TheHoTTGame/Trinitarianism/Quest0.md

Terms and Types

There are three ways of looking at A : Type.

• proof theoretically, 'A is a proposition'
• type theoretically, 'A is a construction'
• categorically, 'A is an object in category Type'

A first example of a type construction is the function type. Given types A and B, we have another type A → B which can be seen as

• the proposition 'A implies B'
• the construction 'ways to convert A recipes to B recipes'
• internal hom of the category Type

To give examples of this, let's make some types first!

-- Here is how we define 'true'
data ⊤ : Type u where
  trivial : ⊤

It reads '⊤ is an inductive type with a constructor trivial', with three interpretations

• ⊤ is a proposition and there is a proof of it, called trivial.
• ⊤ is a construction with a recipe called trivial
• ⊤ is a terminal object: every object has a morphism into ⊤ given by · ↦ trivial

What goes on the right of the : is called a type, and will always be in (some) Type, and what goes on the left is called a term of that term. The above tells you how we make a term of type ⊤, let's see an example of using a term of type ⊤:

TrueToTrue : ⊤ → ⊤
TrueToTrue = {!!}

• press C-c C-l (this means Ctrl-c Ctrl-l) to load the document, and now you can fill the holes
• navigate to the hole { } using C-c C-f (forward) or C-c C-b (backward)
• press C-c C-r and agda will try to help you (r for refine)
• you should see λ x → { }
• navigate to the new hole
• C-c C-, to check the goal (C-c C-comma)
• the Goal area should look like

Goal: ⊤
—————————————————————————
x : ⊤

• you have a proof/recipe/generalized element x : ⊤ and you need to give a p/r/g.e. of ⊤
• you can give it a p/r/g.e. of ⊤ and press C-c C-SPC to fill the hole

There is more than one proof (see solutions) - are they the same? Here is an important one:

TrueToTrue : ⊤ → ⊤
TrueToTrue x = {!!}

• Naviagate to the hole and check the goal.
• Note x is already taken out for you.
• You can try type x in the hole and C-c C-c
• c stands for 'cases on x' and the only case is trivial.

Built into the definition of ⊤ is agda's way of making a map out of ⊤ into another type A, which we have just used. It says to map out of ⊤ it suffices to do the case when x is trivial, or

• the only proof of ⊤ is trivial
• the only recipe for ⊤ is trivial
• the only one generalized element trivial in ⊤

-- Here is how we define 'false'
data ⊥ : Type u where

It reads '⊥ is an inductive type with no constructors', with three interepretations

• ⊥ is a proposition with no proofs
• ⊥ is a construction with no recipes
• There are no generalized elements of ⊥ (it is a strict initial object)

Let's try mapping out of ⊥.

explosion : ⊥ → ⊤
explosion x = {!!}

• Navigate to the hole and do cases on x.

Agda knows that there are no cases so there is nothing to do! This has three interpretations:

• false implies anything (principle of explosion)
• ?
• ⊥ is initial in the category Type u

We can also encode "natural numbers" as a type.

data ℕ : Type where
  zero : ℕ
  suc : ℕ → ℕ

As a construction, this reads '

• ℕ is a type of construction
• zero is a recipe for ℕ
• suc takes an existing recipe for ℕ and gives another recipe for ℕ. '

We can see ℕ as a categorical notion: ℕ is a natural numbers object in the category Type, with zero : ⊤ → ℕ and suc : ℕ → ℕ such that given any ⊤ → A → A there exist a unique morphism ℕ → A such that the diagram commutes:

This has no interpretation as a proposition since there are 'too many terms/proofs' - mathematicians classically didn't distinguish between proofs of the same thing. (ZFC doesn't even mention logic internally, unlike Type Theory!)

To see how to use terms of type ℕ, i.e. induct on ℕ, go to Quest1!

Universes

You may have noticed the notational similarities between zero : ℕ and ℕ : Type. Which may lead to the question Type : ?. We simply assert Type : Type 1, but then we are chasing our tail, asking Type 1 : ?. Type theorists make sure that every type (the thing on the right side of the :) itself is a term, and every term has a type. So what we really need is

Type : Type 1, Type 1 : Type 2, Type 2 : Type 3, ⋯ 

These are called universes. We will see definitions, for example groups that will require multiple universes.