Category Theory
Natural Transformation
1 Category theory
See (Lane 1998)
1.1 Introduction
1.1.1 Natural transformation
Definition (Index set \(\left(A_i\right)_{i \in I}\) ) \(A: I \rightarrow X, i \mapsto A_i\) Definition (Product over the index set \(\left(A_i\right)_{i \in I}\) ) \[ \prod_{i \in I} A_i:=\left\{a: I \rightarrow \bigcup_{i \in I} A_i \mid \forall i \in I, a(i) \in A_i\right\} \]
Definition (Semigroup \((S, *)\) )
\[ *: S \times S \rightarrow S \]
- Axiom 1 (Closure)
\[ \forall a, b \in S, a * b \in S \]
- Axiom 2 (Associativity)
\[ \forall a, b, c \in S,(a * b) * c=a *(b * c) \]
Example (Semigroup)
\[ (\mathbb{N},+), \mathbb{N}=\{1,2,3, \ldots\}, 0 \notin \mathbb{N} \]
Definition (Commutative semigroup)
\[ \forall a, b \in S, a * b=b * a \]
Definition (Non-commutative semigroup)
\[ \left\{\begin{array}{l} \forall a, b, c \in S,(a * b) * c=a *(b * c) \\ \exists x, y \in S, x * y \neq y * x \end{array}\right. \] 查看是否更新