Loop-space

A loop-space in $A$ is the space of identity types from a fixed basepoint $a$: $\Omega (A,a)=a{\equiv }_{A}a$ This gives rise to a higher group, restricting the Weak-Infinity Groupoids

$$\begin{array}{rlr}{\Omega }^0(A,a)& =& (A,a)\\ {\Omega }^{\mathrm{suc}n}(A,a)& =& {\Omega }^n(\Omega (A,a))\end{array}$$

See also

Homotopy Group

References

hott2013-book