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## 数学代写|代数拓扑代考Algebraic Topology代考|Exact Sequence of Groups

This section conveys some results of exact sequences of groups and their homomorphisms which are frequently applied in algebraic topology. For this section the book Adhikari and Adhikari (2014) is referred.
Definition 1.4.1 A sequence of groups and their homomorphisms
$$\cdots \rightarrow G_{n+1} \stackrel{f_{n+1}}{\longrightarrow} G_n \stackrel{f_n}{\longrightarrow} G_{n-1} \rightarrow \cdots$$
is said to be exact if ker $f_n=\operatorname{Im} f_{n+1}$ for all $n$. Clearly, $f_n \circ f_{n+1}=0$ for an exact sequence.

Proposition 1.4.2 (i) In the short exact sequence $0 \rightarrow G \stackrel{f}{\longrightarrow} K$, $f$ is $a$ monomorphism.
(ii) In the short exact sequence $G \stackrel{f}{\longrightarrow} K \rightarrow 0, f$ is an epimorphism.
(iii) The sequence $0 \rightarrow G \stackrel{f}{\longrightarrow} K \rightarrow 0$ is exact if and only if $f$ is an isomorphism;
(iv) If $G$ is a normal subgroup of $K$ and i : $\hookrightarrow K$ is the inclusion map (i.e., $i(x)=x$ for all $x \in G)$, then the sequence
$$0 \rightarrow G \stackrel{i}{\longrightarrow} K \stackrel{p}{\longrightarrow} K / G \rightarrow 0$$
is an exact sequence, where 0 denotes the trivial group and $p$ is the natural homomorphism defined by $p(x)=x+G$ for all $x \in K$.

Proposition 1.4.3 Given exact sequences of groups and homomorphisms
$$0 \rightarrow G_i \stackrel{f_i}{\longrightarrow} K_i \stackrel{g_i}{\longrightarrow} H_i \rightarrow 0$$
for each element $i \in I$, the sequence
$$0 \rightarrow \bigoplus_{i \in I} G_i \stackrel{\oplus f_i}{\longrightarrow} \bigoplus_{i \in I} K_i \stackrel{\oplus g_i}{\longrightarrow} \bigoplus_{i \in I} H_i \rightarrow 0$$
is also exact

## 数学代写|代数拓扑代考Algebraic Topology代考|Free Product and Tensor Product of Groups

This subsection conveys the concept of free product of groups.
Definition 1.5.1 Let $G$ and $H$ be groups (not necessarily abelian). Their free product denoted by $G * H$ is a group satisfying the following condition: if there are homomorphisms $i$ and $j$ such that given a pair of homomorphisms $f: G \rightarrow K$ and $g: H \rightarrow K$ for any group $K$, there exists a unique homomorphism $h: G * H \rightarrow K$ making the diagram in Fig. 1.2 commutative.
For example, $\mathbf{Z} * \mathbf{Z}$ is a free group (of rank 2 ).

Remark 1.5.2 An alternative description of free product $G * H$ may be given with the help of presentations of groups $G$ and $H$.

Definition 1.5.3 Let $G=\langle X: R\rangle$ and $H=\langle Y: S\rangle$ be presentations of the groups $G$ and $H$ in which the sets $X$ and $Y$ are generators (and thus the relations $R$ and $S$ ) are disjoint. Then a presentation of $G * H$ is given by
$$G * H=\langle X \cup Y: R \cup S\rangle$$

This subsection conveys the concept of tensor products of groups.
Definition 1.5.4 Let $G$ and $H$ be two abelian groups. Their tensor product denoted by $G \otimes H$ is the group defined as the abelian group generated by all pairs of the form $(g, h)$ with $g \in G, h \in H$ satisfying the bilinearity relations $\left(g+g^{\prime}, h\right)=(g, h)+$ $\left(g^{\prime}, h\right)$ and $\left(g, h+h^{\prime}\right)=(g, h)+\left(g, h^{\prime}\right)$.

For example, $\mathbf{Z}m \otimes \mathbf{Z}_n=\mathbf{Z}{(m, n)}$, where $(m, n)$ is the gcd of $m$ and $n$ : on the other hand, this tensor product is 0 , if $m$ and $n$ are relatively prime.

## 数学代写|代数拓扑代考Algebraic Topology代考|Exact Sequence of Groups

1.4.1群及其同态的序列
$$\cdots \rightarrow G_{n+1} \stackrel{f_{n+1}}{\longrightarrow} G_n \stackrel{f_n}{\longrightarrow} G_{n-1} \rightarrow \cdots$$

(二)在短的确切序列$G \stackrel{f}{\longrightarrow} K \rightarrow 0, f$是一个外胚。
(iii)序列$0 \rightarrow G \stackrel{f}{\longrightarrow} K \rightarrow 0$是精确的当且仅当$f$是同构的;
(iv)如果$G$是$K$的正子群，并且i: $\hookrightarrow K$是包含图(即$i(x)=x$对于所有$x \in G)$，则序列
$$0 \rightarrow G \stackrel{i}{\longrightarrow} K \stackrel{p}{\longrightarrow} K / G \rightarrow 0$$

$$0 \rightarrow G_i \stackrel{f_i}{\longrightarrow} K_i \stackrel{g_i}{\longrightarrow} H_i \rightarrow 0$$

$$0 \rightarrow \bigoplus_{i \in I} G_i \stackrel{\oplus f_i}{\longrightarrow} \bigoplus_{i \in I} K_i \stackrel{\oplus g_i}{\longrightarrow} \bigoplus_{i \in I} H_i \rightarrow 0$$

## 数学代写|代数拓扑代考Algebraic Topology代考|Free Product and Tensor Product of Groups

1.5.1设$G$和$H$为组(不一定是abel)。它们的自由积表示为$G * H$是满足以下条件的群:如果存在同态$i$和$j$，使得对于任意群$K$给定一对同态$f: G \rightarrow K$和$g: H \rightarrow K$，则存在一个唯一的同态$h: G * H \rightarrow K$，使得图1.2中的图可以交换。

1.5.3设$G=\langle X: R\rangle$和$H=\langle Y: S\rangle$表示组$G$和$H$，其中集合$X$和$Y$是生成器(因此关系$R$和$S$)是不相交的。然后介绍$G * H$是由
$$G * H=\langle X \cup Y: R \cup S\rangle$$

## 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。

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

This section conveys some basic concepts of set theory (naive) initiated around 1870 by the German mathematician Georg Cantor (1845-1918) which are used throughout the book. Set theory occupies an important position in mathematics. Many concrete concepts and examples are based on it. It is assumed that the readers are familiar with the sets
$\mathbf{N}$ (set of natural numbers/positive integers)
$\mathbf{Z}$ (set of integers)
$\mathbf{Q}$ (set of rational numbers)
$\mathbf{R}$ (set of real numbers)
$\mathbf{C}$ (set of complex numbers)
For precise description of many concepts of mathematics and also for mathematical reasoning the concepts of relations(functions) and cardinality of sets are very important, which are discussed first.

A binary relation $\rho$ on a nonempty set $X$ is a subset of $X \times X$, which is said to be an equivalence relation if $\rho$ is reflexive, i.e., $(x, x) \in \rho$ for each $x \in X$; symmetric, i.e., $(x, y) \in \rho$ implies $(y, x) \in \rho$ and transitive i.e., $(x, y) \in \rho$ and $(y, z) \in \rho$ imply $(x, z) \in \rho$ for $x, y, z \in X$.

Definition 1.1.1 Let $X$ be a nonempty set and $\rho$ be an equivalence relation on $X$. The disjoint classes $[x]$ into which the set $X$ is partitioned by $\rho$ constitute a set, called the quotient set of $X$ by $\rho$, denoted by $X / \rho$, where $[x]$ denotes the class (determined by $\rho$ ) containing the element $x$ of $X$. Each element $x$ of the class $[x]$ is called a representative of $[x]$.

Example 1.1.2 Given a positive integer $n$, the quotient set $\mathbf{Z}n$ consists of all $n$ distinct classes $[0],[1], \ldots,[n-1]$. The set $\mathbf{Z}_n$ is called the residue classes of $\mathbf{Z}$ modulo $n$. Remark 1.1.3 The set $\mathbf{Z}_n$ provides very strong different algebraic structures (depending on $n$ ). The visual description of $\mathbf{Z}{12}$ is a 12-h clock.

## 数学代写|代数拓扑代考Algebraic Topology代考|Groups and Fundamental Homomorphism Theorem

This section conveys some basic results of group theory which are used throughout the book. Originally, a group was defined as the set of permutations (i.e., bijections) on a nonempty set $X$ with the property that combination (called composition) of two permutations is also a permutation on $X$. Earlier definition of a group is generalized to the present concept of an abstract group by a set of axioms.

Definition 1.2.1 A group $G$ is a nonempty set $G$ together with a binary operation (called composition), that is, a rule that assigns to each ordered pair $(a, b)$ in $G \times G$, an element of $G$, denoted by $a b$ (or $a \cdot b$ called a multiplication) such that
G(1) $a b(c)=a(b c)$ for all $a, b, c$ in $G$ (associative law);
$\mathbf{G ( 2 )}$ there exists an element $e$ in $G$ such that $a e=e a=a$ for all $a$ in $G$ (existence of identity);
$\mathbf{G ( 3 )}$ for each $a$ in $G$, there is an element $a^{\prime}$ in $G$ such that $a a^{\prime}=a^{\prime} a=e$ (existence of inverse).

Remark 1.2.2 In a group $G, e$ is unique and for each $a$ in $G, a^{\prime}$ is also unique. The element $a^{\prime}$ denoted by $a^{-1}$, is called the inverse of $a$ for each $a \in G$. In additive notation, $a b$ is written as $a+b ; e$ is as 0 (zero) and $a^{-1}$ as $-a$.

A group $G$ is said to be commutative (or abelian) if $a b=b a$ for all $a, b$ in $G$. We usually use the term ‘abelian group’ when the composition law is in additive notation. A group $G$ is said to be finite if its underlying set $G$ is finite; otherwise, it is said to be infinite.

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

$\mathbf{N}$(自然数/正整数的集合)
$\mathbf{Z}$(一组整数)
$\mathbf{Q}$(一组有理数)
$\mathbf{R}$(实数集)
$\mathbf{C}$(复数集合)

1.1.1设$X$为非空集合，$\rho$为$X$上的等价关系。集合$x$被$\rho$分割成的不相交类$[x]$构成一个集合，称为$x$除以$\rho$的商集，表示为$x / \rho$，其中$[x]$表示包含$x$的元素$x$的类(由$\rho$决定)。类$[x]$中的每个元素$x$被称为$[x]$的代表。

## 数学代写|代数拓扑代考Algebraic Topology代考|Groups and Fundamental Homomorphism Theorem

$\mathbf{G(2)}$在$G$中存在一个元素$e$，使得$a e=e a=a$对于$G$中的所有$a$(单位的存在性);
$\mathbf{G(3)}$对于$G$中的每一个$a$，在$G$中存在一个$a^{\素数}$使得$a a^{\素数}=a^{\素数}a=e$(逆的存在性)。

## 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

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

To make the last naturality statement meaningful, we need to discuss in what sense also simplicial homology is functorial. First we describe the morphisms in the category of simplicial complexes.

Definition $3.21$
A map $f: X \rightarrow Y$ of simplicial complexes $X$ and $Y$ is called simplicial if it restricts to a map of the 0 -skeleta $f^0: X^0 \rightarrow Y^0$ such that
(i) Whenever $v_0, \ldots, v_n \in X^0$ span a simplex in $X$, so do $f\left(v_0\right), \ldots, f\left(v_n\right)$ in $Y$.
(ii) The restriction of $f$ to a simplex $\left[v_0, \ldots, v_n\right]$ is the unique affine linear extension of $f^0$ so that $f\left(\sum_{i=0}^n t_i v_i\right)=\sum_{i=0}^n t_i f\left(v_i\right)$.

## 数学代写|代数拓扑代考Algebraic Topology代考|The Definition of Singular Homology

As we just alluded, the construction of singular homology, like simplicial homology, is a two step procedure of defining functors
$$\operatorname{Top}^{(2)} \stackrel{C_(-; R)}{\longrightarrow} R \text {-chain } \stackrel{H_}{\longrightarrow} R \text {-mod }$$
Here a morphism $f_:\left(C_, c_\right) \rightarrow\left(D_, d_*\right)$ in the category $R$-chain of chain complexes of $R$-modules is a chain map, a family of homomorphisms $f_n: C_n \rightarrow$ $D_n$ such that $d_n \circ f_n=f_{n-1} \circ c_n$ for all $n \in \mathbb{Z}$. So a chain map gives rise to a commutative ladder

To construct the second functor, we observe again that $f_n$ maps cycles in $C_n$ to cycles in $D_n$ and boundaries in $C_n$ to boundaries in $D_n$, so a chain map $f_$ induces morphisms $$H_n\left(f_\right): H_n\left(C_\right) \rightarrow H_n\left(D_\right)$$
and we clearly have $H_n\left(f_* \circ g_\right)=H_n\left(f_\right) \circ H_n\left(g_\right)$ and $H_n\left(\mathrm{id}{C}\right)=\operatorname{id}{H_n\left(C*\right)}$. To construct the first functor, let $X$ be any topological space.

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

(i) 每当 $v_0, \ldots, v_n \in X^0$ 跨越一个单纯形 $X$ ，也是 $f\left(v_0\right), \ldots, f\left(v_n\right)$ 在 $Y$.
(ii) 限制 $f$ 到单纯形 $\left[v_0, \ldots, v_n\right]$ 是的唯一仿射线性扩展 $f^0$ 以便 $f\left(\sum_{i=0}^n t_i v_i\right)=\sum_{i=0}^n t_i f\left(v_i\right)$.

## 数学代写|代数拓扑代考Algebraic Topology代考|The Definition of Singular Homology

Vleft 或额外的 iright

## 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|Cofibrations and Homotopy Pushouts

A map $i: A \rightarrow X$ of spaces has the homotopy extension property (HEP) for a space $Y$ if for each homotopy $H: A \times I \rightarrow Y$ and for each map $f: X \rightarrow Y$ with $f(i(a))=H(a, 0)$ for all $a \in A$, there exists a homotopy $H^{\prime}: X \times I \rightarrow Y$ such that $H^{\prime}(i(a), t)=H(a, t)$ and $H^{\prime}(x, 0)=f(x)$ for all $a \in A, x \in X$, and $t \in I$. The homotopy $H^{\prime}$ is called an extension of $H$ with initial condition $f$. A map $i: A \rightarrow X$ is called a cofibration if it has the HEP for all spaces $Y$. Setting $i_0^A(a)=(a, 0)$ and $i_0^X(x)=(x, 0)$, we can express the definition by the diagram in which we require only existence of $H^{\prime}$, not uniqueness. The definition of cofibration is of course impractical to verify directly. Therefore it is good to know that if $i$ has the HEP for its own mapping cylinder $M_i$, then $i$ has the HEP for all spaces. To prove this, we first observe that the pushout diagram uniquely defines the map $s: M_i \rightarrow X \times I$. If $i$ is the inclusion of a subspace, then $s$ is a continuous bijection onto the image $X \times{0} \cup A \times I$. But the topology of $M_i$ might be finer than the subspace topology within $X \times I$. However, we see that $s$ is a homeomorphism onto the image, hence a subspace inclusion, if $i$ and thus also $i \times$ id is the inclusion of a closed subspace. By Proposition $2.12$ (iii) below, $s$ is also a homeomorphism onto the image if $i$ is a cofibration.

## 数学代写|代数拓扑代考Algebraic Topology代考|Higher Homotopy Groups

As a reward for the hard work of the previous section, we obtain the following version of van Kampen’s theorem, which is a powerful tool to carry out actual computations of fundamental groups.

The fundamental group of a pointed space
$$\pi_1\left(X, x_0\right)=\left{\gamma:(I,{0,1}) \rightarrow\left(X, x_0\right)\right} / \simeq$$
is defined in terms of pointed homotopy classes of one-dimensional loops and hence encodes primarily low-dimensional data. It is therefore good at distinguishing lowdimensional spaces, for example we have
$$\pi_1\left(S^1, \bullet\right) \not \pi_1\left(S^2, \bullet\right)$$

## 数学代写|代数拓扑代考Algebraic Topology代考|Higher Homotopy Groups

$\backslash$ left 缺少或无法识别的分隔符

$$\pi_1\left(S^1, \bullet\right) \pi_1\left(S^2, \bullet\right)$$

## 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。

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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。

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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。

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## 数学代写|代数拓扑代考Algebraic Topology代考|The Category of A -Modules

Let $\mathscr{M}$ denote the category of (left) $\mathscr{A}$-modules. This is an abelian category with additional structure; namely, the fact that $\mathscr{A}$ is a Hopf algebra implies that the tensor product (as graded vector spaces) of two $\mathscr{A}$-modules has a natural $\mathscr{A}$-module structure. Explicitly, the Steenrod squares act via:
$$S q^n(x \otimes y)=\sum_{i+j=n} S q^i(x) \otimes S q^j(y)$$
this corresponds to the fact that the diagonal $\Delta: \mathscr{A} \rightarrow \mathscr{A} \otimes \mathscr{A}$ is determined by $\Delta S q^n=\sum_{i+j=n} S q^i \otimes S q^j$

Since $\mathscr{A}$ is a connected algebra (concentrated in non-negative degrees, with $\mathscr{A}^0=\mathbb{F}$ ) the Hopf algebra conjugation (or antipode) $\chi: \mathscr{A}^{\circ} \rightarrow \mathscr{A}$ is determined by the diagonal [MM65] and is an isomorphism of algebras, where $\mathscr{A}^{\circ}$ is $\mathscr{A}$ equipped with the opposite algebra structure ( $\chi$ is an anti-automorphism of $\mathscr{A}$ ).

## 数学代写|代数拓扑代考Algebraic Topology代考|Unstable Modules and Destabilization

Whereas the cohomology of a spectrum (object from stable homotopy theory which represents a cohomology theory) is simply an $\mathscr{A}$-module, the cohomology of a space has further structure; it is an algebra (via the cup product) and the underlying $\mathscr{A}$-module is unstable.

Definition 2.2.3.1 An $\mathscr{A}$-module $M$ is unstable if $S q^i x=0, \forall i>|x|$. The full subcategory of unstable modules is denoted $\mathscr{U} \subset \mathscr{M}$.

Proposition 2.2.3.2 The category $\mathscr{U}$ is an abelian subcategory of $\mathscr{M}$ and is closed under the tensor product $\otimes$ of $\mathscr{M}$.

Proof From the definition of instability, it is clear that a submodule (respectively quotient) of an unstable module is unstable. This implies that $\mathscr{U}$ is an abelian subcategory of $\mathscr{M}$.

Closure under $\otimes$ is seen as follows. By definition, $S q^n(x \otimes y)=$ $\sum_{i+j=n} S q^i(x) \otimes S q^j(y)$; if $n>|x \otimes y|$ and $i+j=n$, then either $i>|x|$ or $j>|y|$, so that the right hand expression is zero, as required.

Remark 2.2.3.3 The duality functor $(-)^{\vee}: \mathscr{M}^{\mathrm{op}} \rightarrow \mathscr{M}$ does not preserve $\mathscr{U}$, since the relation $S q^0=1$ implies that an unstable module is concentrated in degrees $\geq 0$. The dual $M^{\vee}$ of a module $M$ concentrated in degrees $\geq 0$ is concentrated in degrees $\geq 0$ if and only if $M=M^0$; for example, the dual of $\Sigma \mathbb{F}$ is not unstable.
Example 2.2.3.4 For $n \in \mathbb{N}$, the suspension functor $\Sigma^n: \mathscr{M} \rightarrow \mathscr{M}$ restricts to an exact functor $\Sigma^n: \mathscr{U} \rightarrow \mathscr{U}$ (given by $\Sigma^n \mathbb{F} \otimes-$ ). This is not an equivalence of categories if $n>0$.

For later use, the following definition is recalled, which uses the tensor product of $\mathscr{U}$.

## 数学代写|代数拓扑代考Algebraic Topology代考|The Category of A -Modules

$$S q^n(x \otimes y)=\sum_{i+j=n} S q^i(x) \otimes S q^j(y)$$

## 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。

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## 数学代写|代数拓扑代考Algebraic Topology代考|Statement of HKR Theorem

If $X$ is a space, we denote $\widetilde{H}_*(X)$ its reduced homology coalgebra. Recall that a space is formal if its cochain algebra is quasi-isomorphic to its cohomology as a CDGA. This includes all spheres, suspensions, Lie group or Kähler varieties.

Theorem 1.6.1 Assume $X$ is a formal space of finite type in each degree. And let $(\operatorname{Sym}(V), d) \stackrel{\cong}{\longrightarrow} A$ be a cofibrant resolution ${ }^{33}$ of $A$. There are natural $($ in $A, M)$ equivalences
respectively in CDGA and in $\mathbf{C H}_X(A)$-Mod. Further, if $f: X \rightarrow Y$ is $a$ formal $\operatorname{map}^{34}$ we have a commutative diagrams (respectively in CDGA and $\left.\mathbf{C H}_X(A)-\operatorname{Mod}\right):$

## 数学代写|代数拓扑代考Algebraic Topology代考|HKR Isomorphism and Hodge Decomposition

We now relate the HKR isomorphisms from Sect. 1.6.1 with the Hodge filtrations on the various (co)chains functors.

Recall from Example 1.4.6 that $\operatorname{Sym}(V \oplus V[d])$ and $\operatorname{Sym}\left(V \oplus V^{\vee}[d]\right)$ are endowed with canonical dg-multiplicative- $\gamma$-ring with zero multiplication structure for which $V$ is of pure weight 0 and $V[d]$ or $V^{\vee}[d]$ are of pure weight 1 . More generally if $U$ and $W$ are graded modules and $d$ is a differential on $\operatorname{Sym}(U \oplus W)$

such that $d(W) \subset W \otimes \operatorname{Sym}(U)$, then $(\operatorname{Sym}(U \oplus W), d)$ has a dg-multiplicative$\gamma$-ring with zero multiplication structure for which $U$ is on weight 0 and $W$ in weight 1.

Corollary 1.6.8 Assume $X$ is a formal space of finite type in each degree. And let $(\operatorname{Sym}(V), d) \stackrel{\cong}{\longrightarrow} A$ be a cofibrant resolution of $A$.
The HKR quasi-isomorphisms yields natural (in $A$ and $M$ ) equivalences
1.
$H K R: \mathbf{C H}{S^d \times X}(A) \stackrel{\cong}{\longrightarrow} \operatorname{Sym}\left(\left(V \otimes H(X)\right) \oplus\left(V \otimes H_(X)\right)[d]\right), \quad(1.100)$
$H K R: \mathbf{C H}{S^d \wedge X}(A) \stackrel{\cong}{\longrightarrow} \operatorname{Sym}\left(V \oplus\left(V \otimes \widetilde{H}(X)\right)[d]\right)$ of $d g$-multiplicative $\gamma$-ring with trivial multiplication, 2. \begin{aligned} &\mathbf{C H}{S^d \times X}(A, M) \ &\stackrel{\cong}{\stackrel{\cong}{\longrightarrow}} M \underset{\operatorname{Sym}(V)}{\otimes} \operatorname{Sym}\left(\left(V \otimes H(X)\right) \oplus\left(V \otimes H_*(X)\right)[d]\right), \ & \end{aligned}
of $d g-\gamma$-ring with trivial multiplication in $\mathbf{C H}{S^d \times X}(A)-\mathrm{Mod}$ and $\mathrm{CH}{S^d \wedge X}(A)$-Mod respectively

1. as well as
\begin{aligned} &\mathbf{C H}^{S^d \times X}(A, M) \cong \underset{\operatorname{Sym}(V)}{M} \otimes \operatorname{Sym}\left(\left(V^{\vee} \otimes \widetilde{H}^(X) \oplus V\right) \oplus\left(V^{\vee} \otimes H^(X)\right)[-d]\right),(1.104) \ &\mathbf{C H}^{S^d \wedge X}(A, M) \cong M_{\operatorname{Sym}(V)}^{\otimes} \operatorname{Sym}\left(V \oplus\left(V^{\vee} \otimes \widetilde{H}^*(X)\right)[-d]\right) \ & \end{aligned}

## 数学代写|代数石扑代考Algebraic Topology代考|HKR Isomorphism and Hodge Decomposition

$H K R: \mathbf{C H} S^d \wedge X(A) \stackrel{\cong}{\longrightarrow} \operatorname{Sym}(V \oplus(V \otimes \widetilde{H}(X))[d])$ 的 $d g$-乘刧 $\gamma$-环与平凡的乘法，2。

1. $$\left.\left.\mathbf{C H}^{S^{d^d} \times}(A, M) \cong \underset{\operatorname{Sym}(V)}{M} \otimes \operatorname{Sym}\left(\left(V^{\vee} \otimes \widetilde{H}^{(} X\right) \oplus V\right) \oplus\left(V^{\vee} \otimes H^{(} X\right)\right)[-d]\right),(1.104) \quad \mathbf{C H}^{S^d \wedge X}(A, M) \cong M_{\mathrm{Sym}(V)}^{\otimes} \operatorname{Sym}\left(V \oplus \left(V ^ { \vee } \otimes \widetilde { H } ^ { * } \left($$

## 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。

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## 数学代写|代数拓扑代考Algebraic Topology代考|Introduction and Overview

This paper is based on lectures given at the Vietnamese Institute for Advanced Studies in Mathematics. It aims to present both the theory of higher Hochschild (co)homology and its application to higher string topology. It contains detailed proofs of results stated in the note [Gi3] as well as some new results building on our previous work [GTZ3, Gi4] notably. One of the main new result is an application of the techniques of Higher Hochschild (co)homology to study higher string topology ${ }^1$ and prove that, in addition to its already rich algebraic package, the latter inherits an additional Hodge filtration (compatible with the rest of the structure). We also prove that the $E_n$-centralizer of maps of commutative (dg-)algebras are equipped with a Hodge decomposition and a compatible structure of framed $E_n$-algebras ${ }^2$ and study Hodge decompositions suspensions and products by spheres generalizing the ones of [P] and dual results of [TW], see below for more details on these results.

This various results are also a pretext to illustrate the techniques of higher order Hochschild Homology in the case of commutative differential graded algebras, both using its derived (in an $\infty$-categorical sense) interpretation and functoriality and emphasizing on and using its nice combinatorial structure and how to use it. The emphasis on this latter point is another benefit of this paper compared to most of the literature we know ${ }^3$ and a good way to get a feeling on the behavior and benefits of higher Hochschild (co)homology, in, we hope, a gentle way.

Higher Hochschild (co)homology was first emphasized by Pirashvili in [P] in order to understand the Hodge decomposition of Hochschild homology and how to generalize it. Higher Hochschild (co)homology is in fact a joint invariant of both topological spaces (or their homotopy combinatorial avatar : simplicial sets) and commutative differential graded algebras (CDGA for short). As the name suggests, it is a generalization for commutative (dg-)algebras of the standard Hochschild homology of dg-associative algebras. It is also a special case [GTZ2, AF] of factorization homology ${ }^4[\mathrm{BD}, \mathrm{Lu} 3, \mathrm{AF}]$ which get extra-functoriality and is one of the easiest one to compute and manipulate. ${ }^5$

## 数学代写|代数拓扑代考Algebraic Topology代考|Notations, Conventions and a Few Standard Facts

We fix a ground field $k$ of characteristic 0 . We will also use the following notations and conventions

If $\left(C, d_C\right)$ is a cochain complex, $C[i]$ is the cochain complex such that $C[i]^n:=$ $C^{n+i}$ with differential $(-1)^i d_C$. We will mainly work with cochain complexes and adopt the convention that a chain complex is a cochain complex with opposite grading when we need to compare gradings.

An $\infty$-category will be a complete Segal space. Any model category gives rise to an $\infty$-category.

We write k-Mod ${ }^{d g}$ for the category of cochain complexes and k-Mod for its associated $\infty$-category. We will use the abbreviation dg for differential graded. We will use the words (co)homology for an object of these $\infty$-categories (in other words a complex thought up to quasi-isomorphism) and use the words (co)homology groups for the actual groups computed by taking the quotient of the (co)cycles by (co)boundaries (for instance see Definition 1.3.19).

sSet and Top: sSet is the (model) category of simplicial sets, that is functors from $\Delta^{o p} \rightarrow$ Set where $\Delta$ is the simplex category of finite sets $n_{+}:={0, \ldots, n}$ with order preserving maps. We also have the (model) category of topological spaces Top. These two categories are Quillen equivalent: $|-|:$ sSet $\underset{\leftrightarrows}{\leftrightarrows}$ Top : $\Delta_{\bullet}(-)$. Here $\Delta_{\bullet}:$ Top $\rightarrow$ sSet the singular set functor defined by $\Delta_n(X)=$ $\operatorname{Map}{\text {Top }}\left(\Delta^n, X\right)$, where $\Delta^n$ is the standard $n$-dimensional simplex, and $\left|Y{\bullet}\right|$ the geometric realization. Their associated $\infty$-categories, respectively denoted sSet and Top are thus equivalent.

These four $(\infty)$-categories are symmetric monoidal with respect to disjoint union.

## 数学代写|代数拓扑代考Algebraic Topology代考|Introduction and Overview

Pirashvili 在 [P] 中首先强调了更高的 Hochschild (共) 同源性，以了解 Hochschild 同源性的 Hodge 分解以及如何推广它。 高等 Hochschild (共) 同源性实际上是拓扑空间 (或其同伦组合化身：单纯集) 和交换微分梯度代数（简称 CDGA) 的联合不变: 量。顾名思义，它是 dg-结合代数的标准 Hochschild 同调的交换 (dg-) 代数的推广。也是分解同调的特例[GTZ2,AF] [ $[\mathrm{BD}, \mathrm{Lu} 3, \mathrm{AF}]$ 它具有额外的功能性，是最容易计算和操作的一种。

## 数学代写|代数拓扑代考Algebraic Topology代考|Notations, Conventions and a Few Standard Facts

sSet 和 Top: sSet 是单纯集的 (模型) 类别，即来自 $\Delta^{o p} \rightarrow$ 设置位置 $\Delta$ 是有限集的单纯形范帱 $n_{+}:=0, \ldots, n$ 带有保留地图的 顺序。我们还有拓扑空间 Top 的 (模型) 类别。这两个光别是 Quillen 等效的: $|-|$ : 集士最佳： $\Delta_{\bullet}(-)$. 这里 $\Delta_{\bullet}$ :最佳 $\rightarrow$ sSet 定义的奇异集函子 $\Delta_n(X)=\operatorname{Map} \operatorname{Top}\left(\Delta^n, X\right)$ ，在哪里 $\Delta^n$ 是标准 $n$-维单纯形，和 $|Y \bullet| 几$ 何实现。他们的关联 $\infty$-类别，分 别表示为 sSet 和 Top，因此是等价的。

## 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。