category theory

depends on two sorts of objects:

• objects of the theory
• morphism of the category
  • tl/dr: think of arrow to connect relations between two mathematical objects

definition

a category $\mathcal{C}$ consists of the following entities:

• a class $\text{ob}(\mathcal{C})$ whose elements are called objects

• a class $\text{hom}(\mathcal{C})$

  • a morphism $f : a \mapsto b$
  • $\text{hom}(a,b)$, or $\text{hom}_{\mathcal{C}}(a,b), \text{mor}(a,b), \mathcal{C}(a,b)$ denotes hom-class of all morphism from $a$ to $b$

• a binary operator $\circ$, or composition of morphisms such that we have:

  $$\circ : \text{hom}(b,c) \times \text{hom}(a,b) \mapsto \text{hom}(a,c)$$

  • associativity: if $f: a \to b, g: b \to c$ and $h: c \to d$ then we have

    $$h \circ (g \circ f) = (h \circ g) \circ f$$

  • identity: For every object $x$ there exists a identity morphism $1_{x}: x \to x$ for x such that for every $f: a \to b$ we have

    $$1_b \circ f = f = f \circ 1_a$$