TheHoTTGame/0Trinitarianism/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 : Type and B : Type, we have another type A → B : Type 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!

True / Unit / Terminal object

data ⊤ : Type where
  tt : ⊤

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

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

In general, the expression a : A is read 'a is a term of type A', and has three interpretations,

• a is a proof of the proposition A
• a is a recipe for the construction A
• a is a generalised element of the object A in the category Type.

The above tells you how we make a term of type ⊤. Let's see an example of using a term of type ⊤:

TrueToTrue : ⊤ → ⊤
TrueToTrue = { }

• enter C-c C-l (this means Ctrl-c Ctrl-l). Whenever you do this, Agda will check the document is written correctly. This will open the *Agda Information* window looking like

  ?0 : ⊤ → ⊤
  ?1 : ⊤
  ?2 : ⊤
  

  This says you have three unfilled holes.

• Now you can fill the hole { }0.

• navigate to the hole { } using C-c C-f (forward) or C-c C-b (backward)

• enter C-c C-r. The r stands for refine. Whenever you do this whilst having your cursor in a hole, Agda will try to help you.

• you should see λ x → { }. This is agda's notation for x ↦ { } and is called λ abstraction, think λ for 'let'.

• navigate to the new hole

• enter C-c C-, (this means Ctrl-c Ctrl-comma). Whenever you make this command whilst having your cursor in a hole, Agda will check the goal.

• the goal (*Agda information* window) should look like

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

  you have a proof/recipe/generalized element x : ⊤ and you need to give a proof/recipe/generalized element of ⊤

• write the proof/recipe/generalized element x of ⊤ in the hole

• press C-c C-SPC to fill the hole with x. In general when you have some term (and your cursor) in a hole, doing C-c C-SPC will tell Agda to replace the hole with that term. Agda will give you an error if it can't make sense of your term.

• the *Agda Information* window should now only have two unfilled holes left, this means Agda has accepted your proof.

  ?1 : ⊤
  ?2 : ⊤
  

There is more than one proof (see Quest0Solutions.agda). 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'. Doing C-c C-c with x in the hole tells agda to 'do cases on x'. The only case is tt.

One proof says for any term x : ⊤ give x again. The other says it suffices to do the case of tt, for which we just give tt.

Are these proofs 'the same'? What is 'the same'?

(This question is deep and should be unsettling. Sneak peek: they are internally but not externally 'the same'.)

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 tt', or

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

Let's define another type.

False / Empty / Initial object

data ⊥ : Type 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! (See Quest0Solutions.agda) This has three interpretations:

• false implies anything (principle of explosion)
• One can convert recipes of ⊥ to recipes of any other construction since there are no recipes of ⊥.
• ⊥ is initial in the category Type

The natural numbers

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 also see ℕ categorically : ℕ is a natural numbers object in the category Type. This means it is equipped with morphisms zero : ⊤ → ℕ and suc : ℕ → ℕ such that given any ⊤ → A → A there exist a unique morphism ℕ → A such that the diagram commutes:

ℕ has no interpretation as a proposition since there are 'too many proofs' - mathematicians classically don't distinguish between proofs of a single proposition. (ZFC doesn't even mention logic internally, unlike Type Theory!)

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

Universes

You may have noticed the notational similarities between zero : ℕ and ℕ : Type. This may have lead you to the question, Type : ?. In type theory, we simply assert Type : Type₁. But then we are chasing our tail, asking Type₁ : Type₂. Type theorists make sure that every type (i.e. anything the right side of :) itself is a term (i.e. anything on the left of :), and every term has a type. So what we really need is

Type : Type₁, Type₁ : Type₂, Type₂ : Type₃, ⋯

These are called universes. The numberings of universes are called levels. It will be crucial that types can be treated as terms. This will allows us to

• reason about 'structures' such as 'the structure of a group', think 'for all groups'
• do category theory without stepping out of the theory (no need for classes etc. For experts, we have Grothendieck universes.)
• reason about when two types are 'the same', for example when are two definitions of the natural numbers 'the same'? What is 'the same'?