Lecture 3:

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.

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2-graphs: globular sets

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

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• 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))

Adjunction

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.

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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

\[η_B: \begin{cases} B ⟶ GFB ≝ A ⟹ (A×B) \\ b \mapsto λa.(a, b) \end{cases}\] \[ε_B: \begin{cases} FGB ≝ A × (A ⟹ B) ⟶ B \\ (a, f) \mapsto f(a) \end{cases}\]

“All logic is about adjunction”