Category Axioms

def. Category. Category $\mathcal{C}$ consists of collection $\text{Obj}$ and arrow (=morphism) $\text{Arw}$ which can compose. This composition must be:

1. if there is an arrow from $a\stackrel{f}{\to }b$ and $b\stackrel{g}{\to }c$ there must be an arrow $a\to c$ which we call $g\circ f$:
   • Q. If you have $f,g,g\circ f$, then you should have $f\circ g\circ g$, … ad infinitum? The set of morphisms isn’t closed under composition?
     • Well yes, but you also define a “multiplication table”, composition of morphisms such that, e.g. $f\circ g=f$, or such that prevents explosion of morphisms. ⇒ This multiplication table ensures set of morphisms is closed (in most categories).
2. each object $a$ must have an arrow mapping to itself, named ${\text{id}}_a$where composition of it and any other morphism $f$, ${\text{id}}_a\circ f$
   1. ${\text{id}}_b\circ f=f$
   2. $g\circ {\text{id}}_a=g$ where

• Multiple arrows pointing in the same direction can be present. Multiple arrows from an object to itself can be present aside from the identity.
• due to composition we can see compositions are associative:

Clarifying Morphisms §

• Equality of morphism. An arrow or morphism $f,g$ from $A\to B$ (=$f,g\in \text{Hom}(A,B)$) are equal when $\forall a\in A.f(a)=g(a)$.
• Order of composition. $f\circ g$ means that $g$ is applied first, then $f$ is applied. You read from right to left, just like function composition.
• Hom-set. $\text{Hom}(a,b)$ is the set of all arrows going from $a\to b$

Constructing Simple Categories §

• Categories with no objects, $\varnothing$. Yes, it does satisfy axioms
• Category with 1 object. At least one morphism, the identity:

def. Free Construction. Take any Graph, which may not be a category. You add composition morphisms and identity morphisms to satisfy the category theory axioms. This is called free construction, and the category generated by a graph is a free category.

def. Discrete Category. This “has no structure,” i.e. is isomorphic to a simple set.

Programming §

Motivation. Consider objects as types and morphisms as (pure) functions. Programming is basically writing morphisms that go from one type to another, and then composing them in desirable ways.