# Homotopy Type Theory circle (Rev #5, changes)

Showing changes from revision #4 to #5: Added | Removed | Changed

## Idea

The circle is one of the simplist higher inductive types it consists of a single point and a single non-trivial path between.

## Definition

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

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

Alternative definitions include the suspension of $\mathbf{2}$ and as a coequalizer?.

## Properties

Its induction principle? says that for any $P:S^1\to Type$ equipped with a point $base' : P(base)$ and a dependent path? $loop':base'= base'$, there is $f:\prod_{(x:S^1)} P(x)$ such that:

higher inductive type

$f(base)=base'\qquad apd_f(loop) = loop'$

## References

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

HoTT book

$f(base)=x\qquadf(loop)=l$

# Old Page

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.)

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:

### 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.