Idea

Definition

The cricle denoted $S^1$ is defined as the higher inductive type generated by:

• A point $base : S^1$
• A path $loop : base = base$

Properties

See also

higher inductive type

References

HoTT book

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Old Page

The circle $S^1$ is most commonly defined in homotopy type theory as a higher inductive type generated by

• a point $base:S^1$, and
• a path $loop:base=base$.

Its induction principle? says that for any $P:S^1\to Type$ equipped with a point $base^* : P(base)$ and a dependent path? $loop^*:base^*=_{loop} base^*$, there is $f:\prod_{(x:S^1)} P(x)$ such that $f(base)=base^*$ and $apd_f(loop) = loop^*$. As a special case, its recursion principle? says that given any type $X$ with a point $x:X$ and a loop $l:x=x$, there is $f:S^1 \to X$ with $f(base)=x$ and $f(loop)=l$.

Properties

$\Omega(S^1)=\mathbb{Z}$

There are several proofs in the book that the loop space $\Omega(S^1)$ of the circle is the integers $\mathbb{Z}$. (Any such proof requires the univalence axiom, since without that it is consistent that $S^1$ is contractible. Indeed, $S^1$ is contractible if and only if UIP holds.)

Two of them were blogged about:

• The first proof by Mike Shulman
• A simplification by Dan Licata

Alternative definition using torsors

The circle can also be defined without HITs using only univalence, as the type of $\mathbb{Z}$-torsors. One can then prove that this type satisfies the same induction principle (propositionally). This is due to Dan Grayson.