# 破密學專題 0920

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[總共筆](https://hackmd.io/rEa_Y3SPQDO5MNrvx6bE8A)

* Monoid:
* Identity
* 結合律
* Group:
* Inverse
* Identity
* Associativity 結合律
* Homomorphism:
1. $f:G\rightarrow H$
2. $f(g_1,g_2) = f(g_1)f(g_2)$
3. $f(1_G) = 1_H$
4. 由 2.3 可推出: $f(g^{-1}) = f(g)^{-1}$
* Monoid homomorphism:
* Example: $(Z,+),\{0,1,2,3 \}, \text{strings}, ,...$

* Category "Polyoid":
* Object: A,B,C,...
* Functions: $f:A\rightarrow B, g:B\rightarrow C, h:C\rightarrow D$
* Morphisms (arrows): $f,g,h,..$
* Associativity of composition: $(h \circ g) \circ f = h \circ (g \circ f)$
* Identity:
* $\forall c \in \mathbb{C}$ , $\exists1_c:C\rightarrow C \text{ such that } 1_D\circ f = f \circ 1_c = f$

* Category theory:
* Two basic properties
* Compose arrows associavity
* Exists an identity for each object
* Category of groups
* Objects: groups $G_1,G_2,...$
* Arrows:
* group
* homomophisms
* $\phi:G_1 \rightarrow G_2$ and $\psi:G_2 \rightarrow G_3$ can derive $\psi \circ \phi: G_1 \rightarrow G_3$

* [Small category](https://en.wikipedia.org/wiki/Category_(mathematics)#Small_and_large_categories)
* Category of sets:
* sets as objects, functions as arrows
* $\mathbb{S}et \supset \mathbb{G}roup$
* $\mathbb{S}et \supset \mathbb{R}ing$
* Discrete Category:
* Every component is independent.
* E.g.
$S = \{x_1,x_2,...,x_n\}$
$e : object : x_1,x_2,...,x_n$
$arrow : 1_{x1},1_{x2},...,1_{xn}$
* Posetal category;
* Definition: There's only on arrow between two objects.

* [Russell's paradox](https://zh.wikipedia.org/wiki/%E7%BD%97%E7%B4%A0%E6%82%96%E8%AE%BA)

* covariant Functor $F: \mathbb{C} \rightarrow \mathbb{O}$
* $f:A \rightarrow B$, $F(f):F(A) \rightarrow F(B)$
* $F(f \circ g)=F(f) \circ F(g)$
* $F(1_c) = 1_{FC}$
* contravarient functor $F^{-1}: \mathbb{C}^{op} \rightarrow \mathbb{D}$

* ==Power set functor== $P : Set \rightarrow Set$ (Map between categories)
* $S \mapsto P(S) = \{ S' \subset S\}$
* $f: S \rightarrow T \mapsto P(f): P(S) \rightarrow P(T)$

* Vector space $\mathbb{V}ect_k$ $V=\{v_1,v_2,...,\}$
* $(V,+) is an abelian group$
* $k \times V \rightarrow V$
* $0v = 0$
* $1v = 1$
* $(\alpha + \beta)v = \alpha v + \beta v$
* $(\alpha \beta)v = \alpha (\beta v)$

* dual space endofunctor:
* Definition: $V \mapsto V^* = \{\phi : V \rightarrow k\}$

## 打屁哈拉留言區
==**逃避並不可恥但是有用**==
phi 怎麼打 $\phi$
<3