Posted on Categories:Algebraic Topology, 代数拓扑, 数学代写

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## 数学代写|代数拓扑代考Algebraic Topology代考|Categories

In the first few semesters of studying math, one realizes that many constructions and arguments pop up repeatedly in different contexts. For instance, products are defined in virtually the same way, no matter whether we are dealing with products of groups, rings, or vector spaces. This raises the desire to explain the term “product” once and for all in an abstract fashion that would specialize to all the particular cases needed in mathematics. But to come up with a meaningful abstract definition of ” $X \times Y$,” it is indispensable to first convey in one way or another that ” $X$ “‘ and ” $Y$ ” should be two “instances” of the same “type”; or we had better say two objects in the same category.

Definition $1.1$
A category $\mathcal{C}$ consists of a class of objects $\operatorname{ob}(\mathcal{C})$ and a class of morphisms $\operatorname{Hom}{\mathcal{C}}(X, Y)$ associated with any two objects $X, Y \in \mathrm{ob}(\mathcal{C})$. Morphisms are also called arrows $$(f: X \rightarrow Y) \in \operatorname{Hom}{\mathcal{C}}(X, Y)$$
from the domain $X$ to the codomain $Y$. They are subject to two conditions.
(i) Two morphisms can be composed if the codomain of the former is the domain of the latter. Given $f: X \rightarrow Y$ and $g: Y \rightarrow Z$, we obtain $g \circ f: X \rightarrow Z$ and composition is associative: for $X \stackrel{f}{\rightarrow} Y \stackrel{g}{\rightarrow} Z \stackrel{h}{\rightarrow} W$, we have $h \circ(g \circ f)=(h \circ g) \circ f$.
(ii) For every object $X \in \operatorname{ob}(\mathcal{C})$, there exists an identity morphism $\operatorname{id}X \in \operatorname{Hom}{\mathcal{C}}(X, X)$, such that for all $f: X \rightarrow A$ and $g: B \rightarrow X$ we have $f \circ$ id $_X=f$ and id $X \circ g=g$.

## 数学代写|代数拓扑代考Algebraic Topology代考|Functors

A category has objects and arrows with composition and identities. Functors relate one category to another. As such, they should preserve all available structure so that there is no alternative to the following definition.
Definition $1.4$
A (covariant) functor $\mathcal{F}: \mathcal{C} \rightarrow \mathcal{D}$ from a category $\mathcal{C}$ to a category $\mathcal{D}$ assigns to every $X \in \operatorname{ob}(\mathcal{C})$ an object $\mathcal{F}(X) \in \operatorname{ob}(\mathcal{D})$ and to every morphism $f: X \rightarrow Y$ with $X, Y \in$ $\operatorname{ob}(\mathcal{C})$ a morphism $\mathcal{F}(f) \in \operatorname{Hom}{\mathcal{D}}(\mathcal{F}(X), \mathcal{F}(Y))$ such that (i) $\mathcal{F}(g \circ f)=\mathcal{F}(g) \circ \mathcal{F}(f)$ for all $f \in \operatorname{Hom}{\mathcal{C}}(X, Y)$ and $g \in \operatorname{Hom}{\mathcal{C}}(Y, Z)$. (ii) $\mathcal{F}\left(\mathrm{id}_X\right)=\operatorname{id}{\mathcal{F}(X)}$ for all $X \in \mathrm{ob}(\mathcal{C})$.

## 数学代写|代数拓扑代考Algebraic Topology代考|Categories

$$(f: X \rightarrow Y) \in \operatorname{Hom} \mathcal{C}(X, Y)$$

(i) 如果前者的陪域是后者的定义域，则可以组合两个态射。鉴于 $f: X \rightarrow Y$ 和 $g: Y \rightarrow Z$ ，我们获得 $g \circ f: X \rightarrow Z$ 并且组合是结合的: 对于 $X \stackrel{f}{\rightarrow} Y \stackrel{g}{\rightarrow} Z \stackrel{h}{\rightarrow} W$ ，我们有 $h \circ(g \circ f)=(h \circ g) \circ f$.
(ii) 对于每个对象 $X \in \operatorname{ob}(\mathcal{C})$, 存在恒等态射id $X \in \operatorname{Hom} \mathcal{C}(X, X)$, 这样对于所有 $f: X \rightarrow A$ 和 $g: B \rightarrow X$ 我们有 $f \circ \mathrm{DD}_X=f$ 和身份证 $X \circ g=g$.

## 数学代写|代数拓扑代考Algebraic Topology代考|Functors

$\mathcal{F}(g \circ f)=\mathcal{F}(g) \circ \mathcal{F}(f)$ 对全部 $f \in \operatorname{Hom} \mathcal{C}(X, Y)$ 和 $g \in \operatorname{Hom} \mathcal{C}(Y, Z)$. (二) $\mathcal{F}\left(\operatorname{id}_X\right)=$ id $\mathcal{F}(X)$ 对全 部 $X \in \mathrm{ob}(\mathcal{C})$.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Algebraic Topology, 代数拓扑, 数学代写

## avatest™帮您通过考试

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## 数学代写|代数拓扑代考Algebraic Topology代考|Interactions Between Loops and Destabilization

Recall from Proposition 2.2.3.17 that, for $t \in \mathbb{N}$, there is a natural isomorphism between $\Omega^t D, D \Sigma^{-t}: \mathscr{M} \rightrightarrows \mathscr{U}$. The following result is another application of Proposition 2.3.2.1:

Corollary 2.3.3.1 For $M$ an $\mathscr{A}$-module, there is a natural short exact sequence:
$$0 \rightarrow \Omega\left(D_s M\right) \rightarrow D_s\left(\Sigma^{-1} M\right) \rightarrow \Omega_1\left(D_{s-1} M\right) \rightarrow 0$$
Proof Let $F_{\bullet} \rightarrow M$ be a free resolution of $M$ (in $\mathscr{M}$ ) and take $C_{\bullet}=D F_{\bullet}$, which is a complex of projective unstable modules by Proposition 2.2.3.13.

Proposition $2.2 .3 .17$ implies that $\Omega C_{\bullet}$ is naturally isomorphic to $D\left(\Sigma^{-1} F_{\bullet}\right)$; $\Sigma^{-1} F_{\bullet}$ is a projective resolution of $\Sigma^{-1} M$, hence the homology of $\Omega C_{\bullet}$ calculates the derived functors $D_s\left(\Sigma^{-1} M\right)$, whereas the homology of $C_{\bullet}$ calculates the derived functors $D_s M$. The result follows immediately from Proposition 2.3.2.1.

Remark 2.3.3.2 The module $\Omega_1\left(D_{s-1} M\right)$ is the obstruction to $\Omega_s\left(D_s M\right) \rightarrow$ $D_s\left(\Sigma^{-1} M\right)$ being an isomorphism. This is zero if and only if $D_{s-1} M$ is reduced, by Corollary 2.3.1.9.

Remark 2.3.3.3 For $m \in \mathbb{N}$ and an $\mathscr{A}$-module $M$, there is a Grothendieck spectral sequence
$$\Omega_p^m D_q M \Rightarrow D_{p+q} \Sigma^{-m} M .$$
The short exact sequence of Corollary 2.3.3.1 corresponds to the case $m=1$.

## 数学代写|代数拓扑代考Algebraic Topology代考|Connectivity for Ds

The explicit identification of the destabilization functor $D M=M / B M$ (see Exercise 2.2.3.10) leads to the following result:

Lemma 2.3.4.1 For $M$ an $\mathscr{A}$-module, the natural surjection $M \rightarrow D M$ is an isomorphism in degrees $\leq 2(\operatorname{conn} M+1)$.

Proof The lowest degree element (if it exists-i.e. if $\operatorname{conn}(M)$ is finite) of $M$ has degree conn $(M)+1$, hence the lowest degree element of $B M$ has degree at least $2(\operatorname{conn}(M)+1)+1$. The result follows.

The following statement is a general result for connected algebras, stated here for the Steenrod algebra.

Lemma 2.3.4.2 An $\mathscr{A}$-module $M$ has a free resolution $F_{\bullet} \rightarrow M$ in $\mathscr{M}$ with $\operatorname{conn}\left(F_s\right) \geq \operatorname{conn}(M)+s$.
Proof An exercise for the reader.

The following weak result is sufficient for the initial applications; a much stronger result holds (combine Lemma 2.5.1.6 with Theorem 2.5.1.8).
Proposition 2.3.4.3 For $0<s \in \mathbb{N}$ and $M$ an $\mathscr{A}$-module
$$\operatorname{conn}\left(D_s M\right) \geq 2(\operatorname{conn} M+s)$$

## 数学代写|代数拓扑代考Algebraic Topology代考|Interactions Between Loops and Destabilization

$$0 \rightarrow \Omega\left(D_s M\right) \rightarrow D_s\left(\Sigma^{-1} M\right) \rightarrow \Omega_1\left(D_{s-1} M\right) \rightarrow 0$$

$$\Omega_p^m D_q M \Rightarrow D_{p+q} \Sigma^{-m} M .$$

## 数学代写|代数拓扑代考Algebraic Topology代考|Connectivity for Ds

$$\operatorname{conn}\left(D_s M\right) \geq 2(\operatorname{conn} M+s)$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。