 Homotopy pullback

For continuous maps $f:X\to Z$ and $g:Y\to Z$, a homotopy pullback is modelled byUse 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 $X{\times }_{Z}Y$ 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 $\ast \to S^1\leftarrow \ast$, 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.