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

# 数学代写|代数拓扑代考Algebraic Topology代考|MATH6510 Categories

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 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。