Homotopy Type Theory For Dummies

Truncation levels
A contractible → a = A a ′ contractible
Assume that A is contractible, that means we have
a 0 : A
h : Π x:A x = a 0
Now for any a, a ′ : A we have
α 0 (a, a ′ ) : a = a ′
α 0 (a, a ′ ) :≡ h(a); h(a ′ ) −1
We need to show that
Π a,a ′ :AΠ p:a=a ′p = α 0 (a, a ′ )
Using J it is sufficient to show
Π a:A refl(a) = α(a, a) ≡ h(a); h(a) −1
Which is just one of the groupoid laws.
Thorsten Altenkirch (Nottingham) Edinburgh 13 October 30, 2013 18 / 29