Homotopy Type Theory For Dummies
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HITs<br />
Higher inductive types<br />
There are other ways to obtain higher types in HoTT.<br />
A higher inductive type has not only constructors for elements but<br />
also for equalities.<br />
A simple example is S 1 : <strong>Type</strong><br />
base : S 1<br />
loop : base = base<br />
S 1 homotopically behaves like the circle.<br />
Indeed we can embedd i : Z into φ(i) : base = base:<br />
i<br />
↦→loop n<br />
0 ↦→refl<br />
− n<br />
↦→(loop −1 ) n<br />
Thorsten Altenkirch (Nottingham) Edinburgh 13 October 30, 2013 21 / 29