Suspension-loop adjunction
= Suspension-loop adjunction
{title2=$[\Sigma X,Y]\cong[X,\Omega Y]$}
For based spaces there is a natural bijection $[\Sigma X,Y]\cong[X,\Omega Y]$. A map on the suspension sends a point $x$ and suspension coordinate $t$ to a loop $t\mapsto f(x,t)$; the collapsed ends and basepoint give based loops. The reverse construction evaluates such loops. The two constructions commute with based homotopies. Applied to spheres it gives $\pi_{n+1}Y\cong\pi_n\Omega Y$.