Higher algebra and topological quantum field theory
... is a fully faithful embedding, with essential image the subcategory of simplicial sets whose Segal maps are bijections. Moreover, it has a left adjoint τ1 , called the fundamental category functor. Proof. The full faithfulness is clear from the above description of the nerve. The existence of an adj ...
... is a fully faithful embedding, with essential image the subcategory of simplicial sets whose Segal maps are bijections. Moreover, it has a left adjoint τ1 , called the fundamental category functor. Proof. The full faithfulness is clear from the above description of the nerve. The existence of an adj ...
Algebraic Topology Introduction
... Example 1.4. Take X to be a rectangle and glue together the boundary in some particular way, for example we may form a torus or a Klein bottle. Both spaces are quotient spaces formed from a rectangle, taking the quotient by the boundary, but observe that the quotient map is the tool carrying all the ...
... Example 1.4. Take X to be a rectangle and glue together the boundary in some particular way, for example we may form a torus or a Klein bottle. Both spaces are quotient spaces formed from a rectangle, taking the quotient by the boundary, but observe that the quotient map is the tool carrying all the ...
2. For each binary operation ∗ defined on a set below, determine
... Inverses: Let a ∈ G, and let a−1 be its inverse in (G, ·). Then a ∗ a−1 = a−1 · a = e = a · a−1 = a−1 ∗ a, so a−1 is also an inverse to a in (G, ∗). (b) Give examples to show that (G, ∗) may not be the same as (G, ·). An example could be found by setting (G, ·) = GLn (R), the group of n×n invertible ...
... Inverses: Let a ∈ G, and let a−1 be its inverse in (G, ·). Then a ∗ a−1 = a−1 · a = e = a · a−1 = a−1 ∗ a, so a−1 is also an inverse to a in (G, ∗). (b) Give examples to show that (G, ∗) may not be the same as (G, ·). An example could be found by setting (G, ·) = GLn (R), the group of n×n invertible ...
Homology Theory - Section de mathématiques
... Hn 1 → Hn 1 is an isomorphism. The upper row consists simply of the kernels of the corresponding vertical morphisms p∗ (observe that ker (p∗ : Hn (X, A) → Hn (1, 1)) = Hn (X, A) since Hn (1, 1) = 0). By naturality of ∂ the lower squares in the diagram commute and so, since p∗ ◦ ∂ = 0 ◦ p∗ we have im ...
... Hn 1 → Hn 1 is an isomorphism. The upper row consists simply of the kernels of the corresponding vertical morphisms p∗ (observe that ker (p∗ : Hn (X, A) → Hn (1, 1)) = Hn (X, A) since Hn (1, 1) = 0). By naturality of ∂ the lower squares in the diagram commute and so, since p∗ ◦ ∂ = 0 ◦ p∗ we have im ...
A convenient category - VBN
... subcategory of K. An object K of K is called I-generated if the cocone (C → K)C∈I consisting of all morphisms from objects of I to K is U-final. Let KI denote the full subcategory of K consisting of I-generated objects. Using the terminology of [2], KI is the final closure of I in K and I is finally ...
... subcategory of K. An object K of K is called I-generated if the cocone (C → K)C∈I consisting of all morphisms from objects of I to K is U-final. Let KI denote the full subcategory of K consisting of I-generated objects. Using the terminology of [2], KI is the final closure of I in K and I is finally ...
Complex Numbers
... We can look at the field from last example as another approach to complex numbers. We identify complex numbers with points of the Cartesian plane (or vectors anchored at the origin) and we call this “geometrical interpretation of complex numbers”. A point z of the plane can be identified by its Cart ...
... We can look at the field from last example as another approach to complex numbers. We identify complex numbers with points of the Cartesian plane (or vectors anchored at the origin) and we call this “geometrical interpretation of complex numbers”. A point z of the plane can be identified by its Cart ...
MA5209L4 - Maths, NUS - National University of Singapore
... Definition In a topological space X a point p is path connected to a point q if there exists a map f : [0,1] X such that f(0) = p and f(1) = q. Lemma Path conn. is an equivalence relation. Proof. p is connected to p by the constant map f : [0,1] X defined by f(x) = p, x in X. If p is connected t ...
... Definition In a topological space X a point p is path connected to a point q if there exists a map f : [0,1] X such that f(0) = p and f(1) = q. Lemma Path conn. is an equivalence relation. Proof. p is connected to p by the constant map f : [0,1] X defined by f(x) = p, x in X. If p is connected t ...
Note - Math
... be any point. The family of open sets in X containing x form a directed system, with respect to inclusions and thus family {F(U )}x∈U , where U are open in X, is a directed family. We define Fx := lim F(U ), called stalks of F over x. ...
... be any point. The family of open sets in X containing x form a directed system, with respect to inclusions and thus family {F(U )}x∈U , where U are open in X, is a directed family. We define Fx := lim F(U ), called stalks of F over x. ...
III.2 Complete Metric Space
... shows that f ∈ B(X, Rn ). Uniform limit of continuous functions is continuous Suppose that fn → f with fn ∈ C(X, Rn ). We want to show that f is continuous. Fix any x0 ∈ X and > 0. Since fn → f uniformly, ∃N such that n ≥ N ⇒ |fn (x)−f (x)| < /3, ∀x ∈ X. For a such fn with n ≥ N , since fn is con ...
... shows that f ∈ B(X, Rn ). Uniform limit of continuous functions is continuous Suppose that fn → f with fn ∈ C(X, Rn ). We want to show that f is continuous. Fix any x0 ∈ X and > 0. Since fn → f uniformly, ∃N such that n ≥ N ⇒ |fn (x)−f (x)| < /3, ∀x ∈ X. For a such fn with n ≥ N , since fn is con ...
Abelian topological groups and (A/k)C ≈ k 1. Compact
... Proof: Let G be compact. Let E be a small-enough open in S 1 so that E contains no non-trivial subgroups b be the open of G. Noting that G itself is compact, let U ⊂ G b : f (G) ⊂ E} U = {f ∈ G b Since E is small, f (G) = {1}. That is, f is the trivial homomorphism. This proves discreteness of G. Fo ...
... Proof: Let G be compact. Let E be a small-enough open in S 1 so that E contains no non-trivial subgroups b be the open of G. Noting that G itself is compact, let U ⊂ G b : f (G) ⊂ E} U = {f ∈ G b Since E is small, f (G) = {1}. That is, f is the trivial homomorphism. This proves discreteness of G. Fo ...
Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied both through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes.