# Categories and functors

order theory $A≤ B$ provability $A ⟹ B$ Boolean/Heyting algebras
category theory: $A \overset{f}{⟶} B$ proof theory $⊢ π: A ⟹ B$ cartesian closed categories

## Natural transformations

Ex: the category

𝔾 ≝ \begin{xy} \xymatrix{ \ast \ar@/^/[d]^t \ar@/_/[d]_s \\ \ast } \end{xy}

A functor $F: 𝔾 ⟶ Set$ is a graph, and a natural transformation $θ: F ⟶ G$ is a graph homomorphism.

2-graphs: globular sets

𝔾_2 ≝ \begin{xy} \xymatrix{ \ast \ar@/^/[d]^t \ar@/_/[d]_s \\ \ast \ar@/^/[d]^t \ar@/_/[d]_s \\ \ast } \end{xy}

• A 2-category with one single object: a strict monoidal category

• Sets and cartesian product:
• 0-cell: $\ast$
• 1-cells: sets
• 2-cells: functions
• A the 2-category of objects and relations:

• 0-cells: sets
• 1-cells: relations (subsets of cartesian product)
• 2-cells: inclusions (at most one 2-cell between two 1-cells: it’s a property (1 arrow at most))

Property: Two functions between sets $L: A ⟶ B$ and $R: B ⟶ A$ are inverse of one another iff $∀a∈A, b∈B, \; La=_A b ⟺ a=_B Rb$

Proof:

1. If $L, R$ are inverse of one another, the result follows immediatly.

2. With $b = La$: $a = RLa$. Similarly: $b = LR b$, so the result follows.

When considering a set on which a group $G$ acts (each element $g∈G$ behaves as a transition function between states of an automaton).

Adjunction: $𝒞(FX, Y) ≃ 𝒟(X, GY)$ is the set, and morphims of $𝒞$ (resp. $𝒟$) act on these sets, by potentially changing them (from $𝒞(FX, Y)$ to $𝒞(FX’, Y’)$)

⟹ action of a category on a family of sets, and naturality means that the action should preserve the relationships between the maps $𝒞(FX, Y) ≃ 𝒟(X, GY)$.

Ex:

A × \_ : \begin{cases} Set ⟶ Set \\ B \mapsto A × B \end{cases}

is left adjoint to

A ⟹ \_ : \begin{cases} Set ⟶ Set \\ B \mapsto A ⟹ B \end{cases}
• counit = evaluation
• unit = coeval
η_B: \begin{cases} B ⟶ GFB ≝ A ⟹ (A×B) \\ b \mapsto λa.(a, b) \end{cases}
ε_B: \begin{cases} FGB ≝ A × (A ⟹ B) ⟶ B \\ (a, f) \mapsto f(a) \end{cases}