TheHoTTGame/1FundamentalGroup/Quest1Part0.md

Loop Space

In this quest, we continue to formalise the problem statement.

The fundamental group of S¹ is ℤ.

Intuitively, the fundamental group of S¹ at base is consists of loops based as base up to homotopy of paths. In homotopy type theory, we have a native description of loops based at base : it is the space base ≡ base.

In general the loop space of a space A at a point a is defined as follows :

Ω : (A : Type) (a : A) → Type
Ω A a = a ≡ a 

Clearly for each integer n : ℤ we have a path that is 'loop around n times'. Locate loop_times in 1FundamentalGroup/Quest1.agda (note how agda treats underscores)

loop_times : ℤ → Ω S¹ base
loop n times = {!!}

Try casing on n, you should see

loop_times : ℤ → Ω S¹ base
loop pos n times = {!!}
loop negsuc n times = {!!}

It says to map out of ℤ it suffices to map the non-negative integers (pos) and the negative integers (negsuc).

data ℤ : Type where
  pos    : (n : ℕ) → ℤ
  negsuc : (n : ℕ) → ℤ

This definition of ℤ uses the naturals, so try casing on n again, you should see

loop_times : ℤ → Ω S¹ base
loop pos zero times = {!!}
loop pos (suc n) times = {!!}
loop negsuc n times = {!!}

It says to map out of ℕ it suffices to map zero and map each succesive integer suc n inductively. When we loop zero (pos zero) times what should we get? Try filling it in. For looping pos (suc n) times we loop n times and loop once more. For this we need composition of paths.

_∙_ : x ≡ y → y ≡ z → x ≡ z

Try typing _∙_ or ? ∙ ? in the hole (input /.) and refining. Checking the new holes you should see that now you need to give two loops. Try giving it 'loop n times' composed with loop. Then try to also define the map on the negative integers. You will need to invert paths using sym.

Looking up definitions

If you don't know the definition of something you can look up the definition by sticking your cursor on it and pressing M-SPC c d in insert mode or SPC c d in evil mode.

You can use it to find out the definition of ℤ and ℕ.