symmetric monoidal category Examples of closed symmetric
... for 2 a topological category, for 3 a topological G-category (TGC). In each case there is an internal hom, . Let denote the category of TGCs. One example of such is the category of pointed G-spaces and nonequivariant maps. ...
... for 2 a topological category, for 3 a topological G-category (TGC). In each case there is an internal hom, . Let denote the category of TGCs. One example of such is the category of pointed G-spaces and nonequivariant maps. ...
Problem Set 5 - Stony Brook Mathematics
... Problem 1. Show that if X is a finite simplicial complex whose underlying topological space is a homology n-manifold, then (a) X consists entirely of n-simplices and their faces, (b) Every (n − 1)-simplex is a face of precisely two n-simplices. Problem 2. Suppose that X is a compact triangulable hom ...
... Problem 1. Show that if X is a finite simplicial complex whose underlying topological space is a homology n-manifold, then (a) X consists entirely of n-simplices and their faces, (b) Every (n − 1)-simplex is a face of precisely two n-simplices. Problem 2. Suppose that X is a compact triangulable hom ...
Math 643 - Topology III, Homework 1 Due date: February 23. (1
... that, in the above situation, any two of the following properties imply the third: • For each b ∈ B, the induced map Fb → Fg(b) is a weak equivalence; • The map g : X → X 0 is a weak equivalence; • The map f : B → B 0 is a weak equivalence. In fact, if all the spaces involved are path connected, the ...
... that, in the above situation, any two of the following properties imply the third: • For each b ∈ B, the induced map Fb → Fg(b) is a weak equivalence; • The map g : X → X 0 is a weak equivalence; • The map f : B → B 0 is a weak equivalence. In fact, if all the spaces involved are path connected, the ...
LECTURE 30: INDUCED MAPS BETWEEN CLASSIFYING SPACES
... Suppose that α : G → H is a homomorphism of topological groups. We can get an induced map α∗ : BG → BH by providing a natural transformation between the functors that these spaces rep resent: α∗ : {Gbundles over X} → {Hbundles over X}. Namely, send a Gbundle E → X to the Hbundle H ×G E → X wher ...
... Suppose that α : G → H is a homomorphism of topological groups. We can get an induced map α∗ : BG → BH by providing a natural transformation between the functors that these spaces rep resent: α∗ : {Gbundles over X} → {Hbundles over X}. Namely, send a Gbundle E → X to the Hbundle H ×G E → X wher ...
Commutative algebra for the rings of continuous functions
... the morphism C(Y ) −→ C(X). The results obtained, so far, do not indicate that there is a more “ natural ” structure than the others, but if we want to make a development of General Topology parallel to the experimented by Algebraic Geometry (using the concepts and methods of Commutative Algebra), t ...
... the morphism C(Y ) −→ C(X). The results obtained, so far, do not indicate that there is a more “ natural ” structure than the others, but if we want to make a development of General Topology parallel to the experimented by Algebraic Geometry (using the concepts and methods of Commutative Algebra), t ...
Categories and functors
... category A is locally small if A(A, B) is a small collection for each A and B. (Many authors build locally small into their definition of category.) A category A is small if it is locally small and the collection ob(A) is small. c. A harmless convention is that A(A, B) ∩ A(A0 , B 0 ) = ∅ unless A = ...
... category A is locally small if A(A, B) is a small collection for each A and B. (Many authors build locally small into their definition of category.) A category A is small if it is locally small and the collection ob(A) is small. c. A harmless convention is that A(A, B) ∩ A(A0 , B 0 ) = ∅ unless A = ...
AAG, LECTURE 13 If 0 → F 1 → F2 → F3 → 0 is a short exact
... (1) in the category of vector spaces over a field k all objects are injective; (2) Q and Q/Z are injective in the category Ab of abelian groups; (3) if R is a commutative ring then HomAb (R, Q/Z) is an injective R-module. Note that the abelian groups Q and Q/Z are not finitely generated, and the R-m ...
... (1) in the category of vector spaces over a field k all objects are injective; (2) Q and Q/Z are injective in the category Ab of abelian groups; (3) if R is a commutative ring then HomAb (R, Q/Z) is an injective R-module. Note that the abelian groups Q and Q/Z are not finitely generated, and the R-m ...
Problems for Category theory and homological algebra
... (b) An (additively written) abelian group A is called divisible if for each a ∈ A and each n ∈ Z>0 there exists an element b ∈ A such that nb = a. The category of divisible abelian groups (with group homomorphisms as morphisms) is denoted Div. Show that the quotient map Q → Q/Z is a monomorphism in ...
... (b) An (additively written) abelian group A is called divisible if for each a ∈ A and each n ∈ Z>0 there exists an element b ∈ A such that nb = a. The category of divisible abelian groups (with group homomorphisms as morphisms) is denoted Div. Show that the quotient map Q → Q/Z is a monomorphism in ...
Click here
... Here, the notation indicates that we’re talking about ∂n (it’s the only one of these maps that I could apply to such an object, so leaving the n out is unambiguous), and that (P0 , . . . , P̌i , . . . , Pn ) = (P0 , . . . , Pi−1 , Pi+1 , . . . , Pn ). It’s σ with the ith entry removed. So we have a ...
... Here, the notation indicates that we’re talking about ∂n (it’s the only one of these maps that I could apply to such an object, so leaving the n out is unambiguous), and that (P0 , . . . , P̌i , . . . , Pn ) = (P0 , . . . , Pi−1 , Pi+1 , . . . , Pn ). It’s σ with the ith entry removed. So we have a ...
The long exact sequence of a pair and excision
... Suppose c ∈ Cp with ∂c = 0. Choose a lift b ∈ Bp. Then g∂b = ∂gb = ∂c = 0, so there is a unique element a ∈ Ap−1 such that f a = ∂b. Note that f ∂a = f ∂∂b = 0, so ∂a = 0. We set δ([c]) = [a]. Proposition 2. The sequence δ ...
... Suppose c ∈ Cp with ∂c = 0. Choose a lift b ∈ Bp. Then g∂b = ∂gb = ∂c = 0, so there is a unique element a ∈ Ap−1 such that f a = ∂b. Note that f ∂a = f ∂∂b = 0, so ∂a = 0. We set δ([c]) = [a]. Proposition 2. The sequence δ ...
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.