### 数学代考|代数几何代写algebraic geometry代考|Categories, and the Category of Algebraic Varieties

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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代考|代数几何代写algebraic geometry代考|The Definition of a Category, and an Example: The Category of Sets

In a category $C$, we have a class of objects $\operatorname{Obj}(C)$ and a class of morphisms $\operatorname{Mor}(C)$, satisfying certain axioms.

Explaining the need to distinguish between sets and classes takes us on a brief detour into set theory. It comes from the fact that the naive interpretation of the notation
$${X \mid \ldots}$$
as “the set of all sets $X$ such that …” leads to a contradiction in
$${X \mid X \notin X}$$
where neither $X \in X$ nor $X \notin X$ are possible; because of that, we distinguish between sets and classes and interpret (2.1.1) as “the class of all sets $X$ such that …,” and define a set as a class which is an element of another class. Otherwise, it is called a proper class. Note that then (2.1.2) is just an example of a proper class; in fact, it is the class of all sets.
The axioms of a category say that $\operatorname{Obj}(C)$ and $\operatorname{Mor}(C)$ satisfy all the formal properties of the most basic example: the category Sets whose objects are sets and morphisms are mappings of sets. Thus, we have two mappings
$$S, T: \operatorname{Mor}(C) \rightarrow \operatorname{Obj}(C)$$
(called source and target, which in the category of sets are the domain and codomain of a mapping). A morphism $f \in \operatorname{Mor}(C)$ with $S(f)=X, T(f)=Y$ (where $X, Y$ are objects) is called a morphism from $X$ to $Y$, and denoted by
$$f: X \rightarrow Y$$
or
$$X \stackrel{f}{\longrightarrow} Y$$
same as for mappings of sets. We have a mapping $\operatorname{Obj}(C) \rightarrow \operatorname{Mor}(C)$ called the identity morphism
$$I d_{X}: X \rightarrow X$$
Also like for mappings, the structure of a category specifies, for two morphisms
$$f: X \rightarrow Y, g: Y \rightarrow Z,$$
the composition
$$g \circ f: X \rightarrow Z$$

(note the reversal of order of $f$ and $g$, motivated by mappings: when we apply mappings to an element, we write $g \circ f(x)=g(f(x))$, even though we apply $f$ first).

Morphisms, however, may not always be mappings, (although in the category Sets, and many other examples, they are), and so they cannot be, in the context of pure category theory, applied to elements. So instead, we must define a category by axioms. These axioms are simple: they say that the source and target of $I d_{X}$ are equal to $X$, and that the composition of morphisms is associative
$$(h \circ g) \circ f=h \circ(g \circ f)$$
(when applicable) and unital, i.e. for $f: X \rightarrow Y$,
$$I d_{Y} \circ f=f \circ I d_{X}=f$$
Lastly, we require that the class $C(X, Y)$ of all morphisms $f: X \rightarrow Y$ be a set. We call the category $C$ small if the class $O b j(C)$ is a set. (Then necessarily also $M o r(C)$ is a set.)
To see that morphisms do not always have to be mappings, note that to every category $C$, there is the opposite (sometimes also called dual) category $C^{O p}$ which “turns around the arrows”: $O b j\left(C^{O p}\right)=\operatorname{Obj}(C), \operatorname{Mor}\left(C^{O p}\right)=\operatorname{Mor}(C)$ and $I d$ is $C$ and $C^{O p}$ are the same, but $S$ in $C^{O p}$ is $T$ in $C$ and vice versa, and composition of morphisms $\alpha \circ \beta$ in $C^{O p}$ is $\beta \circ \alpha$ in $C$.

## 数学代考|代数几何代写algebraic geometry代考|Categories of Algebraic Structures

One purpose of categories is to be able to discuss, and relate, mathematical structures of the same kind. For example, all sets, all groups, all abelian groups, all rings, all topological spaces, all algebraic varieties. (Recall that a group has one operation which is associative, unital and has an inverse; an abelian group is a group which is also commutative.) So we want a category whose objects are the given structures, i.e. the category of groups, rings, etc. But what should the morphisms be?

Of course, we may be able to define the morphisms in a fairly arbitrary way, as long as they satisfy the axioms of a category, which we learned in Sect. 2.1.1. For example, we could define the only morphisms to be identities, but that would not be very useful for understanding the given mathematical structure. This is why, usually, there is a standard choice of morphisms of mathematical structures of a given kind, which are, vaguely speaking, mappings which preserve the given structure. Making this precise requires different techniques in different cases.

The case which is the easiest to handle are categories of algebraic structures. An algebraic structure comes with operations (example: addition or multiplication). In this case, the default choice of morphisms are homomorphisms of the given algebraic structures, which means mappings which preserve the operations.

For example, a homomorphism of groups $f: G \rightarrow H$, written multiplicatively, is required to satisfy
$$f(x \cdot y)=f(x) \cdot f(y)$$
(Philosophically, the unit and inverse are also operations, so we should include $f(1)=1$ and $f\left(x^{-1}\right)=(f(x))^{-1}$, but in the case of groups, it follows from the axioms.) The category of groups and homomorphisms is denoted by Grp, the category of abelian groups and homomorphisms is denoted by $A b$.
Analogously, a homomorphism of rings satisfies
$$f(x+y)=f(x)+f(y)$$
and
$$f(x y)=f(x) f(y)$$
A non-zero ring is not a group with respect to multiplication (because one cannot divide by 0 ), so we must also require
$$f(1)=1,$$
since it does not follow automatically.
One must be careful not to confuse a homomorphism of rings with a homomorphism of $R$-modules over a fixed ring $R$. (Recall that a module over a commutative ring $R$ is an abelian group $M$ with an operation of taking multiples by elements $r \in R$ which satisfies distributivity from both sides, unitality and associativity; an example of an $R$-module is $R$ itself or more generally an ideal of $R$, which is the same thing as a submodule of the $R$-module $R$.)
Thus, a homomorphism $f: M \rightarrow N$ of $R$-modules satisfies
\begin{aligned} &f(x+y)=f(x)+f(y) \ &f(r x)=r f(x) \text { for } r \in R \end{aligned}
Sometimes, the same algebraic object may be used for two different purposes. For example, as already remarked, a ring $R$ is a module over itself. In such cases, we must be careful to specify which category we are working in.

## 数学代考|代数几何代写algebraic geometry代考|Functors and Natural Transformations

Let $C, D$ be categories. A functor $F: C \rightarrow D$ consists of maps $F=O b j(F)$ : $\operatorname{Obj}(C) \rightarrow \operatorname{Obj}(D), F=\operatorname{Mor}(F): \operatorname{Mor}(C) \rightarrow \operatorname{Mor}(D)$ which preserves identity, source, target and composition: For $X \in O b j(C), f, g \in \operatorname{Mor}(C)$,
$$\begin{gathered} F\left(I d_{X}\right)=I d_{F(X)}, \ F(S(f))=S(F(f)), \ F(T(f))=T(F(f)), \ F(g \circ f)=F(g) \circ F(f) . \end{gathered}$$
when applicable.
A natural transformation $\eta: F \rightarrow G$ is a collection of morphisms
$$\eta_{X}: F(X) \rightarrow G(X), X \in O b j(C),$$
such that for every morphism $f: X \rightarrow Y$ in $C$, we have a commutative diagram:
Commutativity means that the two compositions of arrows (i.e. morphisms) indicated in the diagram are equal.

An equivalence of categories $C, D$ is a pair of functors $F: C \rightarrow D$ and $G: D \rightarrow C$ and natural isomorphisms (i.e. natural transformations which have inverses)
\begin{aligned} &F \circ G \cong I d_{D}, \ &G \circ F \cong I d_{C} . \end{aligned}

## 数学代考|代数几何代写algebraic geometry代考|The Definition of a Category, and an Example: The Category of Sets

X∣…

X∣X∉X

（称为源和目标，在集合的类别中是映射的域和共域）。态射F∈铁道部⁡(C)和小号(F)=X,吨(F)=是（在哪里X,是是对象）称为态射X到是，并表示为
F:X→是

X⟶F是

F:X→是,G:是→从,

G∘F:X→从

（注意顺序颠倒F和G，由映射驱动：当我们将映射应用到一个元素时，我们写G∘F(X)=G(F(X)), 即使我们申请F第一的）。

(H∘G)∘F=H∘(G∘F)
（如适用）和单位，即F:X→是,

## 数学代考|代数几何代写algebraic geometry代考|Categories of Algebraic Structures

F(X⋅是)=F(X)⋅F(是)
（从哲学上讲，单位和逆也是运算，所以我们应该包括F(1)=1和F(X−1)=(F(X))−1，但在群的情况下，它来自公理。）群和同态的范畴由 Grp 表示，阿贝尔群和同态的范畴由一种b.

F(X+是)=F(X)+F(是)

F(X是)=F(X)F(是)

F(1)=1,

F(X+是)=F(X)+F(是) F(rX)=rF(X) 为了 r∈R

## 数学代考|代数几何代写algebraic geometry代考|Functors and Natural Transformations

F(一世dX)=一世dF(X), F(小号(F))=小号(F(F)), F(吨(F))=吨(F(F)), F(G∘F)=F(G)∘F(F).

F∘G≅一世dD, G∘F≅一世dC.

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