The Circle

In this series of quests we will prove that the fundamental group of S¹ is ℤ. In fact, our strategy will also show that the higher homotopy groups of S¹ are all trivial. We begin by formalising the problem statement.

A contruction of 'the circle' is :

a point called base
an edge from that point to itself called loop

Here is our definition of the circle in agda.

data S¹ : Type where
  base : S¹
  loop : base ≡ base

The base ≡ base is the space of paths from base to base. The definition asserts that there is a point called loop in base ≡ base, i.e. a path from base to itself. Whenever we have a colon like S¹ : Type or base : S¹ it says the former is a point in the latter, where the latter is viewed as a space; in the first case Type is the space of spaces.

Further details

This is called a higher inductive type (HIT), which generally follows the format of

data
the name of the HIT - in our case S¹
the type of the HIT, in our case Type
where followed by
the constructors of the HIT, in our case base and loop, which we will think of as vertices, edges, surfaces, and so on

An "edge" is the same as a path. There are other paths in S¹, for example the constant path at base. In 1FundamentalGroup/Quest0.agda naviage to

Refl : base ≡ base
Refl = {!!}

we will guide you through defining it. We are about to construct a path Refl : base ≡ base (read path Refl from base to base) The hole { }0 is where you describe the path. We will fill the hole { }0.

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 : base ≡ base
?1 : (something)
?2 : (something)
...
```
This says you have some unfilled holes.
navigate to the hole { }0 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 now see λ i → { }1. This is agda suggesting that for each i : I (if you like you can think of this as a generic point on the the unit interval I) you give a point in between the start and end of the path. This is all you need to specify a path in agda.
navigate to that 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, i.e. what kind of thing you need to stick in.
the goal (*Agda information* window) should look like

 Goal: S¹
 —————————————————————————
 i : I
 ———— Constraints ——————————————
...

you see that agda knows you have a generic point i : I on the unit interval. All the constraints are saying that when you look at i = 0 and i = 1, whatever you give in between must match up with the end points of the path, namely base and base

write base in the hole, since this is the constant path
press C-c C-SPC to fill the hole with base. In general when you have some text (and your cursor) in a hole, doing C-c C-SPC will tell agda to replace the hole with that text. agda will give you an error if it can't make sense of your text.
the number of holes in the *Agda Information* window should have gone down by one, this means agda has accepted what you filled this hole with. Just to be sure you can also reload the agda file and check that agda has no complaints.
if you want to play around with this you can start again by replacing what you wrote with ? and doing C-c C-l

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The Circle

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