in algebraic and differential geometry. First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space. In such contexts several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general cohomology theory, which encompasses also the ""usual"" topological cohomology theories such as singular cohomology. Especially in algebraic geometry and the theory of complex manifolds, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of D-modules, which provide applications to the theory of differential equations. In addition, generalisations of sheaves to more general settings than topological spaces, such as Grothendieck topology, have provided applications to mathematical logic and number theory.
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