2.3 KiB
Loop Space
In this quest, we continue to formalise the problem statement.
The fundamental group of
S¹isℤ.
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.
sym : x ≡ y → y ≡ x
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.