Homotopy pullback
= Homotopy pullback
{wiki}
For <continuous maps> $f:X\to Z$ and $g:Y\to Z$, a <homotopy pullback> is modelled by
$$
P=\{(x,\gamma,y):\gamma:[0,1]\to Z\text{ continuous},\ \gamma(0)=f(x),\ \gamma(1)=g(y)\}.
$$
Use the <compact-open topology> on the path coordinate. A <continuous map> $W\to P$ gives <continuous maps> $u:W\to X$, $v:W\to Y$ together with a chosen <homotopy> from $f\circ u$ to $g\circ v$. Thus agreement in the target is remembered by a path, rather than required as literal equality.
The ordinary <pullback> $X\times_ZY$ maps to $P$ by constant paths. If one of $f,g$ is a <Serre fibration> in the usual <CW complex> setting, this comparison is a <weak homotopy equivalence>. More generally, replace one map by a <fibration> before taking the ordinary <pullback>.
For the two basepoint maps $*\to S^1\leftarrow*$, the ordinary <pullback> is a <point>, whereas $P$ is the <loop space> $\Omega S^1$, whose <path components> are indexed by the <winding number> in $\mathbb Z$. The paths in the model preserve information that the strict equality condition loses.